# Nontrivial Solutions for Periodic Schrödinger Equations with Sign-Changing Nonlinearities

Authors: Shaowei Chen; Lishan Lin; Liqin Xiao
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove a new infinite-dimensional linking theorem. Using this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing nonlinearities.


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## Body

## 1. Introduction and Statement of Results

In this paper, we consider the following semilinear Schrödinger equation:(1)-Δu+Vxu+fx,u=0,u∈H1RN,
where N≥1. The potential V∈L∞(RN) is 1-periodic in xj for j=1,…,N. In this case, the spectrum of the operator -Δ+V is a purely continuous spectrum bounded from below and consists of closed disjoint intervals ([1, Theorem XIII.100]). Thus, the complement R∖σ(L) consists of open intervals called spectral gaps. More precisely, for V, we assume the following.(v)
V∈L∞(RN) is 1-periodic in xj for j=1,…,N, 0 is in a spectral gap (-μ-1,μ1) of -Δ+V, and -μ-1 and μ1 lie in the essential spectrum of -Δ+V. Denote
(2)μ0≔min⁡μ-1,μ1.For f, we assume the following.(f1)
f is a Caratheodory function in RN×R and it is 1-periodic in xj for j=1,…,N. In addition, there exist constants C>0 and 2<p<2* such that
(3)fx,t≤C1+tp-1,∀x,t∈RN×R,
where
(4)2*≔2NN-2,N≥3∞,N=1,2.(f2)
The limitslim⁡t→0f(x,t)/t=0 and lim⁡|t|→∞f(x,t)/t=+∞ hold uniformly for x∈RN.LetF(x,t)=∫0t‍f(x,s)ds, V-(x)=max⁡{-V(x),0}, x∈RN, and
(5)F~x,t≔12tfx,t-Fx,t.(f3)
There existρ>0 and M>ρ such that
(6)inf⁡t≥M/2,x∈RNfx,tt-V-x>0,(7)inf⁡t≥ρ,x∈RNFx,t≥0,(8)inf⁡ρ≤t≤M,x∈RNF~x,t>0,(9)sup⁡t≤ρ,x∈RNfx,tt+sup⁡ρ≤t≤M,x∈RNf2x,tF~x,t1/2·sup⁡t≤ρ,x∈RNF~x,tt21/2<μ0.A solutionu of (1) is called nontrivial if u≢0. Our main result is the following theorem.Theorem 1.
Suppose that (v) and (f1)–(f3) are satisfied. Then problem (1) has a nontrivial solution.As an application of this theorem, we have the following corollary.Corollary 2.
Suppose that2<p<q<2* and (v) is satisfied. Then, there exists λ0>0 such that, for 0<λ<λ0,
(10)-Δu+Vu=λup-2u-uq-2u
has a nontrivial solution.Remark 3.
This corollary shows that, under assumptions (f1)–(f3), the nonlinearity f may be sign-changing.Under assumption (v), the quadratic form ∫RN‍(|∇u|2+V(x)u2)dx has infinite-dimensional negative and positive spaces. This case is called strongly indefinite. Semilinear periodic Schrödinger equations with strongly indefinite linear parts have attracted considerable attention in recent years due to their numerous applications in mathematical physics. In [2], the authors used a dual variational method to obtain a nontrivial solution of (1) with f(x,t)=-W(x)|t|p-2t, where W is a perturbed periodic function and 2<p<2*. In [3], Troestler and Willem used critical point theory to obtain a nontrivial solution of (1) by assuming that f∈C1(RN×R), |∂tf(x,t)|≤C(|t|p-2+|t|q-2) for some 2<p<q<2* and g=-f satisfy the so-called Ambrosetti-Rabinowitz condition 0<γG(x,t)≤tg(x,t), ∀(x,t)∈RN×(R∖{0}), where G(x,t)=∫0t‍g(x,s)ds. Kryszewski and Szulkin [4] subsequently proved a new infinite-dimensional linking theorem. Using this, they generalized Troestler and Willem’s result by assuming that f∈C(RN×R) and g=-f satisfy the Ambrosetti-Rabinowitz condition. Similar results were obtained by Pankov and Pflüger in [5, 6] by using an approximation method and a variant of the Nehari method. Equation (1), with asymptotically linear nonlinearities and other superlinear nonlinearities, has also been studied by several researchers. The interested readers can see [7–10] for the asymptotically linear case and [9, 11–17] for the superlinear case. Moreover, (1), with 0 belonging to the spectrum of -Δ+V, was investigated in [8, 18, 19]. It is worth mentioning that methods of studying strongly indefinite periodic Schrödinger equations can shed light on other strongly indefinite problems, such as Hamiltonian systems and discrete nonlinear Schrödinger equations. The reader can consult [8, 20] or [21] for more details.All existence results for (1) mentioned above are obtained under the assumption that tf does not change sign in RN×R; that is, tf(x,t)≥0 in RN×R or tf(x,t)≤0 in RN×R. However, under our assumptions (f1)–(f3), tf(x,t) can be negative in {(x,t)∣|t|<ρ}. Together with (7), this implies that tf(x,·) may change sign in R. As we know, this situation has never before been studied. And it is the novelty of our main results of Theorem 1 and Corollary 2. Theorem 1 can be seen as a generalization of Theorem 3 of [22].The difficulties of (1) with regard to sign-changing nonlinearity are due to two aspects. The first is that the classical infinite-dimensional linking theorem (see [23, Theorem 6.10] or [4]) cannot be used to deal with (1) in this case. To use this linking theorem, the functional corresponding to (1) must satisfy a certain upper semicontinuous assumption. However, when f(x,t) is sign-changing, the functional corresponding to (1) does not satisfy such a assumption. The second aspect that causes problems is that sign-changing nonlinearity creates more difficulties in proving the boundedness of the Palais-Smale sequence.In this paper, we provide a variant of the classical infinite-dimensional linking theorem (see Theorem7). This theorem and its corollary replace the upper semicontinuous assumption in the classical infinite-dimensional linking theorem ([23, Theorem 6.10]) with other assumptions. We present the theorem and its proof in Section 2. Using this theorem, a (C¯)c sequence (see Definition 6) of (1) is obtained. Under (f1)–(f3), we can prove that this sequence is bounded in H1(RN) (see Lemma 12). We then obtain a nontrivial solution of (1) from the (C¯)c sequence by using the concentration-compactness principle.Notation. Br(a) denotes an open ball of radius r and center a. For a Banach space E, we denote the dual space of E by E′ and denote strong and weak convergence in E by → and ⇀, respectively. For φ∈C1(E;R), we denote the Fréchet derivative of φ at u by φ′(u). The Gateaux derivative of φ is denoted by 〈φ′(u),v〉, ∀u,v∈E. Lp(RN) denotes the standard Lp space (1≤p≤∞), and H1(RN) denotes the standard Sobolev space with norm uH1=(∫RN‍(|∇u|2+u2)dx)1/2. We use O(h), o(h) to mean |O(h)|≤C|h| and o(h)/|h|→0.


## 2. A New Infinite-Dimensional Linking Theorem

In this section, we give a variant linking theorem which is a generalization of the classical infinite-dimensional linking theorem of [23, Theorem 6.10] (see also [4]).Before stating this theorem, we give some notations and definitions.LetX be a separable Hilbert space with inner product (·,·) and norm ∥·∥, respectively. X± are closed subspaces of X and X=X+⊕X-. Let {ek-} be the total orthonormal sequence in X-. Let
(11)Q:X⟶X+,P:X⟶X-
be the orthogonal projections. We define
(12)u=max⁡Qu,∑k=1∞12k+1Pu,ek-
on X. Then
(13)Qu≤u≤u,∀u∈X,
and if un is bounded and un-u→0, then {un} weakly converges to u in X. The topology generated by · is denoted by τ, and all topological notations related to it will include this symbol.LetR>r>0 and z0∈X+ with z0=1. Set
(14)N=z∈X+∣z=r,M=z+tz0∣z∈X-,t≥0,z+tz0≤R.
Then, M is a submanifold of X-⊕R+u0 with boundary
(15)∂M=z∈X-∣z≤R∪z+tz0∣z∈X-,t>0,z+tz0=R.Definition 4.
A functionalϕ∈C1(X,R) is called τ-upper semicontinuous if, for any a∈R, ϕa≔{u∈X∣ϕ(u)≥a} is a τ-closed set. And ϕ′ is called weakly sequentially continuous if u∈X and {un}⊂X are such that un⇀u; then, for any φ∈X, 〈ϕ′(un),φ〉→〈ϕ′(u),φ〉.Remark 5.
If a functionalϕ∈C1(X,R) has the form
(16)ϕu=c1Qu2-c2Pu2-ψu
with c1>0, c2>0 and ψ∈C1(X,R) is weakly sequentially lower semicontinuous, that is, if un⇀u, then liminf⁡n→∞ψ(un)≥ψ(u), then by Remark 2.1 (iii) of [4], ϕ is τ-upper semicontinuous.Definition 6.
Letϕ∈C1(X,R). A sequence {un}⊂X is called a (C¯)c sequence for ϕ, if
(17)limsup⁡n→∞ϕun≤c,1+unϕ′unX′⟶0,hhhihhhhhhhhhasn⟶∞.The main results of this section are the following theorem and corollary.Theorem 7.
IfH∈C1(X,R) satisfies the following(a)
H′ is weakly sequentially continuous,(b)
there exists aτ-upper semicontinuous functional J such that H(u)≤J(u), ∀u∈X,(c)
there existu0∈X+ with u0=1 and R>r>0 such that
(18)inf⁡NH>sup⁡∂MH,(19)sup⁡MH<+∞,then, for any κ<inf⁡NH, there exists a (C¯)c sequence {un} of H with c=sup⁡MH such that
(20)inf⁡nJun≥κ.Corollary 8.
LetH and J satisfy assumptions (a), (b), and (c) in Theorem 7. If
(21)J0<inf⁡NH,
then there exist δ>0 and a (C¯)c sequence {un} of H with c=sup⁡MH such that
(22)inf⁡nun≥δ.Remark 9.
(i) To use the classical infinite-dimensional linking theorems, such as Theorem 3.4 of [4] or Theorem 6.10 of [23], to get a Palais-Smale sequence or (C¯)c sequence for a functional satisfying the linking condition (18), this functional should be τ-upper semicontinuous. This assumption precludes applying these classical infinite-dimensional linking theorems to problems possessing wider class of nonlinearities, such as nonlinear Schrödinger equations with sign-changing nonlinearities.
(ii)(C¯)c sequence (see Definition 6) can be seen as a weighted variation of Palais-Smale sequence. It plays important role in proving the boundedness. For example, the (C¯)c sequence in Lemma 11 of this paper cannot be replaced with Palais-Smale sequence.
(iii) This theorem and its corollary are new infinite-dimensional linking theorems different from the one published in [24, Theorem 1.3].Proof of Theorem7.
Arguing indirectly, assume that the result does not hold. Then, there existϵ>0 and κ0<inf⁡NH such that
(23)1+uH′uX′≥ϵ,∀u∈E,
where
(24)E=u∈X∣Hu≤d+ϵ∩u∈X∣Ju≥κ0,d=sup⁡MH.
SinceH≤J in X, we deduce that
(25)Hκ0≔u∈X∣Hu≥κ0⊂Jκ0≔u∈X∣Ju≥κ0.
Therefore,
(26)Hκ0d+ϵ≔u∈X∣κ0≤Hu≤d+ϵ⊂E.
Step  1. A vector field in a τ-neighborhood of
(27)Hd+ϵ≔u∈X∣Hu≤d+ϵ.
Let(28)b=inf⁡NH,T=2d+ϵ-κ0ϵ,R′=1+sup⁡u∈Mue2T,(29)BR′=u∈X∣u≤R′.
For everyu∈E∩BR′, there exists ϕu∈X with ϕu=1 such that 〈H′(u),ϕu〉≥(3/4)H′(u)X′. Then, (23) implies that
(30)1+uH′u,ϕu>12ϵ.
From the definition of ·, we deduce that if a sequence {un}⊂E∩BR′  τ-converges to u∈X, that is, un-u→0, then un⇀u in X (see Remark 6.1 of [23]). By the weakly sequential continuity of H′, we get that for any φ∈X, 〈H′(un),φ〉→〈H′(u),φ〉. This implies that H′ is τ-sequentially continuous in E∩BR′. By (30), the τ-sequential continuity of H′ in E∩BR′, and the weakly lower semicontinuity of the norm ∥·∥, we get that there exists a τ-open neighborhood Vu of u such that
(31)H′v,1+uϕu>12ϵ,∀v∈Vu∩E∩BR′,1+uϕu=1+u≤21+v,∀v∈Vu.
BecauseBR′ is a bounded convex closed set in the Hilbert space X, BR′ is a τ-closed set. It follows that X∖BR′ is a τ-open set. Moreover, since J is a τ-upper semicontinuous functional, {u∈X∣J(u)≥κ0} is a τ-closed set. It follows that
(32)U=u∈X∣Ju<κ0
is a τ-open set. Therefore, the family
(33)N=Vu∣u∈E∩BR′∪X∖BR′∪U
is a τ-open covering of Hd+ϵ. Let
(34)V=⋃u∈E∩BR′Vu∪X∖BR′∪U.
Then, V is a τ-open neighborhood of Hd+ϵ.
SinceV is metric, hence paracompact, there exists a local finite τ-open covering M={Mi∣i∈Λ} of V finer than N. If Mi⊂Vui for some ui∈E∩BR′, we choose ϖi=(1+ui)ϕui and if Mi⊂X∖BR′ or if Mi⊂U, we choose ϖi=0. Let {λi∣i∈I} be a τ-Lipschitz continuous partition of unity subordinated to M. And let
(35)ξu≔∑i∈Iλiuϖi,u∈V.
Since the τ-open covering M of V is local finite, each u∈V belongs to only finitely many sets Mi. Therefore, for every u∈V, the sum in (35) is a finite sum. This implies the following.(a)
For anyu∈V, there exist a τ-open neighborhood Uu⊂V of u and Lu>0 such that ξ(Uu) is contained in a finite-dimensional subspace of X and
(36)ξv-ξw≤Luv-w,∀v,w∈Uu.
This means that ξ is locally Lipschitz continuous and τ-locally Lipschitz continuous.Moreover, by the definition of ξ and (31), we get the following.(b)
For everyu∈V,
(37)ξu≤21+u,H′u,ξu≥0
and for every u∈E∩BR′,
(38)H′u,ξu>12ϵ.
Step  2. From (36) and the fact that v≤v, ∀v∈X, we have
(39)ξv-ξw≤Luv-w,∀v,w∈Uu.
This implies that ξ is a local Lipschitz mapping under the ∥·∥ norm. Then, by the standard theory of ordinary differential equation in Banach space, we deduce that the following initial value problem
(40)dηdt=-ξη,η0,u=u∈V,
has a unique solution in V, denoted by η(t,u), with the right maximal interval of existence [0,T(u)). Furthermore, using (36) and the Gronwall inequality (see, e.g., Lemma 6.9 of [23]), the similar argument as the proof of (c) in [23, Lemma 6.8] yields that(A)
η is τ-continuous; that is, if un∈V, u0∈V, 0≤tn<T(un), and 0≤t0<T(u0) satisfy un-u0→0 and tn→t0, then η(tn,un)-η(t0,u0)→0.
From〈H′(u),ξ(u)〉≥0, ∀u∈V (see (37)), we have
(41)ddtHηt,u=H′ηt,u,η′t,u≤0.
Therefore, H is nonincreasing along the flow η. It follows that {η(t,u)∣0≤t≤T(u)}⊂Hd+ϵ if u∈Hd+ϵ; that is, for any u∈Hd+ϵ and any t≥0, η(t,u) is still in Hd+ϵ. Moreover, since V is a neighborhood of Hd+ϵ, η(t,u)∈Hd+ϵ⊂V for any u∈Hd+ϵ and t≥0; that is, the flow η(t,u) with u∈Hd+ϵ cannot leave V. Therefore, Hd+ϵ is an invariant set of the flow η. Then, ξ(u)≤2(1+u), ∀u∈V (see (37)) and Theorem 5.6.1 of [25] implies that, for any u∈Hd+ϵ, T(u)=+∞.
Step  3. We will prove that
(42)ηt,u∣0≤t≤T,u∈M⊂BR′,
where T is defined in (28).
Letu∈Hd+ϵ. By the result in Step 2, we have T(u)=+∞ and
(43)ηt,u=u-∫0tξηs,uds,∀t∈0,+∞.
Together with ξ(u)≤2(1+u), ∀u∈V (see (37)), this yields
(44)ηt,u≤u+∫0tξηs,uds≤u+2∫0t1+ηs,uds.
Then, by the Gronwall inequality (see, e.g., Lemma 6.9 of [23]), we get that
(45)ηt,u≤1+ue2t-1,∀t∈0,+∞.
Since M⊂Hd+ϵ, by (45) and the definition of R′ (see (29)), we obtain (42).
Step  4. From (26) and (38), we deduce that
(46)H′u,-ξu<-12ϵ,∀u∈Hκ0d+ϵ∩BR′.
We show that, for anyu∈M, H(η(T,u))≤κ0. Arguing indirectly, assume that this was not true. Then, there exists u∈M such that H(η(T,u))>κ0. Since H is nonincreasing along the flow η, from (42), we deduce that {η(t,u)∣0≤t≤T}⊂Hκ0d+ϵ∩BR′. Then, by (46),
(47)HηT,u=Hη0,u+∫0TH′ηs,u,-ξηs,uds≤Hη0,u+∫0T-12ϵds≤d+ϵ-12ϵT=κ0.
This contradicts H(η(T,u))>κ0. Therefore, we have(B)
η(T,M)⊂Hκ0≔{u∈X∣H(u)≤κ0}.Moreover, using the result (a) in Step 1 and the fact that η is τ-continuous (see (A)), the similar argument as the proof of the result (b) of [23, Lemma 6.8] yields that(C)
each point(t,u)∈[0,T]×Hd+ϵ has a τ-neighborhood N(t,u) such that
(48)v-ηs,v∣s,v∈Nt,u∩0,T×Hd+ϵ
is contained in a finite-dimensional subspace of X.
Step  5. Let
(49)h:0,T×M⟶X,ht,u=Pηt,u+Qηt,u-ru0,
where P,Q,r, and u0 are defined in (11), (14), and (15). Then
(50)0∈ht,M⟺ηt,M∩N≠∅.
From inf⁡NH>sup⁡∂MH (see (18)) and the fact that, for any u∈X, the function H(η(·,u)) is nonincreasing, we deduce that inf⁡NH>sup⁡u∈∂MH(η(t,u)), ∀t∈[0,T]. Therefore,
(51)0∉ht,∂M,∀t∈0,T.
Since η has properties (A) and (C) obtained in Steps  2 and 4, respectively, and h satisfies (51), there is an appropriate degree theory for deg⁡(h(t,·),M,0) (see Proposition 6.4 and Theorem 6.6 of [23]). Then, the same argument as the proof of Theorem 6.10 of [23] yields that
(52)deg⁡hT,·,M,0=deg⁡h0,·,M,0≠0.
It follows that 0∈h(T,M) and η(T,M)∩N≠∅. Therefore, there exists u∈M such that H(η(T,u))≥b. It contradicts property (B) obtained in Step  4, since κ0<b. This completes the proof of this theorem.Proof of Corollary8.
SinceJ(0)<inf⁡NH, we can choose κ such that J(0)<κ<inf⁡NH. By Theorem 7, there exists a (C¯)c sequence {un} of H such that c=sup⁡MH and inf⁡nJ(un)≥κ. From J(0)<κ<inf⁡NH, we deduce that 0∈{u∈X∣J(u)<κ}. Since J is τ-upper semicontinuous, {u∈X∣J(u)<κ} is a τ-open set. Therefore, there exists δ>0 such that
(53)u∈X∣u<δ⊂u∈X∣Ju<κ.
Since inf⁡nJ(un)≥κ, by (53), we obtain inf⁡nun≥δ.


## 3. Variational and Linking Structure for (1)

Under assumptions (v), (f1) and the first part of (f2), the functional
(54)Ju=12∫RN∇u2dx+12∫RNVxu2dx+∫RNFx,udx
belongs to class C1 for X≔H1(RN). The derivative of J is
(55)J′u,v=∫RN∇u∇v+Vxuvdx+∫RNfx,uvdx,u,v∈H1RN
and the critical points of J are weak solutions of (1).There is a standard variational setting for the quadratic form∫RN‍(|∇u|2+V(x)u2)dx. For the reader’s convenience, we state it here. The interested reader should consult [9] or [8] for more details.Assume that (v) holds and let S=-Δ+V be the self-adjoint operator acting on L2(RN) with domain D(S)=H2(RN). By virtue of (v), we have the orthogonal decomposition
(56)L2=L2RN=L1+L2
such that S is positive (resp., negative) in L1 (resp., in L2). Let X=D(|S|1/2) be equipped with the inner product
(57)u,v=S1/2u,S1/2vL2
and norm u=|S|1/2uL2, where (·,·)L2 denotes the inner product of L2. From (v),
(58)X=H1RN
with equivalent norms. Therefore, X continuously embeds into Lq(RN) for all 2≤q≤2*. In addition, we have the decomposition
(59)X=Y⊕Z,
where Y=X∩L1 and Z=X∩L2 and Y,Z are orthogonal with respect to both (·,·)L2 and (·,·). Let P:X→Y and Q:X→Z be orthogonal projections. Therefore, for every u∈X, there is a unique decomposition
(60)u=Pu+Qu
with (Pu,Qu)=0 and
(61)∫RN∇u2dx+∫RNVxu2dx=Pu2-Qu2,u∈X.
Moreover,
(62)μ1PuL22≤Pu2,∀u∈X,μ-1QuL22≤Qu2,∀u∈X.
Therefore,
(63)μ0uL22≤u2,∀u∈X.
We denote
(64)X+≔Z,X-≔Y,u+≔Qu,u-≔Pu.
Let
(65)Φu=-Ju,u∈X.
Then, by (54) and (61), Φ can be written as
(66)Φu=12u+2-u-2-∫RNFx,udx,u∈X.
The derivative of Φ is given by
(67)Φ′u,v=u+,v-u-,v-∫RNfx,uvdx,∀u,v∈X.Let{ek-} be the total orthonormal sequence in X- and let · be the norm defined by (12). The topology generated by · is denoted by τ, and all topological notations related to it will include this symbol.Lemma 10.
Suppose (v) and (f1)–(f3) are satisfied. Then(i)
there existδ>0, R>r>0, and u0∈X+ with u0=1 such that
(68)inf⁡NΦ>sup⁡∂MΦ,
where N,M, and ∂M are defined in (14) and (15);(ii)
there exists aτ-upper sem-continuous functional (see Definition 4) J defined in X such that J(0)<inf⁡NΦ and Φ≤J in X.Proof.
We divide the proof into several steps.
Step  1. We will prove that there exists r>0 such that inf⁡NΦ>0.
From (f1) and (f2), we deduce that, for any ϵ>0, there exists Cϵ>0 such that
(69)Fx,t≤ϵt2+Cϵtp,∀t∈R.
Then by the Sobolev inequality uLp(RN)≤Cu, ∀u∈X, and the definition of Φ (see (66)), there exists C′>0 such that, for any u∈X+,
(70)Φu≥12u2-C′ϵu2-C′Cϵup=12-C′ɛu2-C′Cϵup.
Choose ϵ=1/4C′ in (70) and let r=(8C′Cϵ)-1/(p-2). We then see that, for N={u∈X+∣u=r},
(71)inf⁡NΦ≥r28>0.
Step  2. We will prove that
(72)Φu≤0,∀u∈X-.
Let
(73)F1x,t=Fx,t,t≤ρ,x∈RN0,t>ρ,x∈RN
and F2=F-F1. From (7), we deduce F2≥0 in RN×R. Let
(74)A≔sup⁡t≤ρ,x∈RNfx,tt.
From (9), we have
(75)A<μ0.
Then,
(76)F1x,t≤A2t2,∀x,t∈RN×R.
From (76), (63), (66), and the fact that F2≥0 in RN×R, we deduce that, for any u∈X-,
(77)Φu=-12u2-∫RNF1x,udx-∫RNF2x,udx≤-12u2+A2uL22≤-121-Aμ0u2≤0.
Now, we prove thatΦ(u)→-∞ as u→∞ and u∈X-⊕R+u0. Together with (72), this yields that there exists R>r such that sup⁡∂MΦ≤0<inf⁡NΦ.
Arguing indirectly, assume that, for some sequenceun∈X-⊕R+u0 with un→+∞, there is L>0 such that Φ(un)≥-L for all n. Then, setting wn=un/un, we have wn=1 and, up to a subsequence, wn⇀w, wn-⇀w-∈X- and wn+→w+∈X+. Dividing both sides of Φ(un)≥-L by un2, we obtain
(78)-Lun2≤Φunun2=12wn+2-12wn-2-∫RNFx,unun2dx=12wn+2-12wn-2-∫RNF1x,unun2-∫RNF2x,unun2dx.
By (76) and (63), we have
(79)∫RNF1x,unun2dx≤A2wnL22≤A2μ0wn2=A2μ0wn+2+A2μ0wn-2.
Combining (78) and (79), we obtain
(80)-Lun2≤12+A2μ0wn+2-12-A2μ0wn-2-∫RNF2x,unun2dx.
We first consider the casew≠0. From lim⁡|t|→∞f(x,t)/t=+∞ (see (f2)), we have
(81)liminf⁡t→∞F2x,tt2=+∞.
Note that, for x∈{x∈RN∣w≠0}, we have |un(x)|→+∞. Together with (81), this implies that
(82)liminf⁡n→∞∫x∈RN∣w≠0F2x,unun2wn2dx=+∞.
By F2≥0, we obtain
(83)∫RNF2x,unun2dx=∫RNF2x,unun2wn2dx≥∫x∈RN∣w≠0F2x,unun2wn2dx.
Combining (82) and (83) yields
(84)liminf⁡n→∞12+A2μ0wn+2-12-A2μ0wn-2∫RNF2x,unun2dxhhhhhhhh-12+A2μ0∫RNF2x,unun2dx=-∞.
This contradicts (80), since -L/un2→0 as n→∞.
We then consider the casew=0. In this case, lim⁡n→∞wn+=0. It follows that
(85)lim⁡n→∞wn-=1,
since wn=1 and wn2=wn+2+wn-2. Therefore, the right hand side of (80) is less than -(1/2)(1/2-A/2μ0) when n is sufficiently large. However, as n→∞, the left side of (80) converges to zero. This also induces a contradiction.
Step  3. From (76) and (63), we deduce that, for any u∈X,
(86)Φu=12u+2-12u-2-∫RNF1x,udx-∫RNF2x,udx≤12u+2-12u-2+A2uL22-∫RNF2x,udx≤12u+2-12u-2+A2μ0u2-∫RNF2x,udx=12+A2μ0u+2-12-A2μ0u-2-∫RNF2x,udx.
Let
(87)Ju=12+A2μ0u+2-12-A2μ0u-2-∫RNF2x,udx,u∈X.
Then by A<μ0, F2≥0 and Remark 5, we deduce that J is τ-upper semicontinuous. Moreover, J(0)=0<inf⁡NΦ, and, by (86), Φ≤J in X.
Combining Steps  1–3, we obtain the desired results of this lemma.


## 4. Boundedness of(C¯)c Sequence and Proof of the Main Results

According to Definition6, a sequence {un}⊂X is called a (C¯)c sequence of Φ if
(88)limsup⁡n→∞Φun≤c,lim⁡n→∞1+unΦ′unX′=0.Lemma 11.
Suppose that (v) and (f1)–(f3) are satisfied. Let {un} be a (C¯)c sequence of Φ. Then
(89)lim⁡n→∞∫ϖnun2dx=0,lim⁡n→∞∫ϖnunpdx=0,
where ϖn={x∈RN∣|un(x)|≥M} and p and M are from (f1) and (f3), respectively.Proof.
Letϖ~n+={x∈RN∣un(x)≥M/2} and vn=max⁡{un-M/2,0}. Then
(90)∫RN∇vn2dx=∫ϖ~n+∇un2dx≤∫RN∇un2dx,∫RNvn2dx=∫ϖ~n+vn2dx≤∫ϖ~n+un2dx≤∫RNun2dx.
It follows that vn=O(un). Together with the fact that {un} is a (C¯)c sequence for Φ=-J, this implies that
(91)o1=Φ′un,vn=-J′un,vn.
By (6),
(92)a≔inf⁡t≥M/2,x∈RNfx,tt-V-x>0.
Then, by (55) and (91) and the fact that un≥vn≥0 on ϖ~n+, we obtain
(93)o1=J′un,vn=∫RN∇un∇vn+Vunvndx+∫RNfx,unvndx=∫ϖ~n+∇vn2dx+∫ϖ~n+V+xunvndx+∫ϖ~n+fx,unun-V-unvndx≥∫ϖ~n+∇vn2dx+a∫ϖ~n+vn2dx=∫RN∇vn2dx+a∫RNvn2dx,
where V+=V+V-≥0 in RN. Together with the Sobolev inequality ∫RN‍|∇vn|2dx+a∫RN‍vn2dx≥C(∫RN‍|vn|pdx)2/p, this yields
(94)lim⁡n→∞∫ϖ~n+vn2dx=lim⁡n→∞∫RNvn2dx=0,lim⁡n→∞∫ϖ~n+vnpdx=lim⁡n→∞∫RNvnpdx=0.
Because
(95)ϖn+≔x∈RN∣unx≥M⊂ϖ~n+
and vn≥un/2>0 on ϖn+, we obtain from (94) the result
(96)lim⁡n→∞∫ϖn+un2dx=0,lim⁡n→∞∫ϖn+unpdx=0.
Similarly, we can prove that
(97)lim⁡n→∞∫ϖn-un2dx=0,lim⁡n→∞∫ϖn-unpdx=0,
where ϖn-≔{x∈RN∣-un(x)≥M}. The result of this lemma follows from (96) and (97).Lemma 12.
Suppose that (v) and (f1)–(f3) are satisfied. Let {un} be a (C¯)c sequence of Φ. Then
(98)sup⁡nun<+∞.Proof.
From(1+un)Φ′(un)X′→0, we get that 〈Φ′(un),un±〉=o(1). Then, by (67), we have
(99)un±2=±∫RNfx,unun±dx+o1.
It follows that
(100)un2=∫RNfx,unun+-un-dx+o1.
From (f1) and (f2), we deduce that there exists C2>0 such that
(101)fx,t≤t+C2tp-1,∀t∈R.
Note that u+ and u- are orthogonal with respect to (·,·)L2. Then, by (63), we have
(102)∫RNu+-u-2dx=∫RNu+2dx+∫RNu-2dx≤μ0-1u2,∀u∈X.
Let
(103)D1=sup⁡ρ≤t≤M,x∈RNf2x,tF~x,t
and recall that A=sup⁡|t|≤ρ,x∈RN|f(x,t)/t| (see (74)). Using the Hölder inequality, from (100), (101), and (102), we have
(104)un2=∫x∣un≤ρ+∫x∣ρ<un≤M+∫x∣un>M·fx,unun+-un-dx+o1≤A∫RNun2dx1/2∫RNun+-un-2dx1/2+∫x∣ρ<un≤Mf2x,undx1/2·∫RNun+-un-2dx1/2+∫x∣un>Mun2dx1/2·∫RNun+-un-2dx1/2+C2∫x∣un>Munpdxp-1/p·∫RNun+-un-pdx1/p≤Aμ0-1un2+D11/2μ0-1/2·∫x∣ρ<un≤MF~x,undx1/2un+μ0-1/2∫x∣un>Mun2dx1/2un+C′C2∫x∣un>Munpdxp-1/pun,
where the positive constant C′ is from the Sobolev inequality uLp(RN)≤C′u, ∀u∈X. By Lemma 11, we have
(105)∫x∣unx≥Mun2dx=o1,∫x∣unx≥Munpdx=o1.
Combining (105) with (104) yields
(106)un2≤Aμ0-1un2+D11/2μ0-1/2∫x∣ρ<un≤MF~x,undx1/2un+oun.
Fromsup⁡nΦ(un)≤c and (1+un)Φ′(un)X′→0, we obtain
(107)o1+c≥Φun-12Φ′un,un=∫RNF~x,undx.
Together with (8), this implies
(108)∫x∣ρ<unx≤MF~x,undx=∫x∣ρ<unx≤MF~x,undx≤-∫x∣unx≤ρF~x,undx-∫x∣unx≥MF~x,undx+c+o1≤∫x∣unx≤ρF~x,undx+∫x∣unx≥MF~x,undx+c+o1.
From (f1) and (f2), we deduce that there exists C3>0 such that
(109)F~x,t≤t2+C3tp,∀t∈R,
Let D2=sup⁡t≤ρ,x∈RN(|F~(x,t)|/t2). Combining (109), (105) with (108) yields
(110)∫x∣ρ<unx≤MF~x,undx≤∫x∣unx≤ρF~x,undx+∫x∣unx≥MF~x,undx+c+o1≤D2∫x∣unx≤ρun2dx+∫x∣unx≥Mun2dx+C3∫x∣unx≥Munpdx+c+o1≤D2μ0-1un2+∫x∣unx≥Mun2dx+C3∫x∣unx≥Munpdx+c+o1=D2μ0-1un2+c+o1.
Together with (106), this implies
(111)un2≤Aμ0-1+D11/2D21/2μ0-1un2+Oun.
From (9), we have
(112)Aμ0-1+D11/2D21/2μ0-1<1.
The boundedness of {un} immediately follows from (111) and (112).Proof of Theorem1.
From the proof of Lemma 6.15 in [23], we know that Φ′ is weakly sequentially continuous. Moreover, it is easy to see that sup⁡MΦ<+∞. Then by Lemmas 10 and 12 and Corollary 8, we deduce that there exists a bounded (C¯)c sequence {un} for Φ with c=sup⁡MΦ and inf⁡nun>0. Up to a subsequence, either(i)
lim⁡n→∞sup⁡y∈RN∫B1(y)‍|un|2dx=0, or(ii)
there existϱ>0 and an∈ZN such that ∫B1(an)‍|un|2dx≥ϱ.If (i) occurs, using the Lions lemma (see, e.g., [23, Lemma 1.21]), a similar argument as for the proof of [20, Lemma 3.6] shows that
(113)lim⁡n→∞∫RNfx,unun±dx=0.
Then by (100), we have un→0. This contradicts inf⁡nun>0. Therefore, case (i) cannot occur. As case (ii) therefore occurs, wn=un(·+an) satisfies wn⇀u0≠0. From (1+wn)Φ′(wn)X′=(1+un)Φ′(un)X′→0 and the weakly sequential continuity of Φ′, we have that Φ′(u0)=0. Therefore, u0 is a nontrivial solution of (1). This completes the proof.Proof of Corollary2.
Let(114)fλt=tq-2t-λtp-2t,t∈R.
Since 2<p<q<2*, we deduce that fλ satisfies (f1) and (f2). Because q>p, there exists M>0 such that
(115)inf⁡t≥M/2tq-2-tp-2>max⁡RNV-.
It follows that, for 0<λ≤1, fλ satisfies (6).
Let(116)ρ=qλp1/q-p.
Then, for |t|≥ρ,
(117)Fλt=∫0tfλsds=1qtq-λptp≥0;
that is, Fλ satisfies (7).
Let(118)F~λt=12tfλt-Fλt=12-1qtq-λ12-1ptp.
If |t|≥ρ, then
(119)F~λt=12-1qtq-λ12-1ptp≥q-p2qtq.
This shows that F~λ satisfies (8).
It follows from(120)0≤tq-2-λtp-2≤tq-2ift≥ρ
that
(121)fλ2t=tq-2-λtp-22t2≤t2q-2ift≥ρ.
Let λ be sufficiently small such that M>ρ. Combining (119) with (121) yields
(122)fλ2tF~λt≤2qq-ptq-2≤2qq-pMq-2ifρ≤t≤M.
Moreover, if |t|≤ρ, we have
(123)fλtt=tq-2-λtp-2≤ρq-2+λρp-2,F~λtt2≤12-1qtq-2+λ12-1ptp-2≤12-1qρq-2+λ12-1pρp-2.
Let λ>0 be sufficiently small such that
(124)ρq-2+λρp-2+2qq-pMq-21/2·12-1qρq-2+λ12-1pρp-21/2<μ0.
It follows from (122)–(124) that fλ and F~λ satisfy (9). Therefore, we verified that fλ,Fλ, and F~λ satisfy (6)–(9) if λ>0 is sufficiently small. The result of this corollary immediately follows from Theorem 1.



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*Source: 757153-2015-01-15.xml*