Citation: Ahsan Iqbal, Ayesha Ayoub, Asad Waqar, Azhar Ul-Haq, Muhammad Zahid, Syed Haider. Voltage stability enhancement in grid-connected microgrid using enhanced dynamic voltage restorer (EDVR)[J]. AIMS Energy, 2021, 9(1): 150-177. doi: 10.3934/energy.2021009
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Consider with the following fourth-order elliptic Navier boundary problem
{Δ2u+cΔu=λa(x)|u|s−2u+f(x,u)inΩ,u=Δu=0on∂Ω, | (1.1) |
where Δ2:=Δ(Δ) denotes the biharmonic operator, Ω⊂RN(N≥4) is a smooth bounded domain, c<λ1 (λ1 is the first eigenvalue of −Δ in H10(Ω)) is a constant, 1<s<2,λ≥0 is a parameter, a∈L∞(Ω),a(x)≥0,a(x)≢0, and f∈C(ˉΩ×R,R). It is well known that some of these fourth order elliptic problems appear in different areas of applied mathematics and physics. In the pioneer paper Lazer and Mckenna [13], they modeled nonlinear oscillations for suspensions bridges. It is worth mentioning that problem (1.1) can describe static deflection of an elastic plate in a fluid, see [21,22]. The static form change of beam or the motion of rigid body can be described by the same problem. Equations of this type have received more and more attentions in recent years. For the case λ=0, we refer the reader to [3,7,11,14,16,17,20,23,27,29,34,35,36,37] and the reference therein. In these papers, existence and multiplicity of solutions have been concerned under some assumptions on the nonlinearity f. Most of them considered the case f(x,u)=b[(u+1)+−1] or f having asymptotically linear growth at infinity or f satisfying the famous Ambrosetti-Rabinowitz condition at infinity. Particularly, in the case λ≠0, that is, the combined nonlinearities for the fourth-order elliptic equations, Wei [33] obtained some existence and multiplicity by using the variational method. However, the author only considered the case that the nonlinearity f is asymptotically linear. When λ=1, Pu et al. [26] did some similar work. There are some latest works for problem (1.1), for example [10,18] and the reference therein. In this paper, we study problem (1.1) from two aspects. One is that we will obtain two multiplicity results when the nonlinearity f is superlinear at infinity and has the standard subcritical polynomial growth but not satisfy the Ambrosetti-Rabinowitz condition, the other is we can establish some existence results of multiple solutions when the nonlinearity f has the exponential growth but still not satisfy the Ambrosetti-Rabinowitz condition. In the first case, the standard methods for the verification of the compactness condition will fail, we will overcome it by using the functional analysis methods, i.e., Hahn-Banach Theorem combined the Resonance Theorem. In the last case, although the original version of the mountain pass theorem of Ambrosetti-Rabinowitz [1] is not directly applied for our purpose. Therefore, we will use a suitable version of mountain pass theorem and some new techniques to finish our goal.
When N>4, there have been substantial lots of works (such as [3,7,11,16,17,26,34,35,36,37]) to study the existence of nontrivial solutions or the existence of sign-changing for problem (1.1). Furthermore, almost all of the works involve the nonlinear term f(x,u) of a standard subcritical polynomial growth, say:
(SCP): There exist positive constants c1 and q∈(1,p∗−1) such that
|f(x,t)|≤c1(1+|t|q)for allt∈Randx∈Ω, |
where p∗=2NN−4 expresses the critical Sobolev exponent. In this case, people can deal with problem (1.1) variationally in the Sobolev space H2(Ω)∩H10(Ω) owing to the some critical point theory, such as, the method of invariant sets of descent flow, mountain pass theorem and symmetric mountain pass theorem. It is worth while to note that since Ambrosetti and Rabinowitz presented the mountain pass theorem in their pioneer paper [1], critical point theory has become one of the main tools on looking for solutions to elliptic equation with variational structure. One of the important condition used in many works is the so-called Ambrosetti-Rabinowitz condition:
(AR) There exist θ>2 and R>0 such that
0<θF(x,t)≤f(x,t)t,forx∈Ωand|t|≥R, |
where F(x,t)=∫t0f(x,s)ds. A simple computation explains that there exist c2,c3>0 such that F(x,t)≥c2|t|θ−c3 for all (x,t)∈ˉΩ×R and f is superlinear at infinity, i.e., limt→∞f(x,t)t=+∞ uniformly in x∈Ω. Thus problem (1.1) is called strict superquadratic if the nonlinearity f satisfies the (AR) condition. Notice that (AR) condition plays an important role in ensuring the boundedness of Palais-Smale sequences. However, there are many nonlinearities which are superlinear at infinity but do not satisfy above (AR) condition such as f(x,t)=tln(1+|t|2)+|sint|t.
In the recent years many authors tried to study problem (1.1) with λ=0 and the standard Laplacian problem where (AR) is not assumed. Instead, they regard the general superquadratic condition:
(WSQC) The following limit holds
lim|t|→+∞F(x,t)t2=+∞,uniformly forx∈Ω |
with additional assumptions (see [3,5,7,11,12,15,17,19,24,26,31,37] and the references therein). In the most of them, there are some kind of monotonicity restrictions on the functions F(x,t) or f(x,t)t, or some convex property for the function tf(x,t)−2F(x,t).
In the case N=4 and c=0, motivated by the Adams inequality, there are a few works devoted to study the existence of nontrivial solutions for problem (1.1) when the nonlinearity f has the exponential growth, for example [15] and the references therein.
Now, we begin to state our main results: Let μ1 be the first eigenvalue of (Δ2−cΔ,H2(Ω)∩H10(Ω)) and suppose that f(x,t) satisfies:
(H1) f(x,t)t≥0,∀(x,t)∈Ω×R;
(H2) limt→0f(x,t)t=f0 uniformly for a.e. x∈Ω, where f0∈[0,+∞);
(H3) limt→∞F(x,t)t2=+∞ uniformly for a.e. x∈Ω, where F(x,t)=∫t0f(x,s)ds.
In the case of N>4, our results are stated as follows:
Theorem 1.1. Assume that f has the standard subcritical polynomial growth on Ω (condition (SCP)) and satisfies (H1)–(H3). If f0<μ1 and a(x)≥a0 (a0 is a positive constant ), then there exists Λ∗>0 such that for λ∈(0,Λ∗), problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.
Theorem 1.2. Assume that f has the standard subcritical polynomial growth on Ω (condition (SCP)) and satisfies (H2) and (H3). If f(x,t) is odd in t.
a) For every λ∈R, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)→∞,k→∞, definition of the functional Iλ will be seen in Section 2.
b) If f0<μ1, for every λ>0, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)<0 and Iλ(uk)→0,k→∞.
Remark. Since our the nonlinear term f(x,u) satisfies more weak condition (H3) compared with the classical condition (AR), our Theorem 1.2 completely contains Theorem 3.20 in [32].
In case of N=4, we have p∗=+∞. So it's necessary to introduce the definition of the subcritical exponential growth and critical exponential growth in this case. By the improved Adams inequality (see [28] and Lemma 2.2 in Section 2) for the fourth-order derivative, namely,
supu∈H2(Ω)∩H10(Ω),‖Δu‖2≤1∫Ωe32π2u2dx≤C|Ω|. |
So, we now define the subcritical exponential growth and critical exponential growth in this case as follows:
(SCE): f satisfies subcritical exponential growth on Ω, i.e., limt→∞|f(x,t)|exp(αt2)=0 uniformly on x∈Ω for all α>0.
(CG): f satisfies critical exponential growth on Ω, i.e., there exists α0>0 such that
limt→∞|f(x,t)|exp(αt2)=0,uniformly onx∈Ω,∀α>α0, |
and
limt→∞|f(x,t)|exp(αt2)=+∞,uniformly onx∈Ω,∀α<α0. |
When N=4 and f satisfies the subcritical exponential growth (SCE), our work is still to consider problem (1.1) where the nonlinearity f satisfies the (WSQC)-condition at infinity. As far as we know, this case is rarely studied by other people for problem (1.1) except for [24]. Hence, our results are new and our methods are technique since we successfully proved the compactness condition by using the Resonance Theorem combined Adams inequality and the truncated technique. In fact, the new idea derives from our work [25]. Our results are as follows:
Theorem 1.3. Assume that f satisfies the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)–(H3). If f0<μ1 and a(x)≥a0 (a0 is a positive constant ), then there exists Λ∗>0 such that for λ∈(0,Λ∗), problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.
Remark. Let F(x,t)=t2e√|t|,∀(x,t)∈Ω×R. Then it satisfies that our conditions (H1)–(H3) but not satisfy the condition (AR). It's worth noting that we do not impose any monotonicity condition on f(x,t)t or some convex property on tf(x,t)−2F(x,t). Hence, our Theorem 1.3 completely extends some results contained in [15,24] when λ=0 in problem (1.1).
Theorem 1.4. Assume that f satisfies the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H2) and (H3). If f0<μ1 and f(x,t) is odd in t.
a) For λ>0 small enough, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)→∞,k→∞.
b) For every λ>0, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)<0 and Iλ(uk)→0,k→∞.
When N=4 and f satisfies the critical exponential growth (CG), the study of problem (1.1) becomes more complicated than in the case of subcritical exponential growth. Similar to the case of the critical polynomial growth in RN(N≥3) for the standard Laplacian studied by Brezis and Nirenberg in their pioneering work [4], our Euler-Lagrange functional does not satisfy the Palais-Smale condition at all level anymore. For the class standard Laplacian problem, the authors [8] used the extremal function sequences related to Moser-Trudinger inequality to complete the verification of compactness of Euler-Lagrange functional at some suitable level. Here, we still adapt the method of choosing the testing functions to study problem (1.1) without (AR) condition. Our result is as follows:
Theorem 1.5. Assume that f has the critical exponential growth on Ω (condition (CG)) and satisfies (H1)–(H3). Furthermore, assume that
(H4) limt→∞f(x,t)exp(−α0t2)t≥β>64α0r40, uniformly in (x,t), where r0 is the inner radius of Ω, i.e., r0:= radius of the largest open ball ⊂Ω. and
(H5) f is in the class (H0), i.e., for any {un} in H2(Ω)∩H10(Ω), if un⇀0 in H2(Ω)∩H10(Ω) and f(x,un)→0 in L1(Ω), then F(x,un)→0 in L1(Ω) (up to a subsequence).
If f0<μ1, then there exists Λ∗>0 such that for λ∈(0,Λ∗), problem (1.1) has at least four nontrivial solutions.
Remark. For standard biharmonic problems with Dirichlet boundary condition, Lam and Lu [15] have recently established the existence of nontrivial nonnegative solutions when the nonlinearity f has the critical exponential growth of order exp(αu2) but without satisfying the Ambrosetti- Rabinowitz condition. However, for problem (1.1) with Navier boundary condition involving critical exponential growth and the concave term, there are few works to study it. Hence our result is new and interesting.
The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge and some important lemmas. In Section 3, we give the proofs for our main results. In Section 4, we give a conclusion.
We let λk (k=1,2,⋅⋅⋅) denote the eigenvalue of −Δ in H10(Ω), then 0<μ1<μ2<⋅⋅⋅<μk<⋅⋅⋅ be the eigenvalues of (Δ2−cΔ,H2(Ω)∩H10(Ω)) and φk(x) be the eigenfunction corresponding to μk. Let Xk denote the eigenspace associated to μk. In fact, μk=λk(λk−c). Throughout this paper, we denote by ‖⋅‖p the Lp(Ω) norm, c<λ1 in Δ2−cΔ and the norm of u in H2(Ω)∩H10(Ω) will be defined by the
‖u‖:=(∫Ω(|Δu|2−c|∇u|2)dx)12. |
We also denote E=H2(Ω)∩H10(Ω).
Definition 2.1. Let (E,||⋅||E) be a real Banach space with its dual space (E∗,||⋅||E∗) and I∈C1(E,R). For c∗∈R, we say that I satisfies the (PS)c∗ condition if for any sequence {xn}⊂E with
I(xn)→c∗,I′(xn)→0 in E∗, |
there is a subsequence {xnk} such that {xnk} converges strongly in E. Also, we say that I satisfy the (C)c∗ condition if for any sequence {xn}⊂E with
I(xn)→c∗, ||I′(xn)||E∗(1+||xn||E)→0, |
there exists subsequence {xnk} such that {xnk} converges strongly in E.
Definition 2.2. We say that u∈E is the solution of problem (1.1) if the identity
∫Ω(ΔuΔφ−c∇u∇φ)dx=λ∫Ωa(x)|u|s−2uφdx+∫Ωf(x,u)φdx |
holds for any φ∈E.
It is obvious that the solutions of problem (1.1) are corresponding with the critical points of the following C1 functional:
Iλ(u)=12‖u‖2−λs∫Ωa(x)|u|sdx−∫ΩF(x,u)dx,u∈E. |
Let u+=max{u,0},u−=min{u,0}.
Now, we concern the following problem
{Δ2u+cΔu=λa(x)|u+|s−2u++f+(x,u)inΩ,u=Δu=0on∂Ω, | (2.1) |
where
f+(x,t)={f(x,t)t≥0,0,t<0. |
Define the corresponding functional I+λ:E→R as follows:
I+λ(u)=12‖u‖2−λs∫Ωa(x)|u+|sdx−∫ΩF+(x,u)dx, |
where F+(x,u)=∫u0f+(x,s)ds. Obviously, the condition (SCP) or (SCE) ((CG)) ensures that I+λ∈C1(E,R). Let u be a critical point of I+λ, which means that u is a weak solution of problem (2.1). Furthermore, since the weak maximum principle (see [34]), it implies that u≥0 in Ω. Thus u is also a solution of problem (1.1) and I+λ(u)=Iλ(u).
Similarly, we define
f−(x,t)={f(x,t)t≤0,0,t>0, |
and
I−λ(u)=12‖u‖2−λs∫Ωa(x)|u−|sdx−∫ΩF−(x,u)dx, |
where F−(x,u)=∫u0f−(x,s)ds. Similarly, we also have I−λ∈C1(E,R) and if v is a critical point of I−λ then it is a solution of problem (1.1) and I−λ(v)=Iλ(v).
Prosition 2.1. ([6,30]). Let E be a real Banach space and suppose that I∈C1(E,R) satisfies the condition
max{I(0),I(u1)}≤α<β≤inf||u||=ρI(u), |
for some α<β, ρ>0 and u1∈E with ||u1||>ρ. Let c∗≥β be characterized by
c∗=infγ∈Γmax0≤t≤1I(γ(t)), |
where Γ={γ∈C([0,1],E),γ(0)=0,γ(1)=u1} is the set of continuous paths joining 0 and u1. Then, there exists a sequence {un}⊂E such that
I(un)→c∗≥β and (1+||un||)||I′(un)||E∗→0 as n→∞. |
Lemma 2.1. ([28]). Let Ω⊂R4 be a bounded domain. Then there exists a constant C>0 such that
supu∈E,‖Δu‖2≤1∫Ωe32π2u2dx<C|Ω|, |
and this inequality is sharp.
Next, we introduce the following a revision of Adams inequality:
Lemma 2.2. Let Ω⊂R4 be a bounded domain. Then there exists a constant C∗>0 such that
supu∈E,‖u‖≤1∫Ωe32π2u2dx<C∗|Ω|, |
and this inequality is also sharp.
Proof. We will give a summarize proof in two different cases. In the case of c≤0 in the definition of ‖.‖, if ‖u‖≤1, we can deduce that ‖Δu‖2≤1 and by using Lemma 2.1 combined with the Proposition 6.1 in [28], the conclusion holds.
In the case of 0<c<λ1 in the definition of ‖.‖, from Lemma 2.1, the proof and remark of Theorem 1 in [2] and the proof of Proposition 6.1 in [28], we still can establish this revised Adams inequality.
Lemma 2.3. Assume (H1) and (H3) hold. If f has the standard subcritical polynomial growth on Ω (condition (SCP)), then I+λ (I−λ) satisfies (C)c∗.
Proof. We only prove the case of I+λ. The arguments for the case of I−λ are similar. Let {un}⊂E be a (C)c∗ sequence such that
I+λ(un)=12||un||2−λs∫Ωa(x)|u+n|sdx−∫ΩF+(x,un)dx=c∗+∘(1), | (2.2) |
(1+||un||)||I+′λ(un)||E∗→0 as n→∞. | (2.3) |
Obviously, (2.3) implies that
⟨I+′λ(un),φ⟩=⟨un,φ⟩−λ∫Ωa(x)|u+n|s−2u+nφdx−∫Ωf+(x,un(x))φdx=∘(1). | (2.4) |
Step 1. We claim that {un} is bounded in E. In fact, assume that
‖un‖→∞, as n→∞. |
Define
vn=un‖un‖. |
Then, ‖vn‖=1, ∀n∈N and then, it is possible to extract a subsequence (denoted also by {vn}) converges weakly to v in E, converges strongly in Lp(Ω)(1≤p<p∗) and converges v a.e. x∈Ω.
Dividing both sides of (2.2) by ‖un‖2, we get
∫ΩF+(x,un)‖un‖2dx→12. | (2.5) |
Set
Ω+={x∈Ω:v(x)>0}. |
By (H3), we imply that
F+(x,un)u2nv2n→∞,x∈Ω+. | (2.6) |
If |Ω+| is positive, since Fatou's lemma, we get
limn→∞∫ΩF+(x,un)‖un‖2dx≥limn→∞∫Ω+F+(x,un)u2nv2ndx=+∞, |
which contradicts with (2.5). Thus, we have v≤0. In fact, we have v=0. Indeed, again using (2.3), we get
(1+‖un‖)|⟨I+′λ(un),v⟩|≤∘(1)‖v‖. |
Thus, we have
∫Ω(ΔunΔv−c∇un∇v)dx≤∫Ω(ΔunΔv−c∇u∇v)dx−λ∫Ωa(x)|u+n|s−2u+nvdx−∫Ωf+(x,un)vdx≤∘(1)‖v‖1+‖un‖, |
by noticing that since v≤0, f+(x,un)v≤0 a.e. x∈Ω, thus −∫Ωf+(x,un)vdx≥0. So we get
∫Ω(ΔvnΔv−c∇vn∇v)dx→0. |
On the other hand, from vn⇀v in E, we have
∫Ω(ΔvnΔv−c∇vn∇v)dx→‖v‖2 |
which implies v=0.
Dividing both sides of (2.4) by ‖un‖, for any φ∈E, then there exists a positive constant M(φ) such that
|∫Ωf+(x,un)‖un‖φdx|≤M(φ),∀n∈N. | (2.7) |
Set
fn(φ)=∫Ωf+(x,un)‖un‖φdx,φ∈E. |
Thus, by (SCP), we know that {fn} is a family bounded linear functionals defined on E. Combing (2.7) with the famous Resonance Theorem, we get that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that
|fn|≤C∗. | (2.8) |
Since E⊂Lp∗p∗−q(Ω), using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lp∗p∗−q(Ω) such that ˆfn is an extension of fn, and
ˆfn(φ)=fn(φ),φ∈E, | (2.9) |
‖ˆfn‖p∗q=|fn|, | (2.10) |
where ‖ˆfn‖p∗q denotes the norm of ˆfn(φ) in Lp∗q(Ω) which is defined on Lp∗p∗−q(Ω).
On the other hand, from the definition of the linear functional on Lp∗p∗−q(Ω), we know that there exists a function Sn(x)∈Lp∗q(Ω) such that
ˆfn(φ)=∫ΩSn(x)φ(x)dx,φ∈Lp∗p∗−q(Ω). | (2.11) |
So, from (2.9) and (2.11), we obtain
∫ΩSn(x)φ(x)dx=∫Ωf+(x,un)‖un‖φdx,φ∈E, |
which implies that
∫Ω(Sn(x)−f+(x,un)‖un‖)φdx=0,φ∈E. |
According to the basic lemma of variational, we can deduce that
Sn(x)=f+(x,un)‖un‖a.e.x∈Ω. |
Thus, by (2.8) and (2.10), we have
‖ˆfn‖p∗q=‖Sn‖p∗q=|fn|<C∗. | (2.12) |
Now, taking φ=vn−v in (2.4), we get
⟨A(vn),vn−v⟩−λ∫Ωa(x)|u+n|s−2u+nvndx−∫Ωf+(x,un)‖un‖vndx→0, | (2.13) |
where A:E→E∗ defined by
⟨A(u),φ⟩=∫ΩΔuΔφdx−c∫Ω∇u∇φdx, u,φ∈E. |
By the H¨older inequality and (2.12), we obtain
∫Ωf+(x,un)‖un‖vndx→0. |
Then from (2.13), we can conclude that
vn→vinE. |
This leads to a contradiction since ‖vn‖=1 and v=0. Thus, {un} is bounded in E.
Step 2. We show that {un} has a convergence subsequence. Without loss of generality, we can suppose that
un⇀u in E,un→u in Lγ(Ω), ∀1≤γ<p∗,un(x)→u(x) a.e. x∈Ω. |
Now, it follows from f satisfies the condition (SCP) that there exist two positive constants c4,c5>0 such that
f+(x,t)≤c4+c5|t|q, ∀(x,t)∈Ω×R, |
then
|∫Ωf+(x,un)(un−u)dx|≤c4∫Ω|un−u|dx+c5∫Ω|un−u||un|qdx≤c4∫Ω|un−u|dx+c5(∫Ω(|un|q)p∗qdx)qp∗(∫Ω|un−u|p∗p∗−qdx)p∗−qp∗. |
Similarly, since un⇀u in E, ∫Ω|un−u|dx→0 and ∫Ω|un−u|p∗p∗−qdx→0.
Thus, from (2.4) and the formula above, we obtain
⟨A(un),un−u⟩→0,asn→∞. |
So, we get ‖un‖→‖u‖. Thus we have un→u in E which implies that I+λ satisfies (C)c∗.
Lemma 2.4. Let φ1>0 be a μ1-eigenfunction with ‖φ1‖=1 and assume that (H1)–(H3) and (SCP) hold. If f0<μ1, then:
(i) For λ>0 small enough, there exist ρ,α>0 such that I±λ(u)≥α for all u∈E with ‖u‖=ρ,
(ii) I±λ(tφ1)→−∞ as t→+∞.
Proof. Since (SCP) and (H1)–(H3), for any ε>0, there exist A=A(ε), M large enough and B=B(ε) such that for all (x,s)∈Ω×R,
F±(x,s)≤12(f0+ϵ)s2+A|s|q, | (2.14) |
F±(x,s)≥M2s2−B. | (2.15) |
Choose ε>0 such that (f0+ε)<μ1. By (2.14), the Poincaré inequality and the Sobolev embedding, we obtain
I±λ(u)≥12‖u‖2−λ‖a‖∞s∫Ω|u|sdx−∫ΩF±(x,u)dx≥12‖u‖2−λ‖a‖∞s∫Ω|u|sdx−f0+ε2‖u‖22−A∫Ω|u|qdx≥12(1−f0+εμ1)‖u‖2−λK‖u‖s−C∗∗‖u‖q≥‖u‖2(12(1−f0+εμ1)−λK‖u‖s−2−C∗∗‖u‖q−2), |
where K,C∗∗ are constant.
Write
h(t)=λKts−2+C∗∗tq−2. |
We can prove that there exists t∗ such that
h(t∗)<12(1−f0+εμ1). |
In fact, letting h′(t)=0, we get
t∗=(λK(2−s)C∗∗(q−2))1q−s. |
According to the knowledge of mathematical analysis, h(t) has a minimum at t=t∗. Denote
ϑ=K(2−s)C∗∗(q−2), ˆϑ=s−2q−s, ˉϑ=q−2q−s, ν=12(1−f0+εμ1). |
Taking t∗ in h(t), we get
h(t∗)<ν,0<λ<Λ∗, |
where Λ∗=(νKϑˆϑ+C∗∗ϑˉϑ)1ˉϑ. So, part (i) holds if we take ρ=t∗.
On the other hand, from (2.15), we get
I+λ(tφ1)≤12(1−Mμ1)t2−tsλs∫Ωa(x)|φ1|sdx+B|Ω|→−∞ as t→+∞. |
Similarly, we have
I−λ(t(−φ1))→−∞, as t→+∞. |
Thus part (ii) holds.
Lemma 2.5. Let φ1>0 be a μ1-eigenfunction with ‖φ1‖=1 and assume that (H1)–(H3) and (SCE)(or (CG)) hold. If f0<μ1, then:
(i) For λ>0 small enough, there exist ρ,α>0 such that I±λ(u)≥α for all u∈E with ‖u‖=ρ,
(ii) I±λ(tφ1)→−∞ as t→+∞.
Proof. From (SCE) (or (CG)) and (H1)-(H3), for any ε>0, there exist A1=A1(ε), M1 large enough, B1=B1(ε), κ1>0 and q1>2 such that for all (x,s)∈Ω×R,
F±(x,s)≤12(f0+ϵ)s2+A1exp(κ1s2)|s|q1, | (2.16) |
F±(x,s)≥M12s2−B1. | (2.17) |
Choose ε>0 such that (f0+ε)<μ1. By (2.16), the Hölder inequality and the Adams inequality (see Lemma 2.2), we obtain
I±λ(u)≥12‖u‖2−λ‖a‖∞s∫Ω|u|sdx−∫ΩF±(x,u)dx≥12‖u‖2−λ‖a‖∞s∫Ω|u|sdx−f0+ε2‖u‖22−A1∫Ωexp(κ1u2)|u|q1dx≥12(1−f0+εμ1)‖u‖2−λK‖u‖s−A1(∫Ωexp(κ1r1‖u‖2(|u|‖u‖)2)dx)1r1(∫Ω|u|r′1qdx)1r′1≥12(1−f0+εμ1)‖u‖2−λK‖u‖s−ˆC∗∗‖u‖q1, |
where r1>1 sufficiently close to 1, ‖u‖≤σ and κ1r1σ2<32π2. Remained proof is completely similar to the proof of part (ⅰ) of Lemma 2.4, we omit it here. So, part (ⅰ) holds if we take ‖u‖=ρ>0 small enough.
On the other hand, from (2.17), we get
I+λ(tφ1)≤12(1−M1μ1)t2−tsλs∫Ωa(x)|φ1|sdx+B1|Ω|→−∞ as t→+∞. |
Similarly, we have
I−λ(t(−φ1))→−∞, as t→+∞. |
Thus part (ii) holds.
Lemma 2.6. Assume (H1) and (H3) hold. If f has the subcritical exponential growth on Ω (condition (SCE)), then I+λ (I−λ) satisfies (C)c∗.
Proof. We only prove the case of I+λ. The arguments for the case of I−λ are similar. Let {un}⊂E be a (C)c∗ sequence such that the formulas (2.2)–(2.4) in Lemma 2.3 hold.
Now, according to the previous section of Step 1 of the proof of Lemma 2.3, we also obtain that the formula (2.7) holds. Set
fn(φ)=∫Ωf+(x,un)‖un‖φdx,φ∈E. |
Then from for any u∈E, eαu2∈L1(Ω) for all α>0, we can draw a conclusion that {fn} is a family bounded linear functionals defined on E. Using (2.7) and the famous Resonance Theorem, we know that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that the formula (2.8) (see the proof of Lemma 2.3) holds.
Since E⊂Lq0(Ω) for some q0>1, using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lq0(Ω) such that ˆfn is an extension of fn, and
ˆfn(φ)=fn(φ),φ∈E, | (2.18) |
‖ˆfn‖q∗0=|fn|, | (2.19) |
where ‖ˆfn‖q∗0 is the norm of ˆfn(φ) in Lq∗0(Ω) which is defined on Lq0(Ω) and q∗0 is the dual number of q0.
By the definition of the linear functional on Lq0(Ω), we know that there is a function Sn(x)∈Lq∗0(Ω) such that
ˆfn(φ)=∫ΩSn(x)φ(x)dx,φ∈Lq0(Ω). | (2.20) |
Similarly to the last section of the Step 1 of the proof of Lemma 2.3, we can prove that (C)c∗ sequence {un} is bounded in E. Next, we show that {un} has a convergence subsequence. Without loss of generality, assume that
‖un‖≤β∗,un⇀u in E,un→u in Lγ(Ω), ∀γ≥1,un(x)→u(x) a.e. x∈Ω. |
Since f has the subcritical exponential growth (SCE) on Ω, we can find a constant Cβ∗>0 such that
|f+(x,t)|≤Cβ∗exp(32π2k(β∗)2t2), ∀(x,t)∈Ω×R. |
Thus, from the revised Adams inequality (see Lemma 2.2),
|∫Ωf+(x,un)(un−u)dx|≤Cβ∗(∫Ωexp(32π2(β∗)2u2n)dx)1k|un−u|k′≤C∗∗|un−u|k′→0, |
where k>1 and k′ is the dual number of k. Similar to the last proof of Lemma 2.3, we have un→u in E which means that I+λ satisfies (C)c∗.
Lemma 2.7. Assume (H3) holds. If f has the standard subcritical polynomial growth on Ω (condition (SCP)), then Iλ satisfies (PS)c∗.
Proof. Let {un}⊂E be a (PS)c∗ sequence such that
‖un‖22−λs∫Ωa(x)|un|sdx−∫ΩF(x,un)dx→c∗, | (2.21) |
∫ΩΔunΔφdx−c∫Ω∇un∇φdx−λ∫Ωa(x)|un|s−2unφdx−∫Ωf(x,un)φdx=∘(1)‖φ‖, φ∈E. | (2.22) |
Step 1. To prove that {un} has a convergence subsequence, we first need to prove that it is a bounded sequence. To do this, argue by contradiction assuming that for a subsequence, which is still denoted by {un}, we have
‖un‖→∞. |
Without loss of generality, assume that ‖un‖≥1 for all n∈N and let
vn=un‖un‖. |
Clearly, ‖vn‖=1, ∀n∈N and then, it is possible to extract a subsequence (denoted also by {vn}) converges weakly to v in E, converges strongly in Lp(Ω)(1≤p<p∗) and converges v a.e. x∈Ω.
Dividing both sides of (2.21) by ‖un‖2, we obtain
∫ΩF(x,un)‖un‖2dx→12. | (2.23) |
Set
Ω0={x∈Ω:v(x)≠0}. |
By (H3), we get that
F(x,un)u2nv2n→∞,x∈Ω0. | (2.24) |
If |Ω0| is positive, from Fatou's lemma, we obtain
limn→∞∫ΩF(x,un)‖un‖2dx≥limn→∞∫Ω0F(x,un)u2nv2ndx=+∞, |
which contradicts with (2.23).
Dividing both sides of (2.22) by ‖un‖, for any φ∈E, then there exists a positive constant M(φ) such that
|∫Ωf(x,un)‖un‖φdx|≤M(φ),∀n∈N. | (2.25) |
Set
fn(φ)=∫Ωf(x,un)‖un‖φdx,φ∈E. |
Thus, by (SCP), we know that {fn} is a family bounded linear functionals defined on E. By (2.25) and the famous Resonance Theorem, we get that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that
|fn|≤˜C∗. | (2.26) |
Since E⊂Lp∗p∗−q(Ω), using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lp∗p∗−q(Ω) such that ˆfn is an extension of {\bf f}_n , and
\begin{equation} \hat{{\bf f}}_n(\varphi) = {\bf f}_n(\varphi), \; \varphi\in E, \end{equation} | (2.27) |
\begin{equation} \|\hat{{\bf f}}_n\|_{\frac{p^*}{q}} = |{\bf f}_n|, \end{equation} | (2.28) |
where \|\hat{{\bf f}}_n\|_{\frac{p^*}{q}} denotes the norm of \hat{{\bf f}}_n(\varphi) in L^{\frac{p^*}{q}}(\Omega) which is defined on L^{\frac{p^*}{p^*-q}}(\Omega) .
Remained proof is completely similar to the last proof of Lemma 2.3, we omit it here.
Lemma 2.8. Assume (H_{3}) holds. If f has the subcritical exponential growth on \Omega (condition \mathrm{(SCE)} ), then I_\lambda satisfies \mathrm{(PS)}_{c^*} .
Proof. Combining the previous section of the proof of Lemma 2.7 with slightly modifying the last section of the proof of Lemma 2.6, we can prove it. So we omit it here.
To prove the next Lemma, we firstly introduce a sequence of nonnegative functions as follows. Let \Phi(t)\in C^\infty[0, 1] such that
\Phi(0) = \Phi'(0) = 0, |
\Phi(1) = \Phi'(1) = 0. |
We let
H(t) = \begin{cases} \frac{1}{n}\Phi(nt), &\quad \text{if}\; t\leq \frac{1}{n}, \\ t, &\quad \text{if}\; \frac{1}{n} < t < 1-\frac{1}{n}, \\ 1-\frac{1}{n}\Phi(n(1-t)), &\quad \text{if}\; 1-\frac{1}{n}\leq t\leq1, \\ 1, &\quad \text{if}\; 1\leq t, \end{cases} |
and \psi_n(r) = H((ln n)^{-1}ln \frac{1}{r}). Notice that \psi_n(x)\in E , B the unit ball in \mathbb{R}^N , \psi_n(x) = 1 for |x|\leq \frac{1}{n} and, as it was proved in [2],
\|\Delta \psi_n\|_2 = 2\sqrt{2}\pi(ln n)^{-\frac{1}{2}}A_n = \|\psi_n\|+\circ(1), \; \text{as}\; n\rightarrow \infty. |
where 0\leq \lim\limits_{n\rightarrow \infty} A_n\leq 1. Thus, we take x_0\in \Omega and r_0 > 0 such that B(x_0, r)\subset \Omega, denote
\Psi_n(x) = \begin{cases} \frac{\psi_n(|x-x_0|)}{\|\psi_n\|}, &\quad \text{if}\; x\in B(x_0, r_0), \\ 0, &\quad \text{if}\; x\in \Omega\backslash B(x_0, r_0).\\ \end{cases} |
Lemma 2.9. Assume (H_{1}) and (H_{4}) hold. If f has the critical exponential growth on \Omega (condition \mathrm{(CG)} ), then there exists n such that
\max\{I_\lambda^{\pm}(\pm t\Psi_n):t\geq 0\} < \frac{16\pi^2}{\alpha_0}. |
Proof. We only prove the case of I_\lambda^+ . The arguments for the case of I_\lambda^- are similar. Assume by contradiction that this is not the case. So, for all n , this maximum is larger or equal to \frac{16\pi^2}{\alpha_0}. Let t_n > 0 be such that
\begin{equation} \mathcal{I_\lambda^+}(t_n\Psi_n)\geq\frac{16\pi^2}{\alpha_0}. \end{equation} | (2.29) |
From (H_1) and (2.29), we conclude that
\begin{equation} t_n^{2}\geq \frac{32\pi^2}{\alpha_0}. \end{equation} | (2.30) |
Also at t = t_n , we have
t_n-t_n^{s-1}\lambda\int_\Omega a(x)|\Psi_n|^{s}dx-\int_\Omega f(x, t_n\Psi_n)\Psi_ndx = 0, |
which implies that
\begin{equation} t_n^{2}\geq t_n^{s}\lambda\int_\Omega a(x)|\Psi_n|^{s}dx+\int_{B(x_0, r_0)} f(x, t_n\Psi_n)t_n\Psi_ndx. \end{equation} | (2.31) |
Since (H_4) , for given \epsilon > 0 there exists R_\epsilon > 0 such that
tf(x, t)\geq (\beta-\epsilon)\exp\left(\alpha_0t^{2}\right), \ t\geq R_\epsilon. |
So by (2.31), we deduce that, for large n
\begin{equation} t_n^{2}\geq t_n^{s}\lambda\int_\Omega a(x)|\Psi_n|^{s}dx+ (\beta-\epsilon)\frac{\pi^2}{2}r_0^4\exp\left[ \left((\frac{t_n}{ A_n})^2\frac{\alpha_0}{32\pi^2}-1\right)4ln n\right]. \end{equation} | (2.32) |
By (2.30), the inequality above is true if, and only if
\begin{equation} \lim\limits_{n\rightarrow \infty}A_n = 1 \ \ \text{and}\ \ t_n\rightarrow \left( \frac{32\pi^2}{\alpha_0}\right)^{\frac{1}{2}}. \end{equation} | (2.33) |
Set
A_n^* = \{x\in B(x_0, r_0): t_n \Psi_n(x)\geq R_{\epsilon}\}, \; \; B_n = B(x_0, r_0)\setminus A_n^*, |
and break the integral in (2.31) into a sum of integrals over A_n^* and B_n . By simple computation, we have
\begin{equation} \left[\frac{32\pi^2}{\alpha_0}\right]\geq (\beta-\epsilon) \lim\limits_{n\rightarrow \infty} \int_{B(x_0, r_0)}\exp\left[\alpha_0t_n^2|\Psi_n(x)|^2\right]dx-(\beta-\epsilon)r_0^4\frac{\pi^2}{2}. \end{equation} | (2.34) |
The last integral in (2.34), denote I_n is evaluated as follows:
I_n\geq (\beta-\epsilon)r_0^4\pi^2. |
Thus, finally from (2.34) we get
\left[\frac{32\pi^2}{\alpha_0}\right]\geq(\beta-\epsilon)r_0^4\frac{\pi^2}{2}, |
which means \beta\leq \frac{64}{\alpha_0r_0^4}. This results in a contradiction with (H_4) .
To conclude this section we state the Fountain Theorem of Bartsch [32].
Define
\begin{equation} Y_k = \oplus_{j = 1}^kX_j, \ \ Z_k = \overline{\oplus_{j\geq k}X_j}. \end{equation} | (2.35) |
Lemma 2.10. (Dual Fountain Theorem). Assume that I_\lambda\in C^1(\mathbb{E}, \mathbb{R}) satisfies the \mathrm{(PS)_c^*} condition (see [32]), I_\lambda(-u) = I_\lambda(u) . If for almost every k\in {\bf N}, there exist \rho_k > r_k > 0 such that
\mathrm{(i)} a_k: = \inf\limits_{u\in Z_k, \|u\| = \rho_k}I_\lambda(u)\geq 0,
\mathrm{(ii)} b_k: = \max\limits_{u\in Y_k, \|u\| = r_k}I_\lambda(u) < 0,
\mathrm{(iii)} b_k = \inf\limits_{u\in Z_k, \|u\| = \rho_k}I_\lambda(u)\rightarrow 0, \ \mathit{\text{as}}\ k\rightarrow \infty,
then I_\lambda has a sequence of negative critical values converging 0 .
Proof of Theorem 1.1. For I_\lambda^{\pm}, we first demonstrate that the existence of local minimum v_{\pm} with I_\lambda^{\pm}(v_{\pm}) < 0 . We only prove the case of I_\lambda^+. The arguments for the case of I_\lambda^- are similar.
For \rho determined in Lemma 2.4, we write
\bar{B}(\rho) = \{u\in E, \ \|u\|\leq \rho\}, \ \ \partial B(\rho) = \{u\in E, \ \|u\| = \rho\}. |
Then \bar{B}(\rho) is a complete metric space with the distance
\text{dist}(u, v) = \|u-v\|, \quad \forall u, v\in \bar{B}(\rho). |
From Lemma 2.4, we have for 0 < \lambda < \Lambda^*,
I_\lambda^+(u)|_{\partial B(\rho)}\geq \alpha > 0. |
Furthermore, we know that I_\lambda^+\in C^1(\bar{B}(\rho), \mathbb{R}), hence I_\lambda^+ is lower semi-continuous and bounded from below on \bar{B}(\rho) . Set
c_1^* = \inf\{ I_\lambda^+(u), u\in \bar{B}(\rho)\}. |
Taking \tilde{\phi}\in C_0^\infty (\Omega) with \tilde{\phi} > 0, and for t > 0 , we get
\begin{eqnarray*} I_\lambda^+(t\tilde{\phi})& = &\frac{t^2}{2}\|\tilde{\phi}\|^2-\frac{\lambda t^s}{s}\int_\Omega a(x) |\tilde{\phi}|^s dx -\int_\Omega{F^+(x, t\tilde{\phi})}dx\\ &\leq& \frac{t^2}{2}\|\tilde{\phi}\|^2-\frac{\lambda t^s}{s}\int_\Omega a(x) |\tilde{\phi}|^s dx\\ & < & 0, \end{eqnarray*} |
for all t > 0 small enough. Hence, c_1^* < 0 .
Since Ekeland's variational principle and Lemma 2.4, for any m > 1 , there exists u_m with \|u_m\| < \rho such that
I_\lambda^+(u_m)\rightarrow c_1^*, \quad I_\lambda^{+'}(u_m)\rightarrow 0. |
Hence, there exists a subsequence still denoted by \{u_m\} such that
u_m\rightarrow v_+, \quad I_\lambda^{+'}(v_+) = 0. |
Thus v_+ is a weak solution of problem (1.1) and I_\lambda^+(v_+) < 0 . In addition, from the maximum principle, we know v_+ > 0 . By a similar way, we obtain a negative solution v_- with I_\lambda^-(v_-) < 0 .
On the other hand, from Lemmas 2.3 and 2.4, the functional I_\lambda^+ has a mountain pass-type critical point u_+ with I_\lambda^+(u_+) > 0. Again using the maximum principle, we have u_+ > 0 . Hence, u_+ is a positive weak solution of problem (1.1). Similarly, we also obtain a negative mountain pass-type critical point u_- for the functional I_\lambda^- . Thus, we have proved that problem (1.1) has four different nontrivial solutions. Next, our method to obtain the fifth solution follows the idea developed in [33] for problem (1.1). We can assume that v_+ and v_- are isolated local minima of I_\lambda . Let us denote by b_\lambda the mountain pass critical level of I_\lambda with base points v_+, v_-:
b_\lambda = \inf\limits_{\gamma\in \Gamma}\max\limits_{0\leq t\leq1}I_\lambda(\gamma(t)), |
where \Gamma = \{\gamma\in C([0, 1], E), \gamma(0) = v_+, \gamma(1) = v_-\} . We will show that b_\lambda < 0 if \lambda is small enough. To this end, we regard
I_\lambda(tv_{\pm}) = \frac{t^2}{2}\|v_{\pm}\|^2-\frac{\lambda t^s}{s}\int_\Omega a(x)|v_{\pm}|^sdx-\int_\Omega F(x, tv_{\pm})dx. |
We claim that there exists \delta > 0 such that
\begin{equation} I_\lambda(tv_{\pm}) < 0, \ \forall t\in (0, 1), \ \forall \lambda\in (0, \delta). \end{equation} | (3.1) |
If not, we have t_0\in (0, 1) such that I_\lambda(t_0v_{\pm})\geq 0 for \lambda small enough. Similarly, we also have I_\lambda(tv_{\pm}) < 0 for t > 0 small enough. Let \rho_0 = t_0\|v_{\pm}\| and \check{c}_*^{\pm} = \inf \{I_\lambda^{\pm}(u), u\in \bar{B}(\rho_0)\}. Since previous arguments, we obtain a solution v_{\pm}^* such that I_\lambda(v_{\pm}^*) < 0, a contradiction. Hence, (3.1) holds.
Now, let us consider the 2 -dimensional plane \Pi_2 containing the straightlines tv_- and tv_+ , and take v\in \Pi_2 with \|v\| = \epsilon. Note that for such v one has \|v\|_s = c_s\epsilon. Then we get
I_\lambda(v)\leq \frac{\epsilon^2}{2}-\frac{\lambda}{s}c_s^sh_0\epsilon^s. |
Thus, for small \epsilon ,
\begin{equation} I_\lambda(v) < 0. \end{equation} | (3.2) |
Consider the path \bar{\gamma} obtained gluing together the segments \{tv_-:\epsilon \|v_-\|^{-1}\leq t\leq 1\}, \{tv_+:\epsilon \|v_+\|^{-1}\leq t\leq 1\} and the arc \{v\in \Pi_2: \|v\| = \epsilon\} . by (3.1)and (3.2), we get
b_\lambda\leq \max\limits_{v\in \bar{\gamma}}I_\lambda(v) < 0, |
which verifies the claim. Since the (PS) condition holds because of Lemma 2.3, the level \{I_\lambda(v) = b_\lambda\} carries a critical point v_3 of I_\lambda , and v_3 is different from v_{\pm} .
Proof of Theorem 1.2. We first use the symmetric mountain pass theorem to prove the case of a) . It follows from our assumptions that the functional I_\lambda is even. Since the condition (SCP), we know that (I_1') of Theorem 9.12 in [30] holds. Furthermore, by condition (H_3) , we easily verify that (I_2') of Theorem 9.12 also holds. Hence, by Lemma 2.7, our theorem is proved.
Next we use the dual fountain theorem (Lemma 2.10) to prove the case of b) . Since Lemma 2.7, we know that the functional I_\lambda satisfies \mathrm{(PS)_c^*} condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.
First, we verify (ⅰ) of Lemma 2.10. Define
\beta_k: = \sup\limits_{u\in Z_k, \|u\| = 1}\|u\|_s. |
From the conditions (SCP) and (H_2) , we get, for u\in Z_k, \|u\|\leq R,
\begin{align} I_\lambda(u)&\geq \frac{\|u\|^2}{2}-\lambda \beta_k^s\frac{\|u\|^s}{s}-\frac{f_0+\epsilon}{2}\|u\|_2^2-c_6\|u\|^q\\ &\geq \frac{1}{4}(1-\frac{f_0+\epsilon}{\mu_1})\|u\|^2-\lambda \beta_k^s\frac{\|u\|^s}{s}. \end{align} | (3.3) |
Here, R is a positive constant and \epsilon > 0 small enough. We take \rho_k = (4\mu_1\lambda \beta_k^s/[(\mu_1-f_0-\epsilon)s])^{\frac{1}{2-s}}. Since \beta_k\rightarrow 0, k\rightarrow \infty, it follows that \rho_k\rightarrow 0, k\rightarrow \infty. There exists k_0 such that \rho_k\leq R when k\geq k_0 . Thus, for k\geq k_0, u\in Z_k and \|u\| = \rho_k, we have I_\lambda(u)\geq 0 and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.
Proof of Theorem 1.3. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum v_{\pm} with I_\lambda^{\pm}(v_{\pm}) < 0 . In addition, by Lemmas 2.5 and 2.6, for I_\lambda^{\pm} , we obtain two mountain pass type critical points u_+ and u_- with positive energy. Similar to the last section of the proof of Theorem 1.1, we can also get another solution u_3 , which is different from v_{\pm} and u_{\pm} . Thus, this proof is completed.
Proof of Theorem 1.4. We first use the symmetric mountain pass theorem to prove the case of a) . It follows from our assumptions that the functional I_\lambda is even. Since the condition (SCE), we know that (I_1') of Theorem 9.12 in [30] holds. In fact, similar to the proof of (ⅰ) of Lemma 2.5, we can conclude it. Furthermore, by condition (H_3) , we easily verify that (I_2') of Theorem 9.12 also holds. Hence, by Lemma 2.8, our theorem is proved.
Next we use the dual fountain theorem (Lemma 2.10) to prove the case of b) . Since Lemma 2.8, we know that the functional I_\lambda satisfies \mathrm{(PS)_c^*} condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.
First, we verify (ⅰ) of Lemma 2.10. Define
\beta_k: = \sup\limits_{u\in Z_k, \|u\| = 1}\|u\|_s. |
From the conditions (SCE), (H_2) and Lemma 2.2, we get, for u\in Z_k, \|u\|\leq R,
\begin{align} I_\lambda(u)&\geq \frac{\|u\|^2}{2}-\lambda \beta_k^s\frac{\|u\|^s}{s}-\frac{f_0+\epsilon}{2}\|u\|_2^2-c_7\|u\|^q\\ &\geq \frac{1}{4}(1-\frac{f_0+\epsilon}{\mu_1})\|u\|^2-\lambda \beta_k^s\frac{\|u\|^s}{s}. \end{align} | (3.4) |
Here, R is a positive constant small enough and \epsilon > 0 small enough. We take \rho_k = (4\mu_1\lambda \beta_k^s/[(\mu_1-f_0-\epsilon)s])^{\frac{1}{2-s}}. Since \beta_k\rightarrow 0, k\rightarrow \infty, it follows that \rho_k\rightarrow 0, k\rightarrow \infty. There exists k_0 such that \rho_k\leq R when k\geq k_0 . Thus, for k\geq k_0, u\in Z_k and \|u\| = \rho_k, we have I_\lambda(u)\geq 0 and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.
Proof of Theorem 1.5. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum v_{\pm} with I_\lambda^{\pm}(v_{\pm}) < 0 . Now, we show that I_\lambda^+ has a positive mountain pass type critical point. Since Lemmas 2.5 and 2.9, then there exists a \mathrm{(C)_{c_M}} sequence \{u_n\} at the level 0 < c_M\leq \frac{16\pi^2}{\alpha_0} . Similar to previous section of the proof of Lemma 2.6, we can prove that \mathrm{(C)_{c_M}} sequence \{u_n\} is bounded in E . Without loss of generality, we can suppose that
u_n\rightharpoonup u_+\; \; \text{in}\; E. |
Following the proof of Lemma 4 in [9], we can imply that u_+ is weak of problem (1.1). So the theorem is proved if u_+ is not trivial. However, we can get this due to our technical assumption (H_5) . Indeed, assume u_+ = 0 , similarly as in [9], we obtain f^+(x, u_n)\rightarrow 0 in L^1(\Omega) . Since (H_5), F^+(x, u_n)\rightarrow 0 in L^1(\Omega) and we get
\lim\limits_{n\rightarrow \infty}\|u_n\|^{2} = 2c_M < \frac{32\pi^2}{\alpha_0} , |
and again following the proof in [9], we get a contradiction.
We claim that v_+ and u_+ are distinct. Since the previous proof, we know that there exist sequence \{u_n\} and \{v_n\} in E such that
\begin{equation} u_n\rightarrow v_+, \ I_\lambda^+(u_n)\rightarrow c_*^+ < 0, \ \langle I_\lambda^{+'}(u_n), u_n\rangle\rightarrow 0, \end{equation} | (3.5) |
and
\begin{equation} v_n\rightharpoonup u_+, \ I_\lambda^+(v_n)\rightarrow c_M > 0, \ \langle I_\lambda^{+'}(v_n), v_n\rangle\rightarrow 0. \end{equation} | (3.6) |
Now, argue by contradiction that v_+ = u_+. Since we also have v_n\rightharpoonup v_+ in E , up to subsequence, \lim\limits_{n\rightarrow \infty}\|v_n\|\geq \|v_+\| > 0. Setting
w_n = \frac{v_n}{\|v_n\|}, \ \ w_0 = \frac{v_+}{\lim\limits_{n\rightarrow \infty}\|v_n\|}, |
we know that \|w_n\| = 1 and w_n\rightharpoonup w_0 in E .
Now, we consider two possibilities:
\mathrm{ (i)} \ \|w_0\| = 1, \quad \mathrm{(ii)}\ \|w_0\| < 1. |
If (ⅰ) happens, we have v_n\rightarrow v_+ in E , so that I_\lambda^+(v_n)\rightarrow I_\lambda^+(v_+) = c_*^+. This is a contradiction with (3.5) and (3.6).
Now, suppose that (ⅱ) happens. We claim that there exists \delta > 0 such that
\begin{equation} h\alpha_0\|v_n\|^2\leq \frac{32\pi^2}{1-\|w_0\|^2}-\delta \end{equation} | (3.7) |
for n large enough. In fact, by the proof of v_+ and Lemma 2.9, we get
\begin{equation} 0 < c_M < c_*^++ \frac{16\pi^2}{\alpha_0}. \end{equation} | (3.8) |
Thus, we can choose h > 1 sufficiently close to 1 and \delta > 0 such that
h\alpha_0\|v_n\|^2\leq \frac{16\pi^2}{c_M-I_\lambda^+(v_+)}\|v_n\|^2-\delta. |
Since v_n\rightharpoonup v_+, by condition (H_5) , up to a subsequence, we conclude that
\begin{equation} \frac{1}{2}\|v_n\|^2 = c_M+\frac{\lambda}{s}\int_\Omega a(x) v_+^s dx +\int_\Omega F^+(x, v_+)dx+\circ (1). \end{equation} | (3.9) |
Thus, for n sufficiently large we get
\begin{equation} h\alpha_0\|v_n\|^2\leq 32\pi^2\frac{ c_M+\frac{\lambda}{s}\int_\Omega a(x) v_+^s dx +\int_\Omega F^+(x, v_+)dx+\circ (1)}{c_M-I_\lambda^+(v_+)}-\delta. \end{equation} | (3.10) |
Thus, from (3.9) and the definition of w_0 , (3.10) implies (3.7) for n large enough.
Now, taking \tilde{h} = (h+\epsilon)\alpha_0\|v_n\|^2, it follows from (3.7) and a revised Adams inequality (see [28]), we have
\begin{equation} \int_\Omega \exp((h+\epsilon)\alpha_0\|v_n\|^2|w_n|^2dx\leq C \end{equation} | (3.11) |
for \epsilon > 0 small enough. Thus, from our assumptions and the Hölder inequality we get v_n\rightarrow v_+ and this is absurd.
Similarly, we can find a negative mountain pass type critical point u_- which is different that v_- . Thus, the proof is completed.
In this research, we mainly studied the existence and multiplicity of nontrivial solutions for the fourth-order elliptic Navier boundary problems with exponential growth. Our method is based on the variational methods, Resonance Theorem together with a revised Adams inequality.
The authors would like to thank the referees for valuable comments and suggestions in improving this article. This research is supported by the NSFC (Nos. 11661070, 11764035 and 12161077), the NSF of Gansu Province (No. 22JR11RE193) and the Nonlinear mathematical physics Equation Innovation Team (No. TDJ2022-03).
There is no conflict of interest.
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