In this paper, by using the Mawhin's continuation theorem, some easily verifiable sufficient conditions are obtained to guarantee the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Our result corrects the result obtained in [
Citation: Li Wang, Hui Zhang, Suying Liu. On the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms[J]. AIMS Mathematics, 2022, 7(1): 925-938. doi: 10.3934/math.2022055
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In this paper, by using the Mawhin's continuation theorem, some easily verifiable sufficient conditions are obtained to guarantee the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Our result corrects the result obtained in [
Let Ω be a bounded and regular enough domain in Rn, let α>0, and let a:Ω→R be a nonnegative and nonidentically zero function. Singular elliptic problems like to
{−Δu=au−α in Ω,u=0 on ∂Ω,u>0 in Ω, | (1.1) |
arise in many applications to physical phenomena, for instance, in chemical catalysts process, non-Newtonian fluids, and in models for the temperature of electrical conductors (see e.g., [3,5,13,16] and the references therein). Starting with the pioneering works [6,13,16,26], and [11], the existence of positive solutions of singular elliptic problems has been intensively studied in the literature.
Bifurcation problems whose model is −Δu=au−α+f(.,λu) in Ω, u=0 on ∂Ω, u>0 in Ω, were studied by Coclite and Palmieri [4], under the assumptions a∈C1(¯Ω), a>0 in ¯Ω, f∈C1(¯Ω×[0,∞)) and λ>0. Problems of the form −Δu=Ku−α+λsp in Ω, u=0 on ∂Ω, u>0 in Ω, were studied by 35 [35], when p∈(0,1), K is a regular enough function that may change sign, and λ∈R. Ghergu and Rădulescu [19] addressed multi-parameter singular bifurcation problems of the form −Δu=g(u)+λ|∇u|p+μf(.,u) in Ω, u=0 on ∂Ω, u>0 in Ω, where g is Hölder continuous, nonincreasingt and positive on (0,∞), and singular at the origin; f:¯Ω×[0,∞)→[0,∞) is Hölder continuous, positive on ¯Ω×(0,∞), and such that f(x,s) is nondecreasing with respect to s, 0<p≤2,and λ>0. Dupaigne, Ghergu and Rădulescu [14] studied Lane–Emden–Fowler equations with convection and singular potential; and Rădulescu [32] addressed the existence, nonexistence, and uniqueness of blow-up boundary solutions of logistic equations and of singular Lane-Emden-Fowler equations with convection term. Cîrstea, Ghergu and Rădulescu [7] considered the problem of the existence of classical positive solutions for problems of the form −Δu=a(x)h(u)+λf(u) in Ω, u=0 on ∂Ω, u>0 in Ω, in the case when Ω is a regular enough domain, f and h are positive Hölder continuous functions on [0,∞) and (0,∞) respectively satisfying some monotonicity assumptions, h singular at the origin, and h(s)≤cs−α for some positive constant c and some α∈(0,1).
Multiplicity results for positive solutions of singular elliptic problems were obtained by Gasiński and Papageorgiou [17] and by Papageorgiou and G. Smyrlis [30]; in both articles the singular term of the considered nonlinearity has the form a(x)s−α, with 0≤a∈L∞(Ω), a≢0 in Ω, and α positive.
Recently, problem (1.1) has been addressed by Chu, Gao and Gao [8], under the assumption that α=α(x) (i.e., with a singular nonlinearity with a variable exponent).
Concerning the existence of nonnegative solutions of singular elliptic problems, Dávila and Montenegro [9] studied the free boundary singular bifurcation problem
{−Δu=χ{u>0}(−u−α+λf(.,u)) in Ω,u=0 on ∂Ω,u≥0 in Ω, u≢0 in Ω, |
where 0<α<1, λ>0, and f:Ω×[0,∞)→[0,∞) is a Carathéodory function f such that, for a.e. x∈Ω, f(x,s) is nondecreasing and concave in s, and satisfies lims→∞f(x,s)/s=0 uniformly on x∈Ω. and where, for h:Ω×(0,∞)→R, χ{s>0}h(x,s) stands for the function defined on Ω×[0,∞) by χ{s>0}h(x,s):=h(x,s) if s>0, and χ{s>0}h(x,s):=0 if s=0. Let us mention also the work [10], where a related singular parabolic problem was treated.
For a systematic study of singular problems and additional references, we refer the reader to [18,32], see also [12].
Our aim in this work is to prove an existence result for nonnegative weak solutions of singular elliptic problems of the form
{−Δu=χ{u>0}(au−α−g(.,u)) in Ω,u=0 on ∂Ω,u≥0 in Ω, u≢0 in Ω, | (1.2) |
where Ω is a bounded domain in Rn with C1,1 boundary, α∈(0,1], a:Ω→R, and g:Ω×[0,∞)→R, with a and g satisfying the following conditions h1)-h4):
h1) 0≤a∈L∞(Ω) and a≢0,
h2) {x∈Ω:a(x)=0}=Ω0∪N for some (possibly empty) open set Ω0⊂Ω and some measurable set N⊂Ω such that |N|=0,
h3) g is a nonnegative Carathéodory function on Ω×[0,∞), i.e., g(.,s) is measurable for any s∈[0,∞), and g(x,.) is continuous on [0,∞) for a.e. x∈Ω,
h4) sup0≤s≤Mg(.,s)∈L∞(Ω) for any M>0.
Here and below, χ{u>0}(au−α−g(.,u)) stands for the function h:Ω→R defined by h(x):=a(x)u−α(x)−g(x,u(x)) if u(x)≠0, and h(x):=0 otherwise; u≢0 in Ω means |{x∈Ω:u(x)≠0}|>0 and, by a weak solution of (1.2), we mean a solution in the sense of the following:
Definition 1.1. Let h:Ω→R be a measurable function such that hφ∈L1(Ω) for all φ in H10(Ω)∩L∞(Ω). We say that u:Ω→R is a weak solution to the problem
{−Δu=h in Ω,u=0 on ∂Ω | (1.3) |
if u∈H10(Ω), and ∫Ω⟨∇u,∇φ⟩=∫Ωhφ for all φ in H10(Ω)∩L∞(Ω).
We will say that, in weak sense,
−Δu≤h in Ω (respectively −Δu≥h in Ω),u=0 on ∂Ω |
if u∈H10(Ω), and ∫Ω⟨∇u,∇φ⟩≤∫Ωhφ (respectively ∫Ω⟨∇u,∇φ⟩≥∫Ωhφ) for all nonnegative φ in H10(Ω)∩L∞(Ω).
Our first result reads as follows:
Theorem 1.2. Let Ω be a bounded domain in Rn with C1,1 boundary. Let α∈(0,1], let a:Ω→[0,∞) and let g:Ω×(0,∞)→R; and assume that a and g satisfy the conditions h1)-h4). Then there exists a nonnegative weak solution u∈H10(Ω)∩L∞(Ω), in the sense of Definition 1.1, to problem (1.2), and such that u>0 a.e. in {a>0}. In particular, χ{u>0}(au−α−g(.,u))≢0 in Ω and χ{u>0}(au−α−g(.,u))φ∈L1(Ω) for any φ∈H10(Ω)∩L∞(Ω)).
Let us mention that in [21] it was proved the existence of weak solutions (in the sense of Definition 1.1) of problem (1.2), in the case when 0≤a∈L∞(Ω), a≢0, 0<α<1, and g(.,u)=−bup, with 0<p<n+2n−2, and 0≤b∈Lr(Ω) for suitable values of r. In addition, existence results for weak solutions of problems of the form
{−Δu=χ{u>0}au−α−h(.,u) in Ω,u=0 on ∂Ω,u≥0 in Ω, and u≢0 in Ω, | (1.4) |
were obtained, in [22] (see Remark 2.1 below), and in ([25], Theorem 1.2), for more general nonlinearities h:Ω×[0,∞)→[0,∞)(x,s), in the case when h is a Carathéodory function on Ω×[0,∞), which satisfies h(.,0)≤0 as well as some additional hypothesis. Then the result of Theorem 1.2 is not covered by those in [22] and [25] because, under the assumptions of Theorem 1.2, the condition g(.,0)≤0 is not required and χ{s>0}g(.,s) is not, in general, a Carathéodory function on Ω×[0,∞) (except when g(.,0)≡0 in Ω).
Our next result says that if the condition h4) is replaced by the stronger condition
h4') a>0 a.e. in Ω and sup0<s≤Ms−1g(.,s)∈L∞(Ω) for any M>0,
then the solution u, given by Theorem 1.2, is positive a.e. in Ω and is a weak solution in the usual sense of H10(Ω).
Theorem 1.3. Let Ω, α, and a be as in Theorem 1.2, and let g:Ω×(0,∞)→R. Assume the conditions h1)-h3) and h4'). Then the solution u of (1.2), given by Theorem 1.2, belongs to C(¯Ω)∩W2,ploc(Ω) for any p∈[1,∞), there exist positive constants c, c′ and τ such that cdΩ≤u≤c′dτΩ in Ω, and u is a weak solution, in the usual H10(Ω) sense, of the problem
{−Δu=au−α−g(.,u) in Ω,u=0 on ∂Ω,u>0 in Ω | (1.5) |
i.e., for any φ∈H10(Ω), (au−α−g(.,u))φ∈L1(Ω) and ∫Ω⟨∇u,∇φ⟩=∫Ω(au−α−g(.,u))φ.
Finally, our last result says that, if in addition to h1)-h4), α is sufficiently small, the set where a>0 is nice enough and, for any s≥0, g(.,s)=0 a.e. in the set where a>0, then the solution obtained in Theorem 1.2, is a weak solution in the usual sense of H10(Ω), and that it is positive on some subset of Ω:
Theorem 1.4. Let Ω be a bounded domain in Rn with C1,1 boundary. Assume the hypothesis h1)-h4) of Theorem 1.2 and that 0<α<12+1n when n>2, and α∈(0,1] when n≤2. Let A+:={x∈Ω:a(x)>0} and assume, in addition, the following two conditions:
h5) g(.,s)=0 a.e. in A+ for any s≥0.
h6) A+=Ω+∪N+ for some open set Ω+ and a measurable set N+ such that |N+|=0, and with Ω+ such that Ω+ has a finite number of connected components {Ω+l}1≤l≤N and each Ω+l is a C1,1 domain.
Then the solution u of problem (1.2), given by Theorem 1.2, is a weak solution, in the usual H10(Ω) sense, to the same problem, and there exist positive constants c, c′ and τ such that u≥cdΩ+ a.e. in Ω+, and u≤c′dτΩ a.e. in Ω.
The article is organized as follows: In Section 2 we study, for ε∈(0,1], the existence of weak solutions to the auxiliary problem
{−Δu=au−α−gε(.,u) in Ω,u=0 on ∂Ω,u>0 in Ω. | (1.6) |
where Ω is a bounded domain in Rn with C1,1 boundary, α∈(0,1], a:Ω→[0,∞) is a nonnegative function in L∞(Ω) such that |{x∈Ω:a(x)>0}|>0, and {gε}ε∈(0,1] is a family of real valued functions defined on Ω×[0,∞) satisfying the following conditions h7)-h9):
h7) gε is a nonnegative Carathéodory function on Ω×[0,∞) for any ε∈(0,1].
h8) sup0<s≤Ms−1gε(.,s)∈L∞(Ω) for any ε∈(0,1] and M>0.
h9) The map ε→gε(x,s) is nonincreasing on (0,1] for any (x,s)∈Ω×[0,∞).
Lemma 2.2 observes that, as a consequence of a result of [22], the problem
{−Δu=χ{u>0}au−α−gε(.,u) in Ω,u=0 on ∂Ω,u≥0 in Ω, u≢0 in Ω | (1.7) |
has (at least) a weak solution u (in the sense of Definition 1.1) which satisfies u>0 a.e. in {a>0}; and this assertion is improved in Lemmas 2.6 and 2.7, which state that any weak solution u (in the sense of Definition 1.1) of problem (1.7) is positive in Ω, belongs to C(¯Ω), and is also a weak solution in the usual sense of H10(Ω). By using a sub-supersolution theorem of [28] as well as an adaptation of arguments of [27], Lemma 2.15 shows that, for any ε∈(0,1], problem (1.6) has a solution uε∈H10(Ω), which is a weak solution in the usual sense of H10(Ω), and is maximal in the sense that, if v is a solution, in the sense of Definition 1.1, of problem (1.6) then v≤uε. Lemma 2.16 states that ε→uε is nondecreasing, Lemma 2.17 says that {uε}ε∈(0,1] is bounded in H10(Ω), and Lemma 2.18 says that the function u:=limε→0+uε belong to H10(Ω)∩L∞(Ω) and is positive in {a>0}.
To prove Theorems 1.2–1.4 we consider, in Section 3, the family {gε}ε∈(0,1] defined by gε(.,s):=s(s+ε)−1g(.,s) and we show that, in each case, the corresponding function u defined above is a solution of problem (1.2) with the desired properties.
We assume, from now on, that Ω is a bounded domain in Rn with C1,1 boundary, α∈(0,1] and a:Ω→[0,∞) is a nonnegative function in L∞(Ω) such that |{x∈Ω:a(x)>0}|>0, and additional conditions will be explicitely impossed on a and α when necessary, at some steps of the paper. Our aim in this section is to study, for ε∈(0,1], the existence of weak solutions to problem (1.6), in the case when {gε}ε∈(0,1] is a family of functions satisfying the conditions h7)-h9).
In order to present, in the next remark, a need result of [22], we need to recall the notion of principal egenvalue with weight function: For b∈L∞(Ω) such that b≢0, we say that λ∈R is a principal eigenvalue for −Δ on Ω, with weight function b and homogeneous Dirichlet boundary condition, if the problem −Δu=λbu in Ω, u=0 on ∂Ω has a solution u wich is positive in Ω. If b∈L∞(Ω) and b+≢0, it is well known that there exists a unique positive principal eigenvalue for the above problem, which we wiill denote by λ1(b). For a proof of this fact and for additional properties of principal eigenvalues and their associated principal eigenfunctions see, for instance [15].
Remark 2.1. (See [22], Theorem 1.2, or, in a more general setting, [25], Theorem 1.2) Let β∈(0,3), ˜a:Ω→R and f:Ω×[0,∞)→R; and assume the following conditions H1)-H6):
H1) 0≤˜a∈L∞(Ω), and ˜a≢0,
H2) f is a Carathéodory function on Ω×[0,∞),
H3) sup0≤s≤M|f(.,s)|∈L1(Ω) for any M>0,
H4) One of the two following conditions holds:
H4') sups>0f(.,s)s≤b for some b∈L∞(Ω) such that b+≢0, and λ1(b)>m for some integer m≥max{2,1+β},
H4") f∈L∞(Ω×(0,σ)) for all σ>0, and ¯lims→∞f(.,s)s≤0 uniformly on Ω, i.e., for any ε>0 there exists s0>0 such that sups≥s0f(.,s)s≤ε, a.e. in Ω,
H5) f(.,0)≥0.
Then the problem
{−Δu=χ{u>0}˜au−β+f(x,u) in Ω,u=0 on ∂Ω, u≥0 in Ω, u≢0 in Ω. | (2.1) |
has a weak solution (in the sense of Definition 1.1) u∈H10(Ω)∩L∞(Ω) such that u>0 a.e. in {˜a>0}.
Lemma 2.2. Let a∈L∞(Ω) be such that a≥0 in Ω and a≢0, let α∈(0,1], and let {gε}ε∈(0,1] be a family of functions defined on Ω×[0,∞) satisfying the conditions h7)-h9) stated at the introduction. Then, for any ε∈(0,1], problem (1.7) has at least a weak solution u∈H10(Ω)∩L∞(Ω), in the sense of Definition 1.1, such that u>0 a.e. in {a>0}.
Proof. Notice that, since gε is a Carathéodory function, we have gε(.,0)=lims→0+gε(.,s)=lims→0+(ss−1gε(.,s))=0, the last inequality by h8). Thus gε(.,0)=0. Taking into account this fact and h7)-h9), the lemma follows immediately from Remark 2.1.
Let us recall, in the next remark, the uniform Hopf maximum principle:
Remark 2.3. ⅰ) (see [2], Lemma 3.2) Suppose that 0≤h∈L∞(Ω); and let v∈∩1≤p<∞(W2,p(Ω)∩W1,p0(Ω)) be the strong solution of −Δv=h in Ω, v=0 on ∂Ω. Then v≥cdΩ∫ΩhdΩ a.e. in Ω, where dΩ:=dist(.,∂Ω), and c is a positive constant depending only on Ω.
ⅱ) (see e.g., [25], Remark 8) Let Ψ be a nonnegative function in L1loc(Ω), and let v be a function in H10(Ω) such that −Δv≥Ψ on Ω in the sense of distributions. Then
v(x)≥cdΩ∫ΩΨdΩa.e. in Ω, | (2.2) |
where c is a positive constant depending only on Ω.
Remark 2.4. (See, e.g., [23], Lemmas 2.9, 2.10 and 2.12) Let a∈L∞(Ω) be such that a≥0 in Ω and a≢0, and let let α∈(0,1]. Then the problem
{−Δz=az−α in Ω,z=0 on ∂Ω,z≥0 in Ω. | (2.3) |
has a unique weak solution, in the sense of Definition 1.1, z∈H10(Ω)∩L∞(Ω). Moreover:
ⅰ) z∈C(¯Ω).
ⅱ) There exists positive constants c1, c2 and τ>0 such that c1dΩ≤z≤c2dτΩ in Ω.
ⅲ) z is a solution of problem (2.3) in the usual weak sense, i.e., for any φ∈H10(Ω), az−αφ∈L1(Ω) and ∫Ω⟨∇z,∇φ⟩=∫Ωaz−αφ.
Lemma 2.5. Let a, α, and {gε}ε∈(0,1] be as in Lemma 2.2, let z be as given in Remark 2.4; and let ε∈(0,1]. If u∈H10(Ω)∩L∞(Ω) is a weak solution, in the sense of Definition 1.1, of problem (1.7), then u≤z a.e. in Ω.
Proof. By h5), gε(.,u)≥0 and so, from Lemma 2.2 and Remark 2.4, we have, in the sense of Definition 1.1,
−Δ(u−z)=au−α−gε(.,u)−az−α≤a(u−α−z−α) in Ω, |
Thus, taking (u−z)+ as a test function, we get
∫Ω|∇(u−z)+|2≤∫Ωa(u−α−z−α)(u−z)+≤0 |
which implies u≤za.e. in Ω.
Lemma 2.6. Let a, α, and {gε}ε∈(0,1] be as in Lemma 2.2. If ε∈(0,1] and u∈H10(Ω)∩L∞(Ω) is a weak solution, in the sense of Definition 1.1, of problem (1.7), then:
i) There exists a positive constant c1 (which may depend on ε) and constants c2 and τ such that c1dΩ≤u≤c2dτΩ a.e. in Ω (and so, in particular, u>0 in Ω).
ii) For any φ∈H10(Ω) we have (au−α−gε(.,u))φ∈L1(Ω) and
∫Ω⟨∇u,∇φ⟩=∫Ω(au−α−gε(.,u))φ, |
i.e., u is a weak solution, in the usual sense of H10(Ω), to the problem −Δu=au−α−gε(.,u) in Ω, u=0 on ∂Ω.
Proof. We have, in the weak sense of Definition 1.1, −Δu=χ{u>0}au−α−gε(.,u) in Ω, u=0 on ∂Ω. Also, u≥0 in Ω and u≢0 in Ω. Let a0:Ω→R be defined by a0(x)=u−1(x)gε(x,u(x)) if u(x)≠0 and by a0(x)=0 otherwise. Since u∈L∞(Ω) and taking into account h7) and h8), we have 0≤a0∈L∞(Ω), and from the definition of a0 we have gε(.,u)=a0u a.e. in Ω. Therefore u satisfies, in the sense of Definition 1.1, −Δu+a0u=χ{u>0}au−α in Ω, u=0 on ∂Ω. Thus, since u is nonidentically zero, it follows that χ{u>0}au−α is nonidentically zero on Ω. Then there exist η>0, and a measurable set E⊂Ω, such that |E|>0 and χ{u>0}au−α≥ηχE in Ω. Let ψ∈∩1≤q<∞W2,,q(Ω)∩W1,,q0(Ω) be the solution of the problem −Δψ+a0ψ=ηχE in Ω, ψ=0 on ∂Ω. By the Hopf maximum principle (as stated, e.g., in [34], Theorem 1.1) there exists a positive constant c1 such that ψ≥c1dΩ in Ω; and, from (1.7) we have −Δu+a0u≥ηχE in D′(Ω). Then, by the weak maximum principle (as stated, e.g., in [20], Theorem 8.1), u≥ψ in Ω. Hence u≥c1dΩ in Ω. Also, by Lemma 2.5, u≤z a.e. in Ω, and so Remark 2.4 gives positive constants c2 and τ (both independent of ε) such that u≤c2dτΩ in Ω. Thus i) holds.
To see ii), consider an arbitrary function φ∈H10(Ω), and for k∈N, let φ+k:=max{k,φ+}. Thus φ+k∈H10(Ω)∩L∞(Ω), {φ+k}k∈N converges to φ+ in H10(Ω) and, after pass to some subsequence if necessary, we can assume also that {φ+k}k∈N converges to φ+ a.e. in Ω. Since u is a weak solution, in the sense of Definition 1.1, of (1.7) and u>0 a.e. in Ω, we have, for all k∈N, (au−α−gε(.,u))φ+k∈L1(Ω), and, by h6), gε(.,u)∈L∞(Ω). Thus gε(.,u)φ+k∈L1(Ω). Then au−αφ+k∈L1(Ω).
From (1.7),
∫Ω⟨∇u,∇φ+k⟩+∫Ωgε(.,u)φ+k=∫Ωau−αφ+k. | (2.4) |
Now, limk→∞∫Ω⟨∇u,∇φ+k⟩=∫Ω⟨∇u,∇φ+⟩. Also, for any k,
0≤gε(.,u)φ+k≤sups∈[0,‖ |
then, by the Lebesgue dominated convergence theorem, \lim_{k\rightarrow \infty}\int_{\Omega}g_{\varepsilon}\left(., u\right) \varphi_{k} ^{+} = \int_{\Omega}g_{\varepsilon}\left(., u\right) \varphi^{+} < \infty. Hence, by (2.4), \lim_{k\rightarrow\infty}\int_{\Omega }au^{-\alpha}\varphi_{k}^{+} exists and is finite. Since \left\{ au^{-\alpha}\varphi_{k}^{+}\right\} _{k\in\mathbb{N}} is nondecreasing and converges to au^{-\alpha}\varphi^{+} a.e. in \Omega, the monotone convergence theorem gives \lim_{k\rightarrow\infty}\int_{\Omega}au^{-\alpha }\varphi_{k}^{+} = \int_{\Omega}au^{-\alpha}\varphi^{+} < \infty. Thus
\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) \varphi ^{+}\in L^{1}\left( \Omega\right) |
and
\begin{equation} \int_{\Omega}\left\langle \nabla u, \nabla\varphi^{+}\right\rangle +\int_{\Omega}g_{\varepsilon}\left( ., u\right) \varphi^{+} = \int_{\Omega }au^{-\alpha}\varphi^{+}. \end{equation} | (2.5) |
Similarly, we have that \left(au^{-\alpha}-g_{\varepsilon}\left(., u\right) \right) \varphi^{-}\in L^{1}\left(\Omega\right), and that (2.5) holds with \varphi^{+} replaced by \varphi^{-} By writing \varphi = \varphi^{+}-\varphi^{-} the lemma follows.
Lemma 2.7. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. For any \varepsilon\in\left(0, 1\right], if u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) is a weak solution, in the sense of Definition 1.1 (and so, by Lemma 2.6, also in the usual sense of H_{0}^{1}\left(\left(\Omega\right) \right) ), of problem (1.7), then u\in C\left(\overline{\Omega}\right).
Proof. By Lemma 2.6 we have u\geq cd_{\Omega} a.e. in \Omega, with c a positive constant and, by h6), 0\leq u^{-1}g_{\varepsilon}\left(., u\right) \in L^{\infty}\left(\Omega\right). Thus au^{-\alpha }-g_{\varepsilon}\left(., u\right) \in L_{loc}^{\infty}\left(\Omega\right). Also, u\in L^{\infty}\left(\Omega\right). Then, by the inner elliptic estimates (as stated, e.g., in [20], Theorem 8.24), u\in W_{loc}^{2, p}\left(\Omega\right) for any p\in\left[1, \infty\right). Thus u\in C\left(\Omega\right), and, since 0\leq u\leq z, z\in C\left(\overline{\Omega}\right) and z = 0 on \partial\Omega, it follows that u is also continuous at \partial\Omega.
Definition 2.8. Let C_{0}^{\infty}\left(\overline{\Omega}\right) : = \left\{ \varphi\in C^{\infty}\left(\overline{\Omega}\right) :\varphi = 0\text{ on }\partial\Omega\right\}. If u\in L^{1}\left(\Omega\right) and h\in L^{1}\left(\Omega\right), we will say that u is a solution, in the sense of \left(C_{0}^{\infty}\left(\overline {\Omega}\right) \right) ^{\prime}, of the problem -\Delta u = h in \Omega, u = 0 on \partial\Omega, if -\int_{\Omega}u\Delta\varphi = \int_{\Omega}h\varphi for any \varphi\in C_{0}^{\infty}\left(\overline{\Omega}\right).
We will say also that -\Delta u\geq h in \left(C_{0}^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime } (respectively -\Delta u\leq h in \left(C_{0}^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime} ) if -\int_{\Omega} u\Delta\varphi\geq\int_{\Omega}h\varphi (resp. -\int_{\Omega}u\Delta \varphi\leq\int_{\Omega}h\varphi ) for any nonnegative \varphi\in C_{0}^{\infty}\left(\overline{\Omega}\right).
Remark 2.9. The following statements hold:
ⅰ) (Maximum principle, [31], Proposition 5.1) If u\in L^{1}\left(\Omega\right), 0\leq h\in L^{1}\left(\Omega\right), and -\Delta u\geq h in the sense of \left(C_{0}^{\infty}\left(\overline{\Omega }\right) \right) ^{\prime}, then u\geq0 \ a.e. in \Omega .
ⅱ) (Kato's inequality, [31], Proposition 5.7) If h\in L^{1}\left(\Omega\right), u\in L^{1}\left(\Omega\right) and if -\Delta u\leq h in D^{\prime}\left(\Omega\right) , then -\Delta\left(u^{+}\right) \leq\chi_{\left\{ u > 0\right\} }h in D^{\prime}\left(\Omega\right).
ⅲ) ([31], Proposition 3.5) For \varepsilon > 0, let A_{\varepsilon}: = \left\{ x\in\Omega:dist\left(x, \partial\Omega\right) < \varepsilon\right\}. If h\in L^{1}\left(\Omega\right) and if u\in L^{1}\left(\Omega\right) is a solution of -\Delta u = h , in the sense of Definition 2.8, then there exists a constant c such that, for all \varepsilon > 0, \int_{A_{\varepsilon}}\left\vert u\right\vert \leq c\varepsilon^{2}\left\Vert h\right\Vert _{1}. In particular, \lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon}\int_{A_{\varepsilon} }\left\vert u\right\vert = 0.
ⅳ) ([31], Proposition 5.2) Let u\in L^{1}\left(\Omega\right) and h\in L^{1}\left(\Omega\right). If -\Delta u\leq h (respectively -\Delta u = h ) in D^{\prime}\left(\Omega\right) and \lim_{\varepsilon\rightarrow0^{+} }\frac{1}{\varepsilon}\int_{A_{\varepsilon}}\left\vert u\right\vert = 0 then -\Delta u\leq h (resp. -\Delta u = h ) in the sense of \left(C_{0} ^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime}.
ⅴ) ([31], Proposition 5.9) Let f_{1}, f_{2}\in L^{1}\left(\Omega\right). If u_{1}, u_{2}\in L^{1}\left(\Omega\right) are such that \Delta u_{1}\geq f_{1} and \Delta u_{2}\geq f_{2} in the sense of distributions in \Omega , then \Delta\max\left\{ u_{1}, u_{2}\right\} \geq\chi_{\left\{ u_{1} > u_{2}\right\} }f_{1} +\chi_{\left\{ u_{2} > u1\right\} }f_{2}+\chi_{\left\{ u_{1} = u_{2}\right\} }\frac{1}{2}\left(f_{1}+f_{2}\right) in the sense of distributions in \Omega.
If h:\Omega\rightarrow\mathbb{R} is a measurable function such that h\varphi\in L^{1}\left(\Omega\right) for any \varphi\in C_{c}^{\infty}\left(\Omega\right), we say that u:\Omega\rightarrow \mathbb{R} is a subsolution (respectively a supersolution), in the sense of distributions, of the problem -\Delta u = h in \Omega, if u\in L_{loc} ^{1}\left(\Omega\right) and -\int_{\Omega}u\Delta\varphi\leq\int _{\Omega}h\varphi (resp. -\int_{\Omega}u\Delta\varphi\geq\int_{\Omega }h\varphi ) for any nonnegative \varphi\in C_{c}^{\infty}\left(\Omega\right).
Remark 2.10. ([28], Theorem 2.4) Let f:\Omega \times\left(0, \infty\right) \rightarrow\mathbb{R} be a Caratheodory function, and let \underline{w} and \overline{w} be two functions, both in L_{loc}^{\infty}\left(\Omega\right) \cap W_{loc}^{1, 2}\left(\Omega\right), and such that f\left(., \underline{w}\right) and f\left(., \overline{w}\right) belong to L_{loc}^{1}\left(\Omega\right). Suppose that \underline{w} is a subsolution and \overline{w} is a supersolution, both in the sense of distributions, of the problem
\begin{equation} -\Delta w = f\left( ., w\right) \text{ in }\Omega. \end{equation} | (2.6) |
Suppose in addition that 0 < \underline{w}\left(x\right) \leq\overline {w}\left(x\right) a.e. x\in \Omega, and that there exists h\in L_{loc}^{\infty}\left(\Omega\right) such that \sup_{s\in\left[\underline{w}\left(x\right), \overline{w}\left(x\right) \right] }\left\vert f\left(x, s\right) \right\vert \leq h\left(x\right) a.e. x\in \Omega. Then (2.6) has a solution w, in the sense of distributions, which satisfies \underline{w}\leq w\leq\overline{w} a.e. in \Omega. Moreover, as obverved in [28], if in addition f\left(., w\right) \in L_{loc}^{\infty}\left(\Omega\right), then, by a density argument, the equality \int_{\Omega}\left\langle \nabla w, \nabla \varphi\right\rangle = \int_{\Omega}f\left(., w\right) \varphi holds also for any \varphi\in W_{loc}^{1, 2}\left(\Omega\right) with compact support.
Remark 2.11. Let us recall the Hardy inequality (as stated, e.g., in [29], Theorem 1.10.15, see also [1], p. 313): There exists a positive constant c such that \left\Vert \frac{\varphi }{d_{\Omega}}\right\Vert _{L^{2}\left(\Omega\right) }\leq c\left\Vert \nabla\varphi\right\Vert _{L^{2}\left(\Omega\right) } for all \varphi\in H_{0}^{1}\left(\Omega\right).
Remark 2.12. Let a and \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2 and assume that \alpha\in\left(0, 1\right]. Let \varepsilon\in\left(0, 1\right]. If u\in L^{\infty}\left(\Omega\right) and if, for some positive constant c, u\geq cd_{\Omega} a.e. in \Omega, then au^{-\alpha}-g_{\varepsilon }\left(., u\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Indeed, for \varphi\in H_{0}^{1}\left(\Omega\right) we have \left\vert au^{-\alpha}\varphi\right\vert \leq c^{-\alpha}d_{\Omega }^{1-\alpha}\left\vert \frac{\varphi}{d_{\Omega}}\right\vert. Since d_{\Omega}^{1-\alpha}\in L^{\infty}\left(\Omega\right) (because \alpha\leq1 ), the Hardy inequality gives a positive constant c^{\prime} independent of \varphi such that \left\Vert au^{-\alpha}\varphi\right\Vert _{1}\leq c^{\prime}\left\Vert \nabla\varphi\right\Vert _{2}. Also, since u\in L^{\infty}\left(\Omega\right), from h6) and the Hardy inequality, \left\Vert g_{\varepsilon}\left(., u\right) \varphi\right\Vert _{1}\leq c^{\prime\prime}\left\Vert \nabla\varphi\right\Vert _{2}, with c^{\prime\prime} a positive constant independent of \varphi.
Lemma 2.13. Let a and \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2 and assume that \alpha\in\left(0, 1\right]. Let \varepsilon\in\left(0, 1\right]. Suppose that u\in W_{loc}^{1, 2}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) is a solution, in the sense of distributions, of the problem
\begin{equation} -\Delta u = au^{-\alpha}-g_{\varepsilon}\left( ., u\right) ~\mathit{\text{in}}~\Omega, \end{equation} | (2.7) |
and that there exist positive constants c, c^{\prime} and \gamma such that c^{\prime}d_{\Omega}\leq u\leq cd_{\Omega}^{\gamma} a.e. in \Omega. Then u\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega }\right), and u is a weak solution, in the usual sense of H_{0} ^{1}\left(\Omega\right), of problem (1.6).
Proof. Since u\in L^{\infty}\left(\Omega\right) and u\geq c^{\prime}d_{\Omega }, we have au^{-\alpha}-g_{\varepsilon}\left(., u\right) \in L_{loc}^{\infty}\left(\Omega\right). Thus, from the inner elliptic estimates in ([20], Theorem 8.24), u\in C\left(\Omega\right) and, from the inequalities c^{\prime}d_{\Omega}\leq u\leq cd_{\Omega}^{\gamma} a.e. in \Omega, u is also continuous on \partial\Omega. Then u\in C\left(\overline{\Omega}\right)
The proof of that u\in H_{0}^{1}\left(\Omega\right) and that u is a weak solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.6), is a slight variation of the proof of ([24], Lemma 2.4). For the convenience of the reader, we give the details: For j\in\mathbb{N}, let h_{j}:\mathbb{R\rightarrow R} be the function defined by h_{j}\left(s\right) : = 0 if s\leq\frac{1}{j}, h_{j}\left(s\right) : = -3j^{2}s^{3}+14js^{2}-19s+\frac{8}{j} if \frac {1}{j} < s < \frac{2}{j} and h\left(s\right) = s for \frac{2}{j}\leq s. Then h_{j}\in C^{1}\left(\mathbb{R}\right), h_{j}^{\prime}\left(s\right) = 0 for s < \frac{1}{j}, h_{j}^{\prime}\left(s\right) \geq0 for \frac{1}{j} < s < \frac{2}{j} and h_{j}^{\prime}\left(s\right) = 1 for \frac{2}{j}\leq s . Moreover, for s\in\left(\frac{1}{j}, \frac{2} {j}\right) we have s^{-1}h_{j}\left(s\right) = -3j^{2}s^{2} +14js-19+\frac{8}{js} < -3j^{2}s^{2}+14js-11 < 5 (the last inequality because -3t^{2}+14t-16 < 0 whenever t\notin\left[\frac{8}{3}, 2\right] ). Thus 0\leq h_{j}\left(s\right) \leq5s for any j\in\mathbb{N} and s\geq 0.
Let h_{j}\left(u\right) : = h_{j}\circ u. Then, for all j, \nabla\left(h_{j}\left(u\right) \right) = h_{j}^{\prime}\left(u\right) \nabla u. Since u\in W_{loc}^{1, 2}\left(\Omega\right), we have h_{j}\left(u\right) \in W_{loc}^{1, 2}\left(\Omega\right), and since h_{j}\left(u\right) has compact support, Remark 2.10 gives, for all j\in\mathbb{N}, \int_{\Omega }\left\langle \nabla u, \nabla\left(h_{j}\left(u\right) \right) \right\rangle = \int_{\Omega}\left(au^{-\alpha}-g_{\varepsilon}\left(., u\right) \right) h_{j}\left(u\right), i.e.,
\begin{equation} \int_{\left\{ u \gt 0\right\} }h_{j}^{\prime}\left( u\right) \left\vert \nabla u\right\vert ^{2} = \int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) . \end{equation} | (2.8) |
Now, h_{j}^{\prime}\left(u\right) \left\vert \nabla u\right\vert ^{2} is a nonnegative function and \lim_{j\rightarrow\infty}h_{j}^{\prime}\left(u\right) \left\vert \nabla u\right\vert ^{2} = \left\vert \nabla u\right\vert ^{2} a.e. in \Omega, and so, by (2.8) and the Fatou's lemma,
\int_{\Omega}\left\vert \nabla u\right\vert ^{2}\leq\underline{\lim }_{j\rightarrow\infty}\int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) . |
Also,
\lim\limits_{j\rightarrow\infty}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) = au^{1-\alpha}-ug_{\varepsilon }\left( ., u\right) \text{ }a.e.\text{ in }\Omega. |
Now, 0\leq au^{-\alpha}h_{j}\left(u\right) \leq5au^{1-\alpha}. Since a and u belong to L^{\infty}\left(\Omega\right) and \alpha\leq1, we have au^{1-\alpha}\in L^{1}\left(\Omega\right). Also,
0\leq g_{\varepsilon}\left( ., u\right) h_{j}\left( u\right) \leq 5ug_{\varepsilon}\left( ., u\right) \leq5\left\Vert u\right\Vert _{\infty }^{2}\sup\limits_{0 \lt s\leq\left\Vert u\right\Vert _{\infty}}s^{-1}g_{\varepsilon }\left( ., s\right) \text{ }a.e.\text{ in }\Omega, |
and, by h6), \sup_{0 < s\leq\left\Vert u\right\Vert _{\infty}} s^{-1}g_{\varepsilon}\left(., s\right) \in L^{\infty}\left(\Omega\right). Then, by the Lebesgue dominated convergence theorem,
\lim\limits_{j\rightarrow\infty}\int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon }\left( ., u\right) \right) h_{j}\left( u\right) = \int_{\Omega}\left( au^{1-\alpha}-ug_{\varepsilon}\left( ., u\right) \right) \lt \infty. |
Thus \int_{\Omega}\left\vert \nabla u\right\vert ^{2} < \infty, and so u\in H^{1}\left(\Omega\right). Since u\in C\left(\overline{\Omega}\right) and u = 0 on \partial\Omega, we conclude that u\in H_{0}^{1}\left(\Omega\right). Also, by Remark 2.12, au^{-\alpha }-g_{\varepsilon}\left(., u\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Then, by a density argument, the equality
\int_{\Omega}\left\langle \nabla u, \nabla\varphi\right\rangle = \int_{\Omega }\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) \varphi |
which holds for \varphi\in C_{c}^{\infty}\left(\Omega\right), holds also for any \varphi\in H_{0}^{1}\left(\Omega\right).
Lemma 2.14. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Let \varepsilon\in\left(0, 1\right] and let f_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} be defined by f_{\varepsilon}\left(., s\right) : = \chi_{\left(0, \infty\right) }\left(s\right) as^{-\alpha}-g_{\varepsilon}\left(., s\right). Let v_{1} and v_{2} be two nonnegative functions in L^{\infty}\left(\Omega\right) \cap H_{0}^{1}\left(\Omega\right) such that f_{\varepsilon}\left(., v_{i}\right) \in L_{loc}^{1}\left(\Omega\right) for i = 1, 2; and let v: = \max\left\{ v_{1}, v_{2}\right\}. Then:
i) f_{\varepsilon}\left(., v\right) \in L_{loc}^{1}\left(\Omega\right).
ii) If v_{1} and v_{2} are subsolutions, in the sense of distributions, to problem (1.7), then v is also a subsolution, in the sense of distributions, to the problem
-\Delta u = \chi_{\left\{ u \gt 0\right\} }au^{-\alpha}-g_{\varepsilon}\left( ., u\right) ~\mathit{\text{in}}~\Omega. |
Proof. Since 0\leq v\in L^{\infty}\left(\Omega\right), from h7) and h8) it follows that g_{\varepsilon}\left(., v\right) \in L^{1}\left(\Omega\right). Similarly, g_{\varepsilon}\left(., v_{1}\right) and g_{\varepsilon }\left(., v_{2}\right) belong to L^{1}\left(\Omega\right) and so, since f_{\varepsilon}\left(., v_{i}\right) \in L_{loc}^{1}\left(\Omega\right) for i = 1, 2; we get that \chi_{\left\{ v_{1} > 0\right\} }av_{1}^{-\alpha} and \chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} belong to L_{loc}^{1}\left(\Omega\right). Therefore, to prove i) it suffices to see that \chi_{\left\{ v > 0\right\} }av^{-\alpha }\in L_{loc}^{1}\left(\Omega\right). Note that if x\in\Omega and v\left(x\right) > 0 then either v_{1}\left(x\right) > 0 or v_{2}\left(x\right) > 0. Now, \chi_{\left\{ v > 0\right\} }av^{-\alpha } = av^{-\alpha}\leq av_{1}^{-\alpha} = \chi_{\left\{ v_{1} > 0\right\} } av_{1}^{-\alpha} in \left\{ v_{1} > 0\right\}, and similarly, \chi_{\left\{ v > 0\right\} }av^{-\alpha}\leq\chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} in \left\{ v_{2} > 0\right\}. Also, \chi_{\left\{ v > 0\right\} }av^{-\alpha} = 0 in \left\{ v = 0\right\}. Then \chi _{\left\{ v > 0\right\} }av^{-\alpha}\leq\chi_{\left\{ v_{1} > 0\right\} }av_{1}^{-\alpha}+\chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} in \Omega and so \chi_{\left\{ v > 0\right\} }av^{-\alpha}\in L_{loc} ^{1}\left(\Omega\right). Thus i) holds.
To see ii), suppose that -\Delta v_{i}\leq f_{\varepsilon}\left(., v_{i}\right) in D^{\prime}\left(\Omega\right) for i = 1, 2; and let \varphi be a nonnegative function in C_{c}^{\infty}\left(\Omega\right). Let \Omega^{\prime} be a C^{1, 1} subdomain of \Omega, such that supp\left(\varphi\right) \subset\Omega^{\prime} and \overline {\Omega^{\prime}}\subset\Omega. Consider the restrictions (still denoted by v_{1} and v_{2} ) of v_{1} and v_{2} to \Omega^{\prime}. For each i = 1, 2, we have v_{i}\in L^{1}\left(\Omega^{\prime}\right), f_{\varepsilon}\left(., v_{i}\right) \in L^{1}\left(\Omega^{\prime }\right) and -\Delta v_{i}\leq f_{\varepsilon}\left(., v_{i}\right) in D^{\prime}\left(\Omega^{\prime}\right). Thus, from Remark 2.9 v),
\begin{array}{l} -\Delta v & \leq\chi_{\left\{ v_{1} \gt v_{2}\right\} }f_{\varepsilon}\left( ., v_{1}\right) +\chi_{\left\{ v_{2} \gt v_{1}\right\} }f_{\varepsilon}\left( ., v_{2}\right) +\chi_{\left\{ v_{1} = v_{2}\right\} }\frac{1}{2}\left( f_{\varepsilon}\left( ., v_{1}\right) +f_{\varepsilon}\left( ., v_{2}\right) \right) \\ & = f_{\varepsilon}\left( ., v\right) \text{ in }D^{\prime}\left( \Omega^{\prime}\right) \end{array} |
and then -\int_{\Omega}v\Delta\varphi\leq\int_{\Omega}f_{\varepsilon}\left(., v\right) \varphi.
Lemma 2.15. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then for any \varepsilon\in\left(0, 1\right] there exists a weak solution u_{\varepsilon} , in the sense of Definition 1.1, of problem (1.7), which is maximal in the following sense: If v is a weak solution, in the sense of Definition 1.1, of problem (1.7), then v\leq u_{\varepsilon} a.e. in \Omega. Moreover, u_{\varepsilon} is a solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.7).
Proof. Let z be as given in Remark 2.4, and let \mathcal{S} be the set of the nonidentically zero weak solutions, in the sense of Definition 1.1, of problem (1.7). By Lemma 2.2, \mathcal{S}\neq\varnothing and, for any u\in\mathcal{S} , by Lemma 2.5 we have u\leq z in \Omega and, by Lemma 2.6, there exists a positive constant c such that u\geq cd_{\Omega } in \Omega. Then 0 < \int_{\Omega}u\leq\int_{\Omega}z < \infty for any u\in\mathcal{S}. Let \beta: = \sup\left\{ \int_{\Omega}u:u\in\mathcal{S} \right\}. Thus 0 < \beta < \infty. Let \left\{ u_{k}\right\} _{k\in\mathbb{N}}\subset\mathcal{S} be a sequence such that \lim _{k\rightarrow\infty}\int_{\Omega}u_{k} = \beta. For k\in\mathbb{N} , let w_{k}: = \max\left\{ u_{j}:1\leq j\leq k\right\}. Thus \left\{ w_{k}\right\} _{k\in\mathbb{N}} is a nondecreasing sequence in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and a repeated use of Lemma 2.14 gives that each w_{k} is a subsolution, in the sense of D^{\prime}\left(\Omega\right) , of the problem
\begin{equation} -\Delta u = au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \text{ in }\Omega. \end{equation} | (2.9) |
Since w_{k}\in L^{\infty}\left(\Omega\right) and w_{k}\geq u_{1}\geq c_{1}d_{\Omega} a.e. in \Omega, Remark 2.12 gives that aw_{k}^{-\alpha}-g_{\varepsilon}\left(., w_{k}\right) \in\left(H_{0} ^{1}\left(\Omega\right) \right) ^{\prime}. Then, by a density argument, the inequality
\begin{equation} \int_{\Omega}\left\langle \nabla w_{k}, \nabla\varphi\right\rangle \leq \int_{\Omega}\left( aw_{k}^{-\alpha}-g_{\varepsilon}\left( ., w_{k}\right) \right) \varphi, \end{equation} | (2.10) |
which holds for \varphi\in C_{c}^{\infty}\left(\Omega\right), holds also for any \varphi\in H_{0}^{1}\left(\Omega\right), i.e., w_{k} is a subsolution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (2.9)
Note that \left\{ \int_{\left\{ a > 0\right\} }aw_{k}^{1-\alpha}\right\} _{k\in\mathbb{N}} is bounded. Indeed, since u_{k}\leq z a.e. in \Omega for any k\in \mathbb{N} , we have w_{k}\leq z a.e. in \Omega for all k, and so \int_{\left\{ a > 0\right\} }aw_{k}^{1-\alpha}\leq\int_{\Omega}az^{1-\alpha } < \infty. Moreover, \left\{ w_{k}\right\} _{k\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right). In fact, taking w_{k} as a test function in (2.10) we get, for any k\in\mathbb{N},
\begin{equation} \int_{\Omega}\left\vert \nabla w_{k}\right\vert ^{2}+\int_{\Omega }g_{\varepsilon}\left( ., w_{k}\right) w_{k}\leq\int_{\left\{ a \gt 0\right\} }aw_{k}^{1-\alpha} \end{equation} | (2.11) |
Then, after pass to a subsequence if necessary, we can assume that there exists w\in H_{0}^{1}\left(\Omega\right) such that \left\{ w_{k}\right\} _{k\in\mathbb{N}} converges in L^{2}\left(\Omega\right) and a.e. in \Omega to w; and \left\{ \nabla w_{k}\right\} _{k\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R} ^{n}\right) to \nabla w. Let us show that w is a subsolution, in the sense of distributions of problem (2.9). Let \varphi be a nonnegative function in C_{c}^{\infty}\left(\Omega\right) and let k\in\mathbb{N}. Since w_{k} is a subsolution, in the sense of distributions, of (2.9), we have
\begin{equation} \int_{\Omega}\left\langle \nabla w_{k}, \nabla\varphi\right\rangle +\int_{\Omega}g_{\varepsilon}\left( ., w_{k}\right) \varphi\leq\int_{\Omega }aw_{k}^{-\alpha}\varphi. \end{equation} | (2.12) |
Since \left\{ \nabla w_{k}\right\} _{k\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla w, we have
\lim\limits_{k\rightarrow\infty}\int_{\Omega}\left\langle \nabla w_{k}, \nabla \varphi\right\rangle = \int_{\Omega}\left\langle \nabla w, \nabla\varphi \right\rangle . |
Also, since \left\{ g_{\varepsilon}\left(., w_{k}\right) \varphi\right\} _{k\in\mathbb{N}} converges to g_{\varepsilon}\left(., w\right) \varphi a.e. in \Omega, and
\left\vert g_{\varepsilon}\left( ., w_{k}\right) \varphi\right\vert \leq \sup\limits_{s\in\left[ 0, \left\Vert z\right\Vert _{\infty}\right] }\left( s^{-1}g_{\varepsilon}\left( ., s\right) \right) w_{k}\left\vert \varphi\right\vert \in L^{1}\left( \Omega\right) , |
the Lebesgue dominated convergence theorem gives
\lim\limits_{k\rightarrow\infty}\int_{\Omega}g_{\varepsilon}\left( ., w_{k}\right) \varphi = \int_{\Omega}g_{\varepsilon}\left( ., w\right) \varphi. |
On the other hand, \left\{ aw_{k}^{-\alpha}\varphi\right\} _{k\in \mathbb{N}} converges to aw^{-\alpha}\varphi a.e. in \Omega; and w_{k}\geq u_{1}\geq cd_{\Omega} a.e. in \Omega , and so \left\vert aw_{k}^{-\alpha}\varphi\right\vert \leq c^{-\alpha}ad_{\Omega}^{1-\alpha }\left\vert d_{\Omega}^{-1}\varphi\right\vert a.e. in \Omega; and, since d_{\Omega}^{1-\alpha}\in L^{\infty}\left(\Omega\right), the Hardy inequality gives that ad_{\Omega}^{1-\alpha}\left\vert d_{\Omega}^{-1} \varphi\right\vert \in L^{1}\left(\Omega\right). Then, by the Lebesgue dominated convergence theorem, \lim_{k\rightarrow\infty}\int_{\Omega} aw_{k}^{-\alpha}\varphi = \int_{\Omega}aw^{-\alpha}\varphi < \infty. Hence, from (2.12),
\int_{\Omega}\left\langle \nabla w, \nabla\varphi\right\rangle +\int_{\Omega }g_{\varepsilon}\left( ., w\right) \varphi\leq\int_{\Omega}aw^{-\alpha }\varphi, |
and so w is a subsolution, in the sense of distributions to problem (2.9). Note that z is a supersolution, in the sense of distributions, of problem (2.9) and that w\leq z a.e. in \Omega (because u_{k}\leq z for all k\in\mathbb{N} ). Also, for some positive constant c and for any k, w\geq w_{k}\geq u_{1}\geq cd_{\Omega} a.e. in \Omega. Then there exists a positive constant c^{\prime} such that
\sup\limits_{s\in\left[ w\left( x\right) , z\left( x\right) \right] }\left( \chi_{\left\{ s \gt 0\right\} }a\left( x\right) s^{-\alpha}-g_{\varepsilon }\left( x, s\right) \right) \leq c^{\prime}d_{\Omega}^{-\alpha}\text{ for }a.e\text{ }x\in\Omega |
and so, by Remark 2.10, there exists a solution u_{\varepsilon }\in W_{loc}^{1, 2}\left(\Omega\right) , in the sense of distributions, of (2.9) such that w\leq u_{\varepsilon}\leq z a.e. a.e. in \Omega. Therefore, by Remark 2.4, cd_{\Omega }\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{\tau} a.e. in \Omega, with c, c^{\prime} and \tau positive constants. Then, by Lemma 2.13, u_{\varepsilon}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) and u_{\varepsilon} is a weak solution, in the sense of Definition 1.1, of problem (1.7). Also, u_{\varepsilon}\geq w\geq w_{k}\geq u_{k} a.e. in \Omega for any k\in\mathbb{N}, and so \int_{\Omega }u_{\varepsilon}\geq\beta which, by the definition of \beta, implies \int_{\Omega}u_{\varepsilon} = \beta.
Let us show that u_{\varepsilon } is the maximal solution of problem (1.7), in the sense required by the lemma. Suppose that w^{\ast} is a nonidentically zero weak solution, in the sense of Definition 1.1, of (1.7). By Lemmas 2.5, 2.7 and 2.6, w^{\ast}\leq z in \Omega, w^{\ast}\in C\left(\overline{\Omega}\right) and w^{\ast}\geq cd_{\Omega} a.e. in \Omega with c a positive constant c. Let w^{\ast\ast}: = \max\left\{ u_{\varepsilon}, w^{\ast}\right\}. Thus w^{\ast\ast} is a subsolution, in the sense of distributions, of problem (2.9), Remark 2.10 applies to obtain a solution \widetilde{w} , in the sense of distributions, of problem (1.7), such that w^{\ast\ast}\leq\widetilde{w}\leq z, and Lemma 2.13 applies to obtain that \widetilde{w}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) and that \widetilde{w} is a weak solution, in the sense of Definition 1.1, to problem (1.7). Then \int_{\Omega}\widetilde{w}\leq\beta. Since u_{\varepsilon}\leq w^{\ast\ast}\leq\widetilde{w} we get \beta = \int_{\Omega}u_{\varepsilon}\leq\int_{\Omega}w^{\ast\ast}\leq\int_{\Omega }\widetilde{w}\leq\beta, and so u_{\varepsilon} = w^{\ast\ast}. Thus u_{\varepsilon}\geq w^{\ast}.
For \varepsilon\in\left(0, 1\right], let u_{\varepsilon} be the maximal weak solution to problem (1.7) given by Lemma 2.15.
Lemma 2.16. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then the map \varepsilon\rightarrow u_{\varepsilon} is nondecreasing on \left(0, 1\right].
Proof. For 0 < \varepsilon < \eta we have, in the sense of definition 1.1,
-\Delta u_{\varepsilon} = au_{\varepsilon}^{-\alpha}-g_{\varepsilon}\left( ., u_{\varepsilon}\right) \leq au_{\varepsilon}^{-\alpha}-g_{\eta}\left( ., u_{\varepsilon}\right) \text{ in }\Omega, |
and so u_{\varepsilon}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) is a subsolution, in the sense of distributions, to the problem
\begin{equation} -\Delta u = au^{-\alpha}-g_{\eta}\left( ., u\right) \text{ in }\Omega. \end{equation} | (2.13) |
Let z be as in Remark 2.4. Thus z is a supersolution, in the sense of distributions, of problem (2.9), and z\leq cd_{\Omega}^{\tau} a.e. in \Omega, with c and \tau positive constants c. Taking into account that, for some positive constant c, u_{\varepsilon}\geq cd_{\Omega} a.e. in \Omega, Remark 2.10 applies, as before, to obtain a weak solution, in the sense of distributions, \widetilde{u}_{\eta}\in W_{loc}^{1, 2}\left(\Omega\right) of (2.13) such that u_{\varepsilon}\leq\widetilde{u}_{\eta}\leq z. Now, Lemma 2.13 gives that \widetilde{u}_{\eta}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) and that \widetilde{u}_{\eta} is a weak solution, in the sense of Definition 1.1, of problem (2.13), which implies \widetilde{u}_{\eta}\leq u_{\eta}. Thus u_{\varepsilon}\leq u_{\eta}.
Lemma 2.17. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then \left\{ u_{\varepsilon}\right\} _{\varepsilon \in\left(0, 1\right] } is bounded in H_{0}^{1}\left(\Omega\right).
Proof. Let z be as in Remark 2.4. by Lemma 2.5 u_{\varepsilon}\leq z in \Omega and so, since 0 < \alpha\leq1, we have \int_{\left\{ a > 0\right\} }au_{\varepsilon}^{1-\alpha}\leq\int_{\Omega }az^{1-\alpha} < \infty. By taking u_{\varepsilon} as a test function in (1.7) we get, for any \varepsilon\in\left(0, 1\right],
\int_{\Omega}\left\vert \nabla u_{\varepsilon}\right\vert ^{2}+\int_{\Omega }u_{\varepsilon}g_{\varepsilon}\left( ., u_{\varepsilon}\right) = \int_{\left\{ a \gt 0\right\} }au_{\varepsilon}^{1-\alpha}. |
Then \int_{\Omega}\left\vert \nabla u_{\varepsilon}\right\vert ^{2}\leq \int_{\Omega}az^{1-\alpha} < \infty.
Lemma 2.18. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Let \boldsymbol{u}: = \lim_{\varepsilon\rightarrow0^{+} }u_{\varepsilon}. Then:
i) \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).
ii) \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}.
iii) \chi_{\left\{ \boldsymbol{u} > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi\in L^{1}\left(\Omega\right) for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).
iv) If \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}} is a decreasing sequence in \left(0, 1\right] such that \lim_{j\rightarrow\infty}\varepsilon _{j} = 0 then \lim_{j\rightarrow\infty}\int_{\left\{ a > 0\right\} }au_{\varepsilon_{j}}^{-\alpha}\varphi = \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).
Proof. To see i), consider a nonincreasing sequence \left\{ \varepsilon _{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right] such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. By Lemma 2.17, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right) and so, after pass to a subsequence if necessary, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges, strongly in L^{2}\left(\Omega\right), and a.e. in \Omega, to some \widetilde{u}\in H_{0}^{1}\left(\Omega\right), and \left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla\widetilde{u}. Since u_{\varepsilon_{j}} converges to \boldsymbol{u} a.e. in \Omega we have \boldsymbol{u} = \widetilde{u} a.e. in \Omega, and so \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right). Also, 0\leq\boldsymbol{u}\leq u_{\varepsilon_{1}}\in L^{\infty}\left(\Omega\right) and then \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Thus i) holds.
To see ii) and iii), consider an arbitrary nonnegative function \varphi\in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). From (1.7) we have, for each j,
\begin{equation} \int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi \right\rangle +\int_{\Omega}g_{\varepsilon_{j}}\left( ., u_{\varepsilon_{j} }\right) \varphi = \int_{\Omega}au_{\varepsilon_{j}}^{-\alpha}\varphi. \end{equation} | (2.14) |
\left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla \boldsymbol{u}, and thus
\lim\limits_{j\rightarrow\infty}\int_{\Omega}\left\langle \nabla u_{\varepsilon_{j} }, \nabla\varphi\right\rangle = \int_{\Omega}\left\langle \nabla\boldsymbol{u} , \nabla\varphi\right\rangle . |
By Lemma 2.16, \left\{ au_{\varepsilon_{j}}^{-\alpha} \varphi\right\} _{j\in\mathbb{N}} is nondecreasing, then, by the monotone convergence theorem, \lim_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon _{j}}^{-\alpha}\varphi = \lim_{j\rightarrow\infty}\int_{\left\{ a > 0\right\} }au_{\varepsilon_{j}}^{-\alpha}\varphi = \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi.
Let z be as in Lemma 2.5. Then u_{\varepsilon_{j}}\leq z in \Omega and so, taking into account h4), \int_{\Omega}g_{\varepsilon_{j}}\left(., u_{\varepsilon_{j}}\right) \varphi\leq\int_{\Omega}\sup_{0\leq s\leq\left\Vert z\right\Vert _{\infty}}g\left(., s\right) \varphi < \infty. Thus
\begin{array}{l} \int_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi & = \lim\limits_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon_{j}}^{-\alpha} \varphi = \lim\limits_{j\rightarrow\infty}\left( \int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi\right\rangle +\int_{\Omega}g_{\varepsilon _{j}}\left( ., u_{\varepsilon_{j}}\right) \varphi\right) \\ & \leq\overline{\lim}_{j\rightarrow\infty}\int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi\right\rangle +\overline{\lim}_{j\rightarrow \infty}\int_{\Omega}g_{\varepsilon_{j}}\left( ., u_{\varepsilon_{j}}\right) \varphi\\ & \leq\int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi \right\rangle +\int_{\Omega}\sup\limits_{0\leq s\leq\left\Vert z\right\Vert _{\infty }}g\left( ., s\right) \varphi \lt \infty. \end{array} |
Therefore \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha} \varphi < \infty . Since this holds for any nonnegative \varphi\in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), we conclude that \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}. Thus ii) holds. Now,
\int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }a\boldsymbol{u} ^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi \lt \infty, |
and then iii) holds for any nonnegative \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Hence, by writing \varphi = \varphi^{+}-\varphi^{-}, iii) holds also for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Finally, observe that, in the case when \varphi\geq0, the monotone convergence theorem gives iv). Then, by writing \varphi = \varphi^{+}-\varphi^{-}, iv), holds also for an arbitrary \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).
Remark 2.19. Assume that a satisfies the conditions h1), h2) and also the condition h6) of Theorem 1.4; and let \Omega^{+} be as in h6). Taking into account h6), Remark 2.4 (applied in each connected component of \Omega^{+} ) gives that the problem
\begin{equation} \left\{ \begin{array} [c]{c} -\Delta\zeta = a\zeta^{-\alpha}\text{ in }\Omega^{+}, \\ \zeta = 0\text{ on }\partial\Omega^{+}, \\ \zeta \gt 0\text{ in }\Omega^{+}, \end{array} \right. \end{equation} | (2.15) |
has a unique weak solution, in the sense of Definition 1.1, \zeta\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and that it satisfies:
ⅰ) \zeta\in C\left(\overline{\Omega^{+}}\right).
ⅱ) There exists a positive constant c such that \zeta\geq cd_{\Omega^{+}} in \Omega^{+}.
ⅲ) \zeta is also a solution of problem (2.15) in the usual sense of H_{0}^{1}\left(\Omega^{+}\right), i.e., a\zeta^{-\alpha }\varphi\in L^{1}\left(\Omega\right) and \int_{\Omega}\left\langle \nabla\zeta, \nabla\varphi\right\rangle = \int_{\Omega}a\zeta^{-\alpha}\varphi for any \varphi\in H_{0}^{1}\left(\Omega^{+}\right).
Lemma 2.20. Assume that a and g satisfy the conditions h1)-h4) and also the condition h6) of Theorem 1.4. Let \Omega^{+} and A^{+} be as in the statement of Theorem 1.4 and assume, in addition, that g\left(., s\right) = 0 a.e. in A^{+} for any s\geq0. Let \zeta be as in Remark 2.19, let \varepsilon\in\left(0, 1\right], and let u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) be a weak solution, in the sense of Definition 1.1, of problem (1.5). Then u\geq\zeta in \Omega^{+}.
Proof. By Remark 2.19 i), \zeta\in C\left(\overline {\Omega^{+}}\right) and, by Lemma 2.7, u\in C\left(\overline{\Omega}\right). Also, since g\left(., s\right) = 0 a.e. in \Omega^{+} for s\geq0, we have -\Delta\left(u-\zeta\right) = a\left(u^{-\alpha}-\zeta^{-\alpha}\right) \geq0 in D^{\prime}\left(\Omega ^{+}\right). We claim that u\geq\zeta in \Omega^{+}. To prove this fact we proceed by the way of contradiction: Let U: = \left\{ x\in\Omega ^{+}:u\left(x\right) < \zeta\left(x\right) \right\} and suppose that U\neq\varnothing. Then U is an open subset of \Omega^{+} and -\Delta\left(u-\zeta\right) = a\left(u^{-\alpha}-\zeta^{-\alpha}\right) \geq0 in D^{\prime}\left(U\right). Notice that u-\zeta\geq0 on \partial U. In fact, if u\left(x\right) < \zeta\left(x\right) for some x\in\partial U we would have, either x\in\Omega^{+} or x\in \partial\Omega^{+}; if x\in\Omega^{+} then, since u and \zeta are continuous on \Omega^{+}, we would have u < \zeta on some ball around x, contradicting the fact that x\in\partial U, and if x\in\partial\Omega^{+}, then u\left(x\right) \geq0 = \zeta\left(x\right) contradicting our assumption that u\left(x\right) < \zeta\left(x\right). Then U = \varnothing and so u\geq\zeta in \Omega^{+}; and then, by continuity, also u\geq\zeta on \partial\Omega^{+}. Therefore, from the weak maximum principle, u\geq\zeta in \Omega^{+}.
Observe that if g:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} satisfies the conditions h3) and h4) stated at the introduction, and if, for \varepsilon\in\left(0, 1\right], g_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} is defined by
\begin{equation} g_{\varepsilon}\left( ., s\right) : = s\left( s+\varepsilon\right) ^{-1}g\left( ., s\right) , \end{equation} | (3.1) |
then, for any s > 0, g\left(., s\right) = \lim_{\varepsilon\rightarrow0^{+} }g_{\varepsilon}\left(., s\right) a.e. in \Omega; and the family \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } satisfies the conditions h7)-h9). Therefore all the results of the Section 2 hold for such a family \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] }.
Lemma 3.1. Let a:\Omega \rightarrow\mathbb{R} and g:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} satisfying the conditions h1)-h4) and, for \varepsilon \in\left(0, 1\right], let g_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} be defined by (3.1), let u_{\varepsilon} be as given by Lemma 2.15, and let \boldsymbol{u}: = \lim_{\varepsilon\rightarrow0^{+} }u_{\varepsilon}. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N} }\subset\left(0, 1\right] be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0 and, for j\in\mathbb{N} , let u_{\varepsilon_{j}} be as given by Lemma 2.15. Let \theta _{j}: = u_{\varepsilon_{j}}\left(u_{\varepsilon_{j}}+\varepsilon_{j}\right) ^{-1} . Then there exist a nonnegative function \theta^{\ast}\in L^{\infty }\left(\Omega\right) and a sequence \left\{ w_{m}\right\} _{m\in\mathbb{N}}\subset L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) with the following properties:
i) for each m\in\mathbb{N}, w_{m} = \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left(\nabla u_{\varepsilon_{l}}, \theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right), where each \mathcal{F}_{m} is a finite subset of \mathbb{N} satisfying \lim_{m\rightarrow\infty}\min\mathcal{F}_{m} = \infty; \ \gamma _{l, m}\in\left[0, 1\right] for any m\in\mathbb{N} and l\in \mathcal{F}_{m}; and \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m} = 1 for any m\in\mathbb{N}.
ii) \left\{ w_{m}\right\} _{m\in\mathbb{N}} converges strongly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) to \left(\nabla\mathbf{u}, \theta^{\ast }\right).
iii) \lim_{m\rightarrow\infty}\sum_{l\in\mathcal{F}_{m} }\gamma_{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) = \theta^{\ast} a.e. in \Omega.
iv) \theta^{\ast} = \chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) a.e. in \left\{ \mathbf{u} > 0\right\}.
Proof. By Lemma 2.17 \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right). Then, after pass to a subsequence if necessary, we can assume that \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges to \mathbf{u} in L^{2}\left(\Omega\right) and that \left\{ \nabla u_{\varepsilon_{j} }\right\} _{j\in\mathbb{N}} converges weakly to \nabla\mathbf{u} in L^{2}\left(\Omega, \mathbb{R}^{n}\right). Moreover, by Lemma 2.5, u_{\varepsilon_{j}}\leq z a.e. in \Omega for all j, and so \mathbf{u}\leq z a.e. in \Omega. Since, for any j, 0 < \theta_{j} < 1 a.e. in \Omega, and, by h3) and h4), 0\leq g\left(., u_{\varepsilon_{j}}\right) \leq\sup_{s\in\left[0, \left\Vert z\right\Vert _{\infty}\right] }g\left(., s\right) \in L^{\infty}\left(\Omega\right), we have that \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is bounded in L^{2}\left(\Omega\right). Thus, after pass to a further subsequence, we can assume that \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is weakly convergent in L^{2}\left(\Omega\right) to a function \theta^{\ast}\in L^{2}\left(\Omega\right), and that \left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is weakly convergent in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla\mathbf{u}. Then \left\{ \left(\nabla u_{\varepsilon_{j}}, \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right) \right\} _{j\in\mathbb{N}} is weakly convergent to \left(\nabla\mathbf{u}, \theta^{\ast}\right) in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right). Thus (see e.g., [33] Theorem 3.13) there exists a sequence \left\{ w_{m}\right\} _{m\in\mathbb{N}} of the form w_{m} = \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left(\nabla u_{\varepsilon_{l}}, \theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right), where each \mathcal{F}_{m} is a finite subset of \mathbb{N} such that \lim_{m\rightarrow\infty}\min\mathcal{F}_{m} = \infty, \gamma_{l, m}\in\left[0, 1\right] for any m\in\mathbb{N} and l\in\mathcal{F}_{m}, for each m, \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m} = 1 and such that \left\{ w_{m}\right\} _{m\in\mathbb{N}} converges strongly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) to \left(\nabla\mathbf{u}, \theta^{\ast}\right). Then i) and ii) hold, and \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right\} _{m\in\mathbb{N}} converges in L^{2}\left(\Omega\right) to \theta^{\ast}. Therefore, after pass to a further subsequence, we can assume that \lim_{m\rightarrow\infty}\sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left(., \boldsymbol{u} _{\varepsilon_{l}}\right) = \theta^{\ast} a.e. in \Omega and, since \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is bounded in L^{\infty}\left(\Omega\right), we have that \theta^{\ast}\in L^{\infty}\left(\Omega\right). Thus iii) holds. Also \left\{ \theta_{j}\right\} _{j\in\mathbb{N}} and \left\{ g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} converge, a.e. in \left\{ \mathbf{u} > 0\right\} , to \chi_{\left\{ \mathbf{u} > 0\right\} } and to g\left(., \mathbf{u}\right) respectively, and then iv) follows from iii).
Proof of Theorem 1.2. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right) be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0, let \theta^{\ast} and \left\{ w_{m}\right\} _{m\in\mathbb{N}}\subset L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) be as given by Lemma 3.1, and let \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Assume temporarily that \varphi \geq0 in \Omega. Then \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma _{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) \varphi\right\} _{m\in\mathbb{N}} and \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma _{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle \right\} _{m\in\mathbb{N}} converge in L^{1}\left(\Omega\right) to \theta^{\ast}\varphi and \left\langle \nabla\mathbf{u}, \nabla \varphi\right\rangle respectively. Thus
\begin{align} \lim\limits_{m\rightarrow\infty}\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma _{l, m}\theta_{l}g\left( ., u_{\varepsilon_{l}}\right) \varphi & = \int_{\Omega}\theta^{\ast}\varphi, \end{align} | (3.2) |
\begin{align} \lim\limits_{m\rightarrow\infty}\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma _{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle & = \int_{\Omega}\left\langle \nabla\mathbf{u}, \nabla\varphi\right\rangle \end{align} | (3.3) |
and both limits are finite. Since \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is nonincreasing we have, for m\in\mathbb{N} and l\in\mathcal{F}_{m},
\begin{equation} au_{\varepsilon_{L_{m}}}^{-\alpha}\varphi\leq a\sum\limits_{l\in\mathcal{F}_{m} }\gamma_{l, m}u_{\varepsilon_{l}}^{-\alpha}\varphi\leq au_{\varepsilon _{L_{m}^{\ast}}}^{-\alpha}\varphi, \end{equation} | (3.4) |
where L_{m}: = \max\mathcal{F}_{m} and L_{m}^{\ast}: = \min\mathcal{F}_{m}. Also, by the monotone convergence theorem,
\begin{equation} \lim\limits_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon_{j}}^{-\alpha} \varphi = \lim\limits_{j\rightarrow\infty}\int_{\left\{ a \gt 0\right\} }au_{\varepsilon _{j}}^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u} ^{-\alpha}\varphi, \end{equation} | (3.5) |
the last equality because, by Lemma 2.18, \mathbf{u} > 0 a.e. in \left\{ a > 0\right\}. Then, since \lim_{m\rightarrow\infty}L_{m}^{\ast } = \infty, (3.4) and (3.5) give
\begin{equation} \lim\limits_{m\rightarrow\infty}\int_{\left\{ a \gt 0\right\} }a\sum\limits_{l\in \mathcal{F}_{m}}\gamma_{l, m}\boldsymbol{u}_{\varepsilon_{l}}^{-\alpha} \varphi = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u} ^{-\alpha}\varphi. \end{equation} | (3.6) |
(notice that, by Lemma 2.18, \int_{\Omega}\chi_{\left\{ \mathbf{u} > 0\right\} }a\mathbf{u}^{-\alpha}\varphi < \infty ). Since \theta_{l}g\left(., u_{\varepsilon_{l}}\right) = g_{\varepsilon_{l}}\left(., u_{\varepsilon_{l}}\right) we have, for any m\in\mathbb{N} , and in the sense of definition 1.1,
\begin{equation} \left\{ \begin{array} [c]{c} -\Delta\left( \sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l} }\right) \\ = a\sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l}}^{-\alpha} -\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left( ., u_{\varepsilon _{l}}\right) \text{ in }\Omega, \\ \sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l}} = 0\text{ on } \partial\Omega \end{array} \right. \end{equation} | (3.7) |
and so
\begin{align} & \int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle \\ & = \int_{\Omega}a\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l} }^{-\alpha}\varphi-\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m} \theta_{l}g\left( ., u_{\varepsilon_{l}}\right) \varphi. \end{align} | (3.8) |
Taking the limit as m\rightarrow\infty in (3.8), and using (3.2), (3.3), (3.6) and recalling that, by Lemma 3.1 iv), \theta^{\ast} = \chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) a.e. in \left\{ \mathbf{u} > 0\right\} , we get that
\begin{align} \int_{\Omega}\left\langle \nabla\mathbf{u}, \nabla\varphi\right\rangle & = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi-\int_{\Omega}\theta^{\ast}\varphi \\ & = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi-\int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }g\left( ., \mathbf{u}\right) \varphi-\int_{\left\{ \mathbf{u} = 0\right\} } \theta^{\ast}\varphi. \end{align} | (3.9) |
for any nonnegative \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and by writing \varphi = \varphi ^{+}-\varphi^{-} it follows that (3.9) holds also for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).
Let \Omega_{0} be as in h3). If \Omega_{0} = \varnothing then \mathbf{u} > 0 a.e. in \Omega (because \mathbf{u} > 0 a.e. in \left\{ a > 0\right\} ) and thus, by (3.9), \mathbf{u} is a solution, in the sense of Definition 1.1, of problem (1.2). Consider now the case when \Omega_{0}\neq\varnothing . We claim that, in this case, \mathbf{u}\in W_{loc}^{2, p}\left(\Omega_{0}\right) for any p\in\left[1, \infty \right). Indeed, let \Omega_{0}^{\prime} be a an arbitrary C^{1, 1} subdomain of \Omega_{0} such that \overline{\Omega_{0}^{\prime}} \subset\Omega_{0}. We have \chi_{\left\{ \mathbf{u} > 0\right\} } a\mathbf{u}^{-\alpha} = 0 on \Omega_{0}, and so, from (3.9), -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} in D^{\prime}\left(\Omega_{0}\right). Also, the restrictions to \Omega_{0} of \mathbf{u} and \theta^{\ast} belong to L^{\infty}\left(\Omega_{0}\right) and so, from the inner elliptic estimates (as stated e.g., in [20], Theorem 8.24), \mathbf{u}\in W^{2, p}\left(\Omega_{0}^{\prime}\right) . Then \mathbf{u}\in W_{loc}^{2, p}\left(\Omega_{0}\right) for any p\in\left[1, \infty\right). Thus, for any p\in\left[1, \infty\right) , \mathbf{u} is a strong solution in W_{loc}^{2, p}\left(\Omega_{0}\right) of -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} in \Omega_{0}.
Taking into account (3.9), in order to complete the proof of the theorem it is enough to see that the set E: = \left\{ \mathbf{u} = 0\right\} \cap\left\{ \theta^{\ast} > 0\right\} has zero measure. Suppose that \left\vert E\right\vert > 0. Since \mathbf{u} > 0 a.e. in \left\{ a > 0\right\}, from h5) it follows that E\subset\Omega_{0}\cup V, for some measurable V\subset\Omega such that \left\vert V\right\vert = 0. Since \left\vert E\right\vert > 0, there exists a subdomain \Omega^{\prime }, with \overline{\Omega^{\prime}}\subset\Omega_{0}, and such that E^{\prime}: = E\cap\Omega^{\prime} has positive measure. Since \mathbf{u} = 0 a.e. in E^{\prime} and \mathbf{u}\in W^{1, p}\left(\Omega^{\prime }\right) we have \nabla\mathbf{u} = 0 a.e. in E^{\prime} (see [20], Lemma 7.7). Thus \frac{\partial\mathbf{u} }{\partial x_{i}} = 0 a.e. in E^{\prime} for each i = 1, 2, ..., n; and since \frac{\partial\mathbf{u}}{\partial x_{i}}\in W^{1, p}\left(\Omega _{0}^{\prime}\right) the same argument gives that also the second order derivatives \frac{\partial^{2}\mathbf{u}}{\partial x_{i}\partial x_{j}} vanish a.e. in E^{\prime}. Then \Delta\mathbf{u} = 0 a.e. in E^{\prime }, which, taking into account that g\left(., \mathbf{u}\right) is nonnegative and \theta^{\ast} > 0 in E^{\prime} , contradicts the fact that -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} a.e. in \Omega_{0}.
Proof of Theorem 1.3. Notice that the condition h4') is stronger than h4) and so Theorem 1.2 gives a weak solution \boldsymbol{u} , in the sense of definition 1.1, of problem (1.2) which satisfies \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}, and so, since a > 0 a.e. in \Omega, by Lemma 2.18, we have \boldsymbol{u} > 0 a.e. in \Omega. Thus \boldsymbol{u} is a weak solution, in the sense of Definition 1.1, of the problem
\left\{ \begin{array} [c]{c} -\Delta\boldsymbol{u} = a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u} \right) \text{ in }\Omega, \\ \boldsymbol{u} = 0\text{ on }\partial\Omega. \end{array} \right. |
Let a_{0}: = \boldsymbol{u}^{-1}g\left(., \boldsymbol{u}\right). Since g\geq0 and \boldsymbol{u}\in L^{\infty}\left(\Omega\right), h4') gives 0\leq a_{0}\in L^{\infty}\left(\Omega\right). Now, in the sense of Definition 1.1, -\Delta\boldsymbol{u} +a_{0}\boldsymbol{u} = a\boldsymbol{u}^{-\alpha} in \Omega, \boldsymbol{u} = 0 on \partial\Omega, and \boldsymbol{u} > 0 a.e. in \Omega; Then, for some \eta > 0 and some measurable set E\subset\Omega with \left\vert E\right\vert > 0, we have \chi_{\left\{ u > 0\right\} }a\boldsymbol{u} ^{-\alpha}\geq\eta\chi_{E} a.e. in \Omega. Let \psi\in\cap_{1\leq q < \infty}W^{2, , q}\left(\Omega\right) \cap W_{0}^{1, , q}\left(\Omega\right) be the solution of the problem -\Delta\psi+a_{0}\psi = \eta\chi_{E} in \Omega, \psi = 0 on \partial\Omega. By the Hopf maximum principle (as stated, e.g., in [34], Theorem 1.1) there exists a positive constant c_{1} such that \psi\geq c_{1}d_{\Omega} in \Omega; and, from (1.7) we have -\Delta\boldsymbol{u} +a_{0}\boldsymbol{u}\geq\eta\chi_{E} in D^{\prime}\left(\Omega\right). Then, by the weak maximum principle (as stated, e.g., in [20], Theorem 8.1), \boldsymbol{u}\geq\psi a.e. in \Omega. Therefore, \boldsymbol{u}\geq c_{1}d_{\Omega} a.e. in \Omega. Thus, for some positive constant c^{\prime}, a\boldsymbol{u}^{-\alpha}\leq c^{\prime }d_{\Omega}^{-\alpha} a.e. in \Omega. Also, g\left(., \boldsymbol{u} \right) \in L^{\infty}\left(\Omega\right) and so, for a larger c^{\prime} if necessary, we have \left\vert a\boldsymbol{u}^{-\alpha }-g\left(., \boldsymbol{u}\right) \right\vert \leq c^{\prime}d_{\Omega }^{-\alpha} a.e. in \Omega. Then, taking into account that \alpha\leq1, the Hardy inequality gives, for any \varphi\in H_{0}^{1}\left(\Omega\right),
\int_{\Omega}\left\vert \left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi\right\vert \leq\int_{\Omega }c^{\prime}d_{\Omega}^{1-\alpha}\left\vert d_{\Omega}^{-1}\varphi\right\vert \leq c^{\prime\prime}\left\Vert \varphi\right\Vert _{H_{0}^{1}\left( \Omega\right) }. |
with c^{\prime\prime} a positive constant independent of \varphi. Thus a\boldsymbol{u}^{-\alpha}-g\left(., \boldsymbol{u}\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Let z be as in Lemma 2.5. Since \boldsymbol{u}\leq u_{\varepsilon_{j}}\leq z, Lemma 2.5 gives that \boldsymbol{u}\leq c^{\prime\prime\prime} d_{\Omega}^{\tau} for some positive constants c^{\prime\prime\prime} and \tau. Therefore, by Lemma 2.13, \boldsymbol{u} is a weak solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.2). Moreover, since
\begin{equation} cd_{\Omega}\leq\boldsymbol{u}\leq c^{\prime\prime\prime}d_{\Omega}^{\tau }\text{ }a.e.\text{ in }\Omega, \end{equation} | (3.10) |
then a\boldsymbol{u}^{-\alpha}-g\left(., \boldsymbol{u}\right) \in L_{loc}^{\infty}\left(\Omega\right), also \boldsymbol{u}\in L^{\infty }\left(\Omega\right) and then, by the inner elliptic estimates, \boldsymbol{u}\in W_{loc}^{2, p}\left(\Omega\right) for any p\in\left[1, \infty\right). Thus \boldsymbol{u}\in C\left(\Omega\right) and from (3.10), u is also continuous at \partial\Omega. Thus u\in C\left(\overline{\Omega}\right).
Proof of Theorem 1.4. Suppose that 0 < \alpha < \frac{1}{2} +\frac{1}{n} when n > 2, that and 0 < \alpha\leq1 when n\leq2. Assume also that g\left(., s\right) = 0 on \Omega^{+} and that h1)-h4) and h5) hold. Let z be as in Remark 2.4, let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right) be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0, and let \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} be as in Theorem 1.2. Let \boldsymbol{u}: = \lim_{j\rightarrow\infty }u_{\varepsilon_{j}} . By Lemma 2.5 we have, u_{\varepsilon _{j}}\leq z in \Omega for all j\in\mathbb{N}, and so \boldsymbol{u}\leq z a.e. in \Omega. Thus, by Remark 2.4, there exist positive constants c and \tau such that \boldsymbol{u}\leq cd_{\Omega }^{\tau} a.e. in \Omega. Let \Omega^{+} as given by h6), and let \zeta:\Omega^{+}\rightarrow\mathbb{R} be as given by Remark 2.19. Thus, by Remark 2.19 ii), there exists a positive constant c^{\prime} such that \zeta\geq c^{\prime }d_{\Omega^{+}} in \Omega^{+}, and by Remark 2.20, u_{\varepsilon_{j}}\geq\zeta in \Omega^{+} for all j\in\mathbb{N} . Then u_{\varepsilon_{j}}\geq c^{\prime}d_{\Omega^{+}} in \Omega^{+} for all j, and so \boldsymbol{u}\geq cd_{\Omega^{+}} a.e. in \Omega^{+}.
Let \varphi\in H_{0}^{1}\left(\Omega\right) and, for k\in\mathbb{N}, let \varphi_{k}:\Omega\rightarrow\mathbb{R} be defined by \varphi_{k}\left(x\right) = \varphi\left(x\right) if \left\vert \varphi\left(x\right) \right\vert \leq k, \varphi_{k}\left(x\right) = k if \varphi\left(x\right) > k and \varphi_{k}\left(x\right) = -k if \varphi\left(x\right) < -k. Thus \varphi_{k}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) and \left\{ \varphi_{k}\right\} _{k\in\mathbb{N}} converges to \varphi in H_{0} ^{1}\left(\Omega\right). By Theorem 1.2, u is a weak solution, in the sense of definition 1.1, of problem (1.2). Then, for all k\in\mathbb{N},
\begin{align} \int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi_{k}\right\rangle & = \int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }\left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi_{k} \\ & = \int_{\Omega}\left( a\boldsymbol{u}^{-\alpha}-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u}\right) \right) \varphi_{k}\\ & = \int_{\Omega}\left( \chi_{\left\{ a \gt 0\right\} }a\boldsymbol{u} ^{-\alpha}-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u} \right) \right) \varphi_{k}. \end{align} | (3.11) |
Note that \chi_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha} -\chi_{\left\{ \boldsymbol{u} > 0\right\} }g\left(., \boldsymbol{u}\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Indeed, by h4), \chi_{\left\{ \boldsymbol{u} > 0\right\} }g\left(., \boldsymbol{u}\right) \in L^{\infty}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}, and, since \boldsymbol{u}\geq cd_{\Omega^{+}} a.e. in \Omega^{+} and a = 0 a.e. in \Omega\setminus\Omega^{+}, we have \chi_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\in L^{\left(2^{\ast}\right) ^{\prime}}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime} when n > 2 (because \ 0 < \alpha < \frac{1}{2}+\frac{1}{n} if n > 2 ), and, in the case n\leq2, \chi_{\left\{ a > 0\right\} }a\boldsymbol{u} ^{-\alpha}\in L^{\frac{1}{\alpha}-\eta}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime} for \eta positive and small enough, (because 0 < \alpha\leq1 if n\leq2 ). Now, we take \lim_{k\rightarrow\infty} in (3.11), to obtain
\begin{array}{l} \int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi\right\rangle & = \int_{\Omega}\left( \chi_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha }-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u}\right) \right) \varphi\\ & = \int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }\left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi, \end{array} |
the last equality because u > 0 a.e. in \left\{ a > 0\right\}.
The author wish to thank an anonymous referee for his/her helpful suggestions and critical comments, which led to a substantial improvement of the paper.
The author declare no conflicts of interest in this paper
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