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In this paper, we consider the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations
−div(gp(u)|∇u|p−2∇u)+gp−1(u)g′(u)|∇u|p+V(x)|u|p−2u=h(x,u),x∈RN, | (1.1) |
where N≥3, 1<p<N, g:R→R+ is an even differential function with g′(t)≥0 for all t≥0 and g(0)=1, V:RN→R and h:RN×R→R are continuous functions.
Notice that if we take p=2 and g2(u)=1+[(l(u2))′]22, where l is a given real function, then (1.1) turns into
−△u+V(x)u−△(l(u2))l′(u2)u=h(x,u),x∈RN. | (1.2) |
Equation (1.2) is related to the existence of solitary wave solutions for quasilinear Schrödinger equations
izt=−△z+W(x)z−h(x,z)−△(l(|z|2))l′(|z|2)z,x∈RN, | (1.3) |
where z:R×RN→C, W:RN→R is a given potential, h:RN×R→R and l:R→R are suitable functions. The form of (1.3) has many applications in physics. For instance, the case l(s)=s was used to model the time evolution of the condensate wave function in superfluid film [16,17]. In the case of l(s)=√1+s, Eq (1.3) was used as a model of the self-channeling of a high-power ultrashort laser in matter [3,29]. For more details on the physical background, we can refer to [2,4] and references therein.
Putting z(t,x)=exp(−iEt)u(x) in (1.3), where E∈R and u is a real function, we obtain a corresponding equation of elliptic type (1.2).
If we take p=2, then (1.1) turns into
−div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈RN. | (1.4) |
In recent years, many researchers have studied (1.4) under various hypotheses on the potential and nonlinearity, for example [6,7,8,9,24,28,38].
If we set p=2,g2(u)=1+2u2, i.e., l(s)=s, we can get the superfluid film equation in plasma physics
−△u+V(x)u−△(u2)u=h(x,u),x∈RN. | (1.5) |
Equation (1.5) has been extensively studied, see [5,20,27,30,31].
If we take p=2,g2(u)=1+u22(1+u2), i.e., l(s)=√1+s, (1.2) derives the following equation
−△u+V(x)u−[△(1+u2)12]u2(1+u2)12=h(x,u),x∈RN, | (1.6) |
which models of the self-channeling of a high-power ultrashort laser in matter. For (1.6), there are many papers studying the existence of solutions, see [10,13,33] and references therein.
Furthermore, if we set gp(u)=1+2p−1up in (1.1), then we get the following quasilinear elliptic equations
−△p(u)+V(x)|u|p−2u−△p(u2)u=h(x,u),u∈W1,p(RN), | (1.7) |
where △p=div(|∇u|p−2∇u) is the p-Laplacian operator with 1<p≤N. In [11], where h(x,u)=h(u), the authors constructed infinitely many nodal solutions for (1.7) under suitable assumptions.
We point out that the related semilinear elliptic equations with the asymptotically periodic condition have been extensively researched, see [12,19,22,23,25,34,35,36,37] and their references.
Especially, in [12], Lins and Silva considered the following asymptotically periodic p-laplacian equations
{−△pu+V(x)up−1=K(x)up∗−1+g(x,u),x∈RN,u∈W1,p(RN),u≥0. | (1.8) |
Assume that the potential V satisfies
(V0) there exist a constant a0>0 and a function ˉV∈C(RN,R), 1-periodic in xi, 1≤i≤N, such that ˉV(x)≥V(x)≥a0>0 and ˉV−V∈K, where
K:={h∈C(RN)⋂L∞(RN):∀ε>0,meas{x∈RN:|h(x)|≥ε}<+∞}, |
and the asymptotically periodic of g at infinity was assumed to the following condition
(g0) there exist a constant p<q1<p∗ and functions h∈K, g0∈C(RN,R), 1-periodic in xi, 1≤i≤N, such that
|g(x,s)−g0(x,s)|≤h(x)|s|q1−1,forall(x,s)∈RN×[0,+∞). |
For the other conditions on g, please see [12].
In recent paper [36], Xue and Tang studied the following quasilinear Schrodinger equation
−△u+V(x)u−△(u2)u=K(x)|u|22∗−2u+g(x,u),x∈RN, | (1.9) |
they proposed reformative conditions, which unify the asymptotic process of the potential and nonlinear term at infinity, see the below condition (V1) and (i) of (f5). It is easy to see that this conditions contains more elements than those in [12]. To the best of our knowledge, there is no work concerning with the unified asymptotic process of the potential and nonlinear term at infinity for general quasilinear elliptic equations.
Motivated by above papers, under the asymptotically periodic conditions, we establish the existence of a nontrivial positive solution for Eq (1.1) with critical nonlinearity. We assume h(x,u)=f(x,u)+K(x)g(u)|G(u)|p∗−2G(u). Equation (1.1) can be rewritten in the following form:
−div(gp(u)|∇u|p−2∇u)+gp−1(u)g′(u)|∇u|p+V(x)|u|p−2u=f(x,u)+K(x)g(u)|G(u)|p∗−2G(u),x∈RN, | (1.10) |
where p∗=NpN−p for N≥3, G(t)=∫t0g(τ)dτ and f:RN×R→R is continuous function.
We observe that the energy functional associate with (1.10) is given by
I(u)=1p∫RNgp(u)|∇u|pdx+1p∫RNV(x)|u|pdx−∫RNF(x,u)dx−1p∗∫RNK(x)|G(u)|p∗dx, |
where F(x,u)=∫u0f(x,τ)dτ. However, I may be not well defined in W1,p(RN) because of the term ∫RNgp(u)|∇u|pdx. To overcome this difficulty, we make use of a change of variable constructed by [32],
v=G(u)=∫u0g(t)dt. |
Then we obtain the following functional
J(v)=1p∫RN[|∇v|p+V(x)|G−1(v)|p]dx−∫RNF(x,G−1(v))dx−1p∗∫RNK(x)|v|p∗dx. |
Since g is a nondecreasing positive function, we can get |G−1(v)|≤1g(0)|v|. From this and our hypotheses, it is clear that J is well defined in W1,p(RN) and J∈C1.
If u is said to be a weak solution for Eq (1.10), then it should satisfy
∫RN[gp(u)|∇u|p−2∇u∇ψ+gp−1(u)g′(u)|∇u|pψ+V(x)|u|p−2uψ−f(x,u)ψ−K(x)g(u)|G(u)|p∗−2G(u)ψ]dx=0,forallψ∈C∞0(RN). | (1.11) |
Let ψ=1g(u)φ, we know that (1.11) is equivalent to
⟨J′(v),φ⟩=∫RN[|∇v|p−2∇v∇φ+V(x)|G−1(v)|p−2G−1(v)g(G−1(v))φ−f(x,G−1(v))g(G−1(v))φ−K(x)|v|p∗−2vφ]dx=0, |
for any φ∈C∞0(RN).
Therefore, in order to obtain a nontrivial solution of (1.1), it suffices to study the following equations
−Δpv+V(x)|G−1(v)|p−2G−1(v)g(G−1(v))−f(x,G−1(v))g(G−1(v))−K(x)|v|p∗−2v=0. |
Obviously, if v is a nontrivial critical point of the functional J, then u=G−1(v) is a nontrivial critical point of the functional I, i.e., u=G−1(v) is a nontrivial solution of equation (1.1).
In the asymptotically periodic potential case, the functional J loses the ZN-translation invariance due to the asymptotically periodic potential. For this reason, many effective methods applied in periodic problems become invalid in asymptotically periodic problems. In this paper, we adopt some tricks to overcome the difficulties caused by the dropping of periodicity of V(x).
Before stating our results, we introduce some hypotheses on the potential V,K:
(V1)0<Vmin≤V(x)≤V0(x)∈L∞(RN)andV(x)−V0(x)∈F0,whereF0:={k(x):∀ε>0,lim|y|→∞meas{x∈B1(y):|k(x)|≥ε}=0},VminisapositiveconstantandV0(x)is1−periodicinxi,1≤i≤N. |
(K1) The function K∈C(RN,R) is 1-periodic in xi,1≤i≤N and there exists a point x0∈RN such that
(i)K(x)≥infx∈RNK(x)>0,forallx∈RN;(ii)K(x)=||K||∞+O(|x−x0|N−pp−1),asx→x0. |
Moreover, the nonlinear term f∈C(RN×R,R) should satisfy the following assumptions:
(f0)f(x,t)=0,t≤0uniformlyforx∈N;(f1)limt→0+f(x,t)g(t)|G(t)|p−1=0uniformlyforx∈N;(f2)limt→+∞f(x,t)g(t)|G(t)|p∗−1=0uniformlyforx∈N;(f3)f(x,G−1(t))tg(G−1(t))−pF(x,G−1(t))≥f(x,G−1(ts))tsg(G−1(ts))−pF(x,G−1(ts))forallt∈+ands∈[0,1];(f4)thereexistsanopenboundedsetΩ⊂N,containingx0givenby(K1)suchthat |
limt→+∞F(x,t)|G(t)|μ=+∞uniformlyforx∈Ω,whereμ=p∗−pp−1,ifN<p2;limt→+∞F(x,t)|G(t)|plog|G(t)|=+∞uniformlyforx∈Ω,ifN=p2;limt→+∞F(x,t)|G(t)|p=+∞uniformlyforx∈Ω,ifN>p2,whereF(x,t)=∫t0f(x,τ)dτ;(f5)Thereexistsaperiodicfunctionf0∈C(RN×R+,R+),whichis1−periodicinxi,1≤i≤N,suchthat(i)f(x,t)≥f0(x,t)forall(x,t)∈RN×R+andf(x,t)−f0(x,t)∈F,whereF:={k(x,t):∀ε>0,lim|y|→∞meas{x∈B1(y):|k(x,t)|≥ε}=0uniformlyfor|t|bound};(ii)f0(x,G−1(t))tg(G−1(t))−pF0(x,G−1(t))≥f0(x,G−1(ts))tsg(G−1(ts))−pF0(x,G−1(ts))forallt∈R+ands∈[0,1],whereF0(x,t)=∫t0f0(x,τ)dτ. |
In the asymptotically periodic case, we establish the following theorem.
Theorem 1.1 Assume that (V1),(K1) and (f1)−(f5) hold. Then Eq (1.1) has a nontrivial positive solution.
In the special case: V=V0,f=f0, we can get a nontrivial positive solution for the periodic equation from Theorem 1.1. Indeed, considering the periodic equation
−div(gp(u)|∇u|p−2∇u)+gp−1(u)g′(u)|∇u|p+V0(x)|u|p−2u=f0(x,u)+K(x)g(u)|G(u)|p∗−2G(u), | (1.12) |
under the hypothesis:
(V2) The function V0(x) is 1-periodic in xi,1≤i≤N and there exists a constant Vmin>0 such that
0<Vmin≤V0(x)∈L∞(RN),forallx∈RN. |
In the periodic case, we obtain the following theorem.
Theorem 1.2 Assume that (V2),(K1) hold, f0 satisfies (f1)−(f4). Then Eq (1.12) has a nontrivial positive solution.
Remark 1.3 Compared with the results obtained by [12,21,23,36], our results are new and different due to the following some facts:
(i) Compared with Eq (1.8) and Eq (1.9), Eq (1.10) is more general. In our results, there is no need to assume f(x,t)∈C1(RN,R). To some extent, our results extends the results of the work [12,23,36,38].
(ii) We choose condition (f3) to be weaker than Ambrosetti-Rabinowitz type condition (see [21]).
(iii) The aim of (f3) is to ensure that the Cerami sequence is bounded, which is different from the conditions of [12].
The rest of this paper is organized as follows: in Section 2, we present some preliminary lemmas. We will prove Theorems 1.1 and 1.2 in Sections 3 and 4, respectively.
In this section, we present some useful lemmas.
Let us recall some basic notions. W:=W1,p(RN) is the usual Sobolev space endowed with the norm ||u||W=(∫RN(|∇u|p+|u|p)dx)1p, we denote by Ls(RN) the usual Lebesgue space endowed with the norm ||u||s=(∫RN|u|sdx)1s,∀s∈[1,+∞) and let C denote positive constants. Next, we define the following working space
X:={u∈Lp∗(RN):|∇u|∈Lp(RN),∫RNV(x)|u|pdx<∞} |
endowed with the norm ||u||=(∫RN(|∇u|p+V(x)|u|p)dx)1p. According to [22], it is easy to verify that the norms ||⋅|| and ||⋅||W are equivalent under the assumption (V1).
Next, we summarize some properties of g,G and G−1.
Lemma 2.1 [32] The functions g,G and G−1 satisfy the following properties:
(i)thefunctionsG(⋅)andG−1(⋅)arestrictlyincreasingandodd;(ii)G(s)≤g(s)sforalls≥0;G(s)≥g(s)sforalls≤0;(iii)g(G−1(s))≥g(0)=1foralls∈R;(iv)G−1(s)sisdecreasingon(0,+∞)andincreasingon(−∞,0);(v)|G−1(s)|≤1g(0)|s|=|s|foralls∈R;(vi)|G−1(s)|g(G−1(s))≤1g2(0)|s|=|s|foralls∈R;(vii)G−1(s)sg(G−1(s))≤|G−1(s)|2foralls∈R;(viii)lim|s|→0G−1(s)s=1g(0)=1and |
lim|s|→∞G−1(s)s={1g(∞),ifgisbounded,0,ifgisunbounded. |
Denote
ˉf(x,s)=V(x)|s|p−2s−V(x)|G−1(s)|p−2G−1(s)g(G−1(s))+f(x,G−1(s))g(G−1(s)). | (2.1) |
Then
ˉF(x,s)=∫s0ˉf(x,τ)dτ=1pV(x)|s|p−1pV(x)|G−1(s)|p+F(x,G−1(s)). | (2.2) |
Therefore,
J(v)=1p∫RN(|∇v|p+V(x)|v|p)dx−∫RNˉF(x,v)dx−1p∗∫RNK(x)|v|p∗dx. | (2.3) |
Lemma 2.2 The functions ˉf(x,s) and ˉF(x,s) satisfy the following properties under (f1)−(f3),(f5):
(i) lims→0+ˉf(x,s)|s|p−1 = 0 and lims→0+ˉF(x,s)|s|p=0 uniformly in x∈RN;
(ii) lims→+∞ˉf(x,s)|s|p∗−1 = 0 and lims→+∞ˉF(x,s)|s|p∗=0 uniformly in x∈RN;
(iii) tˉf(x,t)−pˉF(x,t)≥stˉf(x,st)−pˉF(x,st) for all t∈R+ and s∈[0,1];
(iv) ˉF(x,s)≥0 for all (x,s)∈RN×R+.
Proof From (f1) and Lemma 2.1-(8), we have
lims→0+ˉf(x,s)sp−1=lims→0+[V(x)(1−(G−1(s)s)p−1⋅1g(G−1(s)))]+lims→0+f(x,G−1(s))sp−1g(G−1(s))=0, |
uniformly in x∈RN. Moreover, by (f2) and Lemma 2.1-(8), one has
lims→+∞ˉf(x,s)sp∗−1=−lims→+∞[V(x)(G−1(s)s)p−1⋅1g(G−1(s))⋅1sp∗−p]+lims→+∞f(x,G−1(s))sp∗−1g(G−1(s))=0, |
uniformly in x∈RN. Then using the L'Hospital rule, we obtain
lims→0+ˉF(x,s)|s|p=0andlims→+∞ˉF(x,s)|s|p∗=0, |
uniformly in x∈RN. Hence, (i) and (ii) hold.
Let H(x,t)=tˉf(x,t)−pˉF(x,t) for (x,t)∈(RN,R+).
We claim that H(x,t) is an increasing function with respect to t.
By (2.1) and (2.2), we directly calculate
H(x,t)=tˉf(x,t)−pˉF(x,t)=V(x)|G−1(t)|p−2[G−1(t)2−G−1(t)tg(G−1(t))]+f(x,G−1(t))tg(G−1(t))−pF(x,G−1(t)). |
Next, set η(t)=G−1(t)2−G−1(t)tg(G−1(t)) for any t∈R+. To compete the claim, combining with (f3) and Lemma 2.1-(1), We only need to prove that η(t) is an increasing function on R+. Please see literature [26], for the reader's convenience, we give a brief proof.
By Lemma 2.1-(2) and g′(t)≥0 for all t≥0, one has
G(t)[g(t)−g′(t)tg2(t)]≤t, |
which deduces that
G(t)g(t)(tg(t))′≤tg(t), |
for all t≥0. Set ξ=G(t). Then
G(t)ddξ(tg(t))≤tg(t), |
and thus
ξ[G−1(ξ)g(G−1(ξ))]′≤G−1(ξ)g(G−1(ξ)), |
for all ξ≥0. Therefore,
η′(t)=G−1(t)g(G−1(t))−[G−1(t)g(G−1(t))]′t≥0, |
for all t≥0. It follows that η(t) is increasing with respect to t≥0. Thus, η(st)≤η(t) for all s∈[0,1] and t≥0, and then
G−1(st)2−G−1(st)stg(G−1(st))≤G−1(t)2−G−1(t)tg(G−1(t)), |
for all s∈[0,1] and t≥0. So (iii) holds. Moreover, Lemma 2.1-(5) and (f5)−(i) imply that ˉF(x,s)≥0 for all (x,s)∈RN×R+. This completes the proof.
Lemma 2.3 Assume that (V1),(K1) and (f1)−(f2) are satisfied. Then the functional J satisfies the following mountain pass geometry structure:
(i) there exist positive constants ρ and b such that J(v)≥b for ||v||=ρ;
(ii) there exists a function v0∈X such that ||v0||>ρ and J(v0)<0.
Proof By (f1),(f2), Lemma 2.2-(1), (2), for any ε>0, there exists Cε>0 and q∈(p,p∗) such that
ˉF(x,s)≤ε(|s|p+|s|p∗)+Cε|s|q, | (2.4) |
for all (x,s)∈RN×R+. Therefore, by (2.3) and (2.4), we have
J(v)=1p∫RN[|∇v|p+V(x)|v|p]dx−∫RNˉF(x,v)dx−1p∗∫RNK(x)|v|p∗dx≥1p∫RN[|∇v|p+V(x)|v|p]dx−ε∫RN|v|pdx−ε∫RN|v|p∗dx−Cε∫RN|v|qdx−||K||∞p∗∫RN|v|p∗dx=1p||v||p−ε∫RN|v|pdx−(ε+||K||∞p∗)∫RN|v|p∗dx−Cε∫RN|v|qdx≥(1p−εC)||v||p−C||v||p∗−C||v||q, |
where ε is small enough, thus (i) is proved because p<q<p∗.
It follows from (2.3) that Lemma 2.2-(4), for any fixed v∈X with v≥0 and v≢0, we obtain
J(tv)=tpp∫RN[|∇v|p+V(x)|v|p]dx−∫RNˉF(x,tv)dx−tp∗p∗∫RNK(x)|v|p∗dx≤tpp||v||p−tp∗p∗∫RNK(x)|v|p∗dx. |
Obviously, J(tv)→−∞ as t→+∞. Thus there exists a t0>0 large enough such that J(t0v)<0 with ||t0v||>ρ. Hence, we take v0=t0v, (ii) is proved.
Lemma 2.4 Assume that (V1),(K1) and (f1)−(f3) hold, then there exists a bounded Cerami sequence {vn}⊂X for J.
Proof From Lemma 2.3, we know that J satisfies the mountain pass geometry structure. By the mountain pass theorem (see [1]), there exists a Cerami sequence {vn}⊂X such that
J(vn)→cand(1+||vn||)||J′(vn)||∗→0, |
where
c:=infγ∈Γsupt∈[0,1]J(γ(t)),Γ:={γ∈C([0,1],X):γ(0)=0,J(γ(1))<0}. |
As in [14], we can take a sequence {tn}⊂[0,1] satisfying
J(tnvn):=maxt∈[0,1]J(tvn). | (2.5) |
We claim that {J(tnvn)} is bounded from above.
Indeed, without loss of the generality, we may assume that tn∈(0,1) for all n∈N. Thus, by Lemma 2.2-(3), we have
J(tnvn)−1p⟨J′(tnvn),tnvn⟩=∫RN[1ptnvnˉf(x,tnvn)−ˉF(x,tnvn)]dx+(1p−1p∗)tp∗n∫RNK(x)|vn|p∗dx≤∫RN[1pvnˉf(x,vn)−ˉF(x,vn)]dx+(1p−1p∗)∫RNK(x)|vn|p∗dx=J(vn)−1p⟨J′(vn),vn⟩ =c+on(1), |
which implies that {J(tnvn)} is bounded from above.
Now, we prove that {vn} is bounded in X. Assume by contradiction that {vn} is unbounded, then up to a subsequence, we may assume that ||vn||→+∞. Set wn=vn||vn||. Clearly, wn is bounded in X and ||wn||=1. Then, there exists w∈X such that wn⇀w in X. Set Λ={x∈RN:w(x)≠0}. If meas(Λ)>0, the Fatou lemma and Lemma 2.2-(4) implies
0=limsupn→∞J(vn)||vn||p=1p−liminfn→∞[∫RNˉF(x,vn)||vn||pdx+1p∗∫ΛK(x)|wn|p|vn|p∗−pdx]≤1p−infx∈RNK(x)p∗liminfn→∞∫Λ|wn|p|vn|p∗−pdx→−∞, |
which is a contradiction. Thus w=0. For any B>0, by ||vn||→+∞ and (2.5), we have
J(tnvn)≥J(B||vn||vn)=J(Bwn)=Bpp−∫RNˉF(x,Bwn)dx−Bp∗p∗∫RNK(x)|wn|p∗dx. |
By (f1),(f2), Lemma 2.2-(1), (2), for ε>0, there exists Cε>0 and q∈(p,p∗) such that
ˉf(x,s)s≤ε(|s|p+|s|p∗)+Cε|s|q, | (2.6) |
for all (x,s)∈RN×R+. Hence, by (2.6), we get
infx∈RNK(x)∫RN|wn|p∗dx≤∫RNK(x)|wn|p∗dx=1||vn||p∗−p−1||vn||p∗∫RNˉf(x,vn)vndx+on(1)→0, |
as n→∞, namely, ∫RN|wn|p∗dx→0 as n→∞. Then by interpolation inequality, we have ∫RN|wn|qdx→0 as n→∞. Moreover, from (2.4), we have
|∫RNˉF(x,Bwn)dx|≤εBp∫RN|wn|pdx+εBp∗∫RN|wn|p∗dx+CεBq∫RN|wn|qdx. |
By the arbitrariness of ε, we can get ∫RNˉF(x,Bwn)dx→0 as n→∞. Consequently,
liminfn→∞J(tnvn)≥Bpp,∀B>0. |
This contradicts the fact that J(tnvn) is bounded above. Therefore, {vn} is bounded in X. The proof of Lemma 2.4 is complete
Next, we do an estimate on c and follow the approach presented in [12]. Given ε>0, we consider the function
Uε(x)=[N(N−pp−1)p−1ε]N−pp2(ε+|x−x0|pp−1)N−pp, |
which is a solution of the following equation
−△pu=|u|p∗−2u,inRN |
and
S:=infu∈D1,p(RN)∖{0}∫RN|∇u|pdx(∫RN|u|p∗dx)pp∗. |
can be achieved at Uε.
Let ϕ∈C∞0(RN,[0,1]) be a cut-off function such that ϕ≡1 in B1(0), ϕ≡0 in RN∖B2(0). Define
uε=ϕUε,vε=uε(∫RNK(x)|u|p∗εdx)1p∗, | (2.7) |
then by a direct computation, there exist positive constants l1,l2 and ε0 such that
l1<∫RNK(x)|u|p∗εdx<l2,forall0<ε<ε0, | (2.8) |
∫RN|∇vε|p≤||K||p−NN∞S+O(εN−pp),asε→0+, | (2.9) |
and as ε→0, we have
∫RN|vε|pdx={O(εN−pp),ifN<p2,O(εp−1|logε|),ifN=p2,O(εp−1),ifN>p2. | (2.10) |
Lemma 2.5 Assume that (V1),(K1) and (f1)−(f2),(f4) are satisfied. Then c<1N||K||p−Np∞SNp.
Proof For t>0, vε defined by above (2.7), we have
J(tvε)=1p∫RN[tp|∇vε|p+V(x)|G−1(tvε)|p]dx−∫RNF(x,G−1(tvε))dx−tp∗p∗∫RNK(x)|vε|p∗dx. | (2.11) |
Lemma 2.3 implies that there exists tε>0 such that J(tεvε)=maxt≥0J(tvε).
We claim that there exist T1,T2>0 such that T1≤tε≤T2 for ε small enough.
Indeed, if tε→0 as ε→0, we have 0<J(tεvε)→J(0)=0, which is a contradiction.
On the other hand, from (2.11) and (f4), one has
0<J(tεvε)=1p∫RN[tpε|∇vε|p+V(x)|G−1(tεvε)|p]dx−∫RNF(x,G−1(tεvε))dx−tp∗εp∗∫RNK(x)|vε|p∗dx≤tpεp∫RN[|∇vε|p+V(x)|vε|p]dx−tp∗εp∗∫RNK(x)|vε|p∗dx→−∞, |
as tε→+∞, a contradiction, which implies that the claim holds.
To complete the proof, it suffices to show that J(tεvε)<1N||K||p−Np∞SNp. Therefore,
J(tεvε)=1p∫RN[tpε|∇vε|p+V(x)|G−1(tεvε)|p]dx−∫RNF(x,G−1(tεvε))dx−tp∗εp∗∫RNK(x)|vε|p∗dx≤tpεp∫RN|∇vε|pdx+tpεp∫RNV(x)|vε|pdx−∫RNF(x,G−1(tεvε))dx−tp∗εp∗≤1N(∫RN|∇vε|pdx)Np+C∫RN|vε|pdx−∫RNF(x,G−1(tεvε))dx, |
for some constant C=Tp2||V0||∞p>0.
Indeed, for t>0, define l(t)=tpp∫RN|∇vε|pdx−tp∗p∗, we have that t0=(∫RN|∇vε|pdx)1p∗−p is a maximum point of l and l(t0)=1N(∫RN|∇vε|pdx)Np. Applying the inequality
(a+b)κ≤aκ+κ(a+b)κ−1b,a,b≥0,κ≥1. | (2.12) |
By (2.9) and (2.12), we have
J(tεvε)≤1N(||K||p−NN∞S+O(εN−pp))Np+C∫RN|vε|pdx−∫RNF(x,G−1(tεvε))dx≤1N||K||p−Np∞SNp+C∫RN|vε|pdx−∫RNF(x,G−1(tεvε))dx+O(εN−pp). | (2.13) |
Now consider
r(ε)={εN−pp,ifN<p2,εp−1|logε|,ifN=p2,εp−1,ifN>p2. | (2.14) |
From (2.13) and (2.14), we have
J(tεvε)≤1N||K||p−Np∞SNp+r(ε)[C−∫RNF(x,G−1(tεvε))dxr(ε)]. |
Next we claim that
limε→0+∫RNF(x,G−1(tεvε))dxr(ε)=+∞. | (2.15) |
It follows (f4) that any A>0, there exists R=RA>0 such that for all (x,s)∈Ω×[RA,+∞),
F(x,G−1(s))≥{A|s|μ,ifN<p2,A|s|plog|s|,ifN=p2,A|s|p,ifN>p2, | (2.16) |
where μ=p∗−pp−1. Now consider the function ηε:[0,+∞)→R defined by
ηε(r)=εN−pp2(ε+rpp−1)N−pp. |
Since ϕ≡1 in B1(0), due to (2.8), we choose a constant C>0 such that vε(x)≥Cηε(|x|) for |x|<1. Note that ηε is decreasing and G−1 is increasing, there exists a positive constant C such that, for |x|<εp−1p,
G−1(tεvε)≥G−1(T1Cηε(|x|))≥G−1(T1Cηε(εp−1p))≥G−1(Cε(N−p)(1−p)p2). |
Then we can choose ε1>0 such that
Cε(N−p)(1−p)p2≥1,G−1(tεvε)≥G−1(Cε(N−p)(1−p)p2)≥R, | (2.17) |
for |x|<εp−1p, 0<ε<ε1. It follows from (2.16) and (2.17) that
F(x,G−1(s))≥{CAε(N−p)(1−p)μp2,ifN<p2,CAε(N−p)(1−p)plogε,ifN=p2,CAε(N−p)(1−p)p,ifN>p2, | (2.18) |
for |x|<εp−1p, 0<ε<ε1.
Using (f5)-(i), one has
F(x,G−1(s))+|s|p≥0,x∈Ω,s≥0. | (2.19) |
Since B2(0)⊂Ω, by (2.18) and (2.19), for 0<ε<ε1, we have
∫RNF(x,G−1(tεvε))dx=∫Bεp−1pF(x,G−1(tεvε))dx+∫Ω∖Bεp−1pF(x,G−1(tεvε))dx≥∫|x|<εp−1pF(x,G−1(tεvε))dx−Tp2||vε||pp, | (2.20) |
where
∫|x|<εp−1pF(x,G−1(tεvε))dx≥{CA∫|x|<εp−1pε(N−p)(1−p)μp2dx=CAεN−pp,ifN<p2,CA∫|x|<εp−1pε(N−p)(1−p)plogεdx=CAεp−1logε,ifN=p2,CA∫|x|<εp−1pε(N−p)(1−p)pdx=CAεp−1,ifN>p2. |
Consequently, by (2.20), we obtain
limε→0+∫RNF(x,G−1(tεvε))dxr(ε)≥CA−Tp2. | (2.21) |
Choosing A>0 sufficiently large, (2.21) establishes (2.15). Lemma 2.5 is proved.
In this section, in order to overcome the difficulties caused by the loss of translation invariance due to the asymptotically periodic potential, we need to state the following technical convergence results. The detailed proofs can be found in [23,36], where p=2.
Lemma 3.1 Assume that (V1), (K1), (f1) and (i) of (f5) hold. Suppose that {vn} is bounded in X, {yn}⊂ZN with |yn|→+∞ and vn→0 in Lαloc(RN), for any α∈[p,p∗). Then up to a subsequence, one has
∫RN(V(x)−V0(x))|G−1(vn)|pdx→on(1); | (3.1) |
∫RN(V(x)−V0(x))|G−1(vn)|p−2G−1(vn)g(G−1(vn))φ(x−yn)dx→on(1),∀φ∈C∞0(RN); | (3.2) |
∫RN[F(x,G−1(vn))−F0(x,G−1(vn))]dx→on(1); | (3.3) |
∫RNf(x,G−1(vn))−f0(x,G−1(vn))g(G−1(vn))φ(x−yn)dx→on(1),∀φ∈C∞0(RN). | (3.4) |
Proof of Theorem 1.1 Lemma 2.3 implies the existence of a Cerami sequence {vn}⊂X. By Lemma 2.4, {vn} is bounded in X. Thus, there exists v∈X such that vn⇀v in X, vn→v in Lploc(RN), vn(x)→v(x) a.e. in RN. For any φ∈C∞0(RN), one has
0=⟨J′(vn),φ⟩+on(1)=⟨J′(v),φ⟩, |
that is, v is a weak solution of Eq (1.10).
Now we prove that v is nontrivial. By contradiction, we assume that v=0. We divide the proof four steps.
Step 1: We claim that {vn}⊂X is also a Cerami sequence for the functional J0:X→R, where
J0(v)=1p∫RN[|∇v|p+V0(x)|G−1(v)|p]dx−∫RNF0(x,G−1(v))dx−1p∗∫RNK(x)|v|p∗dx. |
From (3.1) and (3.3), we can deduce that
|J(vn)−J0(vn)|≤1p∫RN|(V(x)−V0(x))|G−1(vn)|p|dx+∫RN|F(x,G−1(vn))−F0(x,G−1(vn))|dx=on(1), | (3.5) |
and taking φ∈X with ||φ||=1, by (3.2) and (3.4), we obtain that
||⟨J′(vn)−J′0(vn)||∗≤supφ∈X,||φ||=1[∫RN|(V(x)−V0(x))|G−1(vn)|p−2G−1(vn)g(G−1(vn))φ|dx+∫RN|f(x,G−1(vn))−f0(x,G−1(vn))g(G−1(vn))φ|dx]=on(1). | (3.6) |
From (3.5) and (3.6), we can get that {vn} is also a Cerami sequence for J0.
Step 2: We prove that {vn} is non-vanishing i.e.,
β:=limsupn→∞supy∈RN∫B1(y)|vn|pdx>0. | (3.7) |
If β=0, the Lions lemma [18], we have vn→0 in Lq(RN) for all q∈(p,p∗).
Note that
on(1)=⟨J′(vn),vn⟩=∫RN[|∇vn|p+V(x)|vn|p]dx−∫RNˉf(vn,v)vndx−∫RNK(x)|vn|p∗dx, |
which combining with (2.6) leads to
∫RN[|∇vn|p+V(x)|vn|p]dx−∫RNK(x)|vn|p∗dx=on(1). |
Therefore, there exists a constant l≥0 such that
∫RN[|∇vn|p+V(x)|vn|p]dx→l,∫RNK(x)|vn|p∗dx→l. |
Obviously, l>0. Otherwise, J(vn)→0 as n→∞, which contradicts with c>0. Since
l=limn→∞∫RNK(x)|vn|p∗dx≤||K||∞limn→∞∫RN|vn|p∗dx≤||K||∞S−p∗plimn→∞(∫RN|∇vn|pdx)p∗p≤||K||∞S−p∗plimn→∞||∇vn||p∗≤||K||∞S−p∗plp∗p, |
that is, l≥||K||p−Np∞SNp. Consequently, (2.4) implies that
c+on(1)=J(vn)=1p∫RN(|∇vn|p+V(x)|vn|p)dx−∫RNˉF(x,vn)dx−1p∗∫RNK(x)|vn|p∗dx→(1p−1p∗)l=1Nl≥1N||K||p−Np∞SNp, |
as n→∞, which deduces that c≥1N||K||p−Np∞SNp, a contradiction.
Step 3: After a translation of {vn} denoted {wn}, then {wn} converges weakly to a nonzero critical point of J0.
Choose {yn}⊂ZN such that |yn|→+∞ as n→∞ and denote wn(x)=vn(x+yn). Then
||wn||=||vn||,J0(wn)=J0(vn),J′0(wn)=J′0(vn). |
Thus, {wn} is a bounded (PS)c0 of J0, where c0 is defined below. Going if necessary to a subsequence, we get that wn⇀w in X and J′0(w)=0. So by Step 2 we get w≠0. Therefore, by (f5)-(ii), Lemma 2.1-(8) and Fatou Lemma, we have
c=liminfn→∞[J0(wn)−1p⟨J′0(wn),wn⟩]=liminfn→∞1p∫RNV0(x)|G−1(wn)|p−2[(G−1(wn))2−G−1(wn)wng(G−1(wn)]dx+liminfn→∞∫RN[f0(x,G−1(wn))wnpg(G−1(wn))−F0(x,G−1(wn))]dx+(1p−1p∗)liminfn→∞∫RNK(x)|wn|p∗dx≥1p∫RNV0(x)|G−1(w)|p−2[(G−1(w))2−G−1(w)wg(G−1(w)]dx+∫RN[f0(x,G−1(w))wpg(G−1(w))−F0(x,G−1(w))]dx+(1p−1p∗)∫RNK(x)|w|p∗dx=J0(w)−1p⟨J′0(w),w⟩=J0(w), |
which implies that J0(w)≤c.
Step 4: We use w to construct a path which allows us to get a contradiction with the definition of mountain pass level c.
Define the mountain pass level
c0:=infγ∈ˉΓsupt∈[0,1]J0(γ(t))>0, |
where ˉΓ:={γ∈C([0,1],X):γ(0)=0,J0(γ(1))<0}. Applying similar arguments used in [15], we can construct a path γ:[0,1]→X such that
{γ(0)=0,J0(γ(1))<0,w∈γ([0,1]),γ(t)(x)>0,∀x∈RN,t∈[0,1],maxt∈[0,1]J0(γ(t))=J0(w). |
Then c0≤maxt∈[0,1]J0(γ(t))=J0(w). Due to the fact that V(x)≤V0(x) but V(x)≢V0(x), we take the path γ given by above and by γ∈ˉΓ⊂Γ, we have
c≤maxt∈[0,1]J(γ(t))=J(γ(ˉt))<J0(γ(ˉt))≤maxt∈[0,1]J0(γ(t))=J0(w)≤c, |
which is a contradiction. Consequently, v is a nontrivial solution of Eq (1.10), then using the strong maximun principle, we obtain v>0, namely, Eq (1.1) possesses a nontrivial positive solution u=G−1(v). This completes the proof of Theorem 1.1.
In this section, we give the proof of Theorem 1.2.
Proof of Theorem 1.2 By Lemma 2.3, there exists a Cerami sequence {vn}⊂X such that
J0(vn)→cand(1+||vn||)||J′0(vn)||∗→0. |
Applying Lemma 2.4, the Cerami sequence {vn} is bounded in X. Similar to (3.7), it is easy to verify that {vn} is non-vanishing.
As in the step 3 of Theorem 1.1, set wn(x)=vn(x+yn). It is easy to know that ||wn||=||vn|| and {wn} is bounded and non-vanishing. Going if necessary to a subsequence, we have
wn⇀w≠0inX,wn→winLploc(RN). |
Moreover, since V0(x),K(x) and f0(x,u) are periodic on X, we see that {wn} is also a a Cerami sequence of J0. Then for any φ∈C∞0(RN),
⟨J′0(w),φ⟩=limn→∞⟨J′0(wn),φ⟩. |
That is J′0(w)=0 and w is a nontrivial solution to (1.12). By the strong maximun principle, we obtain w>0. This completes the proof of Theorem 1.2.
In [38], we discussed a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential, where p=2 and the nonlinear term is subcritical. In this current work, we have established the existence of nontrivial positive solutions for a class of generalized quasilinear elliptic equations with critical growth. In the next work, we will extend the study to the case of variable exponent p=p(t).
The author expresses his appreciation to the reviewers and the handling editor whose careful reading of the manuscript and valuable comments greatly improved the original manuscript.
The author declare no conflicts of interest in this paper.
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