We study the Schrödinger equation: -Δu+Vxu+fx,u=0,u∈H1(RN), where V is 1-periodic and f is 1-periodic in the x-variables; 0 is in a gap of the spectrum of the operator -Δ+V. We prove that, under some new assumptions for f, this equation has a nontrivial solution. Our assumptions for the nonlinearity f are very weak and greatly different from the known assumptions in the literature.

1. Introduction and Statement of Results

In this paper, we consider the following Schrödinger equation:
(1)-Δu+V(x)u+f(x,u)=0,u∈H1(RN),
where N≥1. For V and f, we assume the following.

(v) V∈C(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}.

(f1) f∈C(RN×R) is 1-periodic in xj for j=1,…,N. And there exist constants C>0 and 2<p<2* such that
(3)|f(x,t)|≤C(1+|t|p-1),∀(x,t)∈RN×R,

where
(4)2*:={2N(N-2),N≥3∞,N=1,2.

(f2) The limit limt→0f(x,t)/t=0 holds uniformly for x∈RN. And there exists D>0 such that
(5)infx∈RN,|t|≥Df(x,t)t>maxRNV-,

where V±(x)=max{±V(x),0}, ∀x∈RN.

(f3) For any (x,t)∈RN×R, F~(x,t)≥0, where
(6)F~(x,t):=12tf(x,t)-F(x,t),F(x,t)=∫0tf(x,s)ds.

(f4) There exist 0<κ<D and ν∈(0,μ0) such that, for every (x,t)∈RN×R with |t|<κ,
(7)|f(x,t)|≤ν|t|

and, for every (x,t)∈RN×R with κ≤|t|≤D,
(8)F~(x,t)>0.

Remark 1.

By the definitions of F and F~, it is easy to verify that, for all (x,t)∈RN×(R∖{0}),
(9)∂∂t(F(x,t)t2)=2F~(x,t)t3.
Together with f(x,t)=o(t) as |t|→0 and (f3), this implies that
(10)F(x,t)≥0∀(x,t)∈RN×R.

Remark 2.

There are many functions satisfying (f1)–(f4). We give several examples here.

Example 1.

D=1+μ0/2+e1+maxRNV-, κ=1+μ0/2, ν=μ0/2, and
(11)f(x,t)={0,|t|≤1,tln|t|,|t|>1.

Example 2.

D=3+μ0/2+2maxRNV-, κ=3/2, ν=μ0/2, and
(12)f(x,t)={0,|t|≤1,D(t-1),t>1,D(t+1),t<-1.

Example 3.

D=μ0/2+e1+maxRNV-, κ=ν=μ0/2, and f(x,t)=tln(1+|t|).

A solution u of (1) is called nontrivial if u≢0. Our main results are as follows.

Theorem 3.

Suppose (v) and (f1)–(f4) are satisfied. Then (1) has a nontrivial solution.

Note that

(f2′) the limits limt→0f(x,t)/t=0 and lim|t|→∞(f(x,t)/t)=+∞ hold uniformly for x∈RN.

Implying (f2), we have the following corollary.

Corollary 4.

Suppose (v), (f1), (f2′), (f3), and (f4) are satisfied. Then (1) has a nontrivial solution.

It is easy to verify that the condition

(f4′)F~(x,t)>0, for every (x,t)∈RN×R.

And the assumption that f(x,t)/t→0 as t→0 uniformly for x∈RN imply (f3) and (f4). Therefore, we have the following corollary.

Corollary 5.

Suppose (v), (f1), (f2), and (f4′) are satisfied. Then (1) has a nontrivial solution.

Semilinear Schrödinger equations with periodic coefficients have attracted much attention in recent years due to its numerous applications. One can see [1–24] and the references therein. In [2], the authors used the dual variational method to obtain a nontrivial solution of (1) with f(x,t)=±W(x)|t|p-2t, where W is an asymptotically periodic function. In [20], Troestler and Willem firstly obtained nontrivial solutions for (1) with f being a C1 function satisfying the Ambrosetti-Rabinowitz condition:

there exists α>2 such that, for every u≠0, 0<αG(x,u)≤g(x,u)u, where g(x,u)=-f(x,u), G(x,u)=-F(x,u), and
(13)|∂f(x,u)∂u|≤C(|u|p-2+|u|q-2)

with 2<p<q<2*. Then, in [9], Kryszewski and Szulkin developed some infinite-dimensional linking theorems. Using these theorems, they improved Troestler and Willem’s results and obtained nontrivial solutions for (1) with f only satisfying (f1) and the (AR) condition. These generalized linking theorems were also used by Li and Szulkin to obtain nontrivial solution for (1) under some asymptotically linear assumptions for f (see [11]). In [13] (see also [14]), existence of nontrivial solutions for (1) under (f1) and the (AR) condition was also obtained by Pankov and Pflüger through approximating (1) by a sequence of equations defined in bounded domains. In the celebrated paper [17], Schechter and Zou combined a generalized linking theorem with the monotonicity methods of Jeanjean (see [8]). They obtained a nontrivial solution of (1) when f exhibits the critical growth. A similar approach was applied by Szulkin and Zou to obtain homoclinic orbits of asymptotically linear Hamiltonian systems (see [19]). Moreover, in [5] (see also [6]), Ding and Lee obtained nontrivial solutions for (1) under some new superlinear assumptions on f different from the classical (AR) conditions.

Our assumptions on f are very weak and greatly different from the assumptions mentioned above. In fact, our assumptions (f1)–(f4) do not involve the properties of f at infinity. It may be asymptotically linear growth at infinity, that is, limsup|t|→∞(f(x,t)/t)<+∞, or superlinear growth at infinity as well, that is, liminf|t|→∞(f(x,t)/t)=+∞. Moreover, the assumptions (f1)–(f4) allow f(x,t)≡0 in a neighborhood of t=0 (see Remark 2).

In this paper, we use the generalized linking theorem for a class of parameter-dependent functionals (see [17, Theorem 2.1] or Proposition 8 in the present paper) to obtain a sequence of approximate solutions for (1). Then, we prove that these approximate solutions are bounded in L∞(RN) and H1(RN) (see Lemmas 13 and 14). Finally, using the concentration-compactness principle, we obtain a nontrivial solution of (1).

Notation. Br(a) denotes the 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 ∥u∥H1=(∫RN(|∇u|2+u2)dx)1/2. We use O(h), o(h) to mean |O(h)|≤C|h|, o(h)/|h|→0.

2. Existence of Approximate Solutions for (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)

Under the assumptions (v), (f1), and (f2), the functional
(14)Φ(u)=12∫RN|∇u|2dx+12∫RNV(x)u2dx+∫RNF(x,u)dx
is of class C1 on X:=H1(RN), and the critical points of Φ 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. One can consult [5] or [6] 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
(15)L2=L2(RN)=L++L-
such that S is negative (resp., positive) in L- (resp., in L+). As in [5, Section 2] (see also [6, Chapter 6.2]), let X=D(|S|1/2) be equipped with the inner product
(16)(u,v)=(|S|1/2u,|S|1/2v)L2
and norm ∥u∥=∥|S|1/2u∥L2, where (·,·)L2 denotes the inner product of L2. From (v),
(17)X=H1(RN)
with equivalent norms. Therefore, X continuously embeds in Lq(RN) for all 2≤q≤2N/(N-2) if N≥3 and for all q≥2 if N=1,2. In addition, we have the decomposition
(18)X=X++X-,
where X±=X∩L± is orthogonal with respect to both (·,·)L2 and (·,·). Therefore, for every u∈X, there is a unique decomposition
(19)u=u++u-,u±∈X±
with (u+,u-)=0 and
(20)∫RN|∇u|2dx+∫RNV(x)u2dx=∥u+∥2-∥u-∥2,u∈X.
Moreover,
(21)μ-1∥u-∥L22≤∥u-∥2,∀u∈X,(22)μ1∥u+∥L22≤∥u+∥2,∀u∈X.
The functional Φ defined by (14) can be rewritten as
(23)Φ(u)=12(∥u+∥2-∥u-∥2)+ψ(u),
where
(24)ψ(u)=∫RNF(x,u)dx.

The above variational setting for the functional (14) is standard. One can consult [5] or [6] for more details.

Let {ek±} be the total orthonormal sequence in X±. Let P:X→X-, Q:X→X+ be the orthogonal projections. We define
(25)|||u|||=max{∥Pu∥,∑k=1∞12k+1|(Qu,ek+)|}
on X. The topology generated by |||·||| is denoted by τ, and all topological notation related to it will include this symbol.

Lemma 6.

Suppose that (v) holds. Then

maxRNV-≥μ-1, where μ-1 is defined in (v);

for any C>μ-1, there exists u0∈X- with ∥u0∥=1 such that C∥u0∥L2>1.

Proof.

(a) We apply an indirect argument, and assume by contradiction that
(26)maxRNV-<μ-1.
From assumption (v), -μ-1 is in the essential spectrum of the operator (with domain D(L)=H2(RN)):
(27)L=-Δ+V:L2(RN)⟶L2(RN).
Then, by Weyl’s criterion (see, e.g., [25, Theorem VII.12] or [26, Theorem 7.2]), there exists a sequence {un}⊂H2(RN) with the properties that ∥un∥L2=1, ∀n and ∥-Δun+Vun+μ-1un∥L2→0.

Since μ-1>maxRNV_, we deduce that -V-(x)+μ-1>0 for all x∈RN. Together with the facts that V is a continuous periodic function and V=V+-V-, this implies
(28)infx∈RN(V(x)+μ-1)>0.
It follows that there exists a constant C′>0 such that
(29)∫RN(|∇u|2+(V(x)+μ-1)u2)dx≥C′∥u∥2,∀u∈X.
Note that
(30)∫RN(-Δun+V(x)un+μ-1un)undx=∫RN(|∇un|2+(V(x)+μ-1)un2)dx.
Together with (29) and the fact that ∥-Δun+Vun+μ-1un∥L2→0 and ∥un∥L2=1, this implies ∥un∥→0. It contradicts ∥un∥L2=1, ∀n. Therefore, maxRNV-≥μ-1.

(b) It suffices to prove that
(31)μ-1=C-:=inf{∥u∥2∣u∈X-,∥u∥L2=1}.
From (21), we deduce that μ-1≤C-. From assumption (v), -μ-1 is in the essential spectrum of L. By Weyl’s criterion, there exists {un}⊂H2(RN) such that ∥un∥L2=1 and ∥-Δun+Vun+μ-1un∥L2→0. Multiplying -Δun+Vun+μ-1un by un+ and then integrating it into RN, by (20) and (22), we get that
(32)(μ1+μ-1)∥un+∥L22≤∫RN(|∇un+|2+V(x)(un+)2+μ-1(un+)2)dx=∫RN(-Δun+V(x)un+μ-1un)un+dx⟶0.
It follows that ∥un-∥L2→1. Multiplying -Δun+Vun+μ-1un by un- and then integrating it into RN, we get that
(33)-∥un-∥2+μ-1∥un-∥L22=∫RN(|∇un-|2+V(x)(un-)2+μ-1(un-)2)dx=∫RN(-Δun+Vun+μ-1un)un-dx⟶0.
It implies that μ-1≥C-. This together with μ-1≤C- implies μ-1=C-.

Let R>r>0 and
(34)A:=infx∈RN,|t|≥Df(x,t)t.
From assumption (5), we have A>maxRNV-. Together with the result (a) of Lemma 6, this implies that (1/2)(A+μ-1)>μ-1. Choose
(35)γ∈(μ-1,(A+μ-1)2).
Then by the result (b) of Lemma 6, there exists u0∈X- with ∥u0∥=1 such that
(36)γ∥u0∥L2>1.
Set
(37)N={u∈X-∣∥u∥=r},M={u∈X+⊕R+u0∣∥u∥≤R}.
Then, M is a submanifold of X+⊕R+u0 with boundary
(38)∂M={u∈X-∣∥u∥≤R}∪{u-+tu0∣u-∈X-,t>0,∥u-+tu0∥=R}.

Definition 7.

Let ϕ∈C1(X;R). A sequence {un}⊂X is called a Palais-Smale sequence at level c ((PS)c-sequence for short) for ϕ, if ϕ(un)→c and ∥ϕ′(un)∥X′→0 as n→∞.

The following proposition is proved in [17] (see [17, Theorem 2.1]).

Proposition 8.

Let 0<K<1. The family of C1-functional {Hλ} has the form
(39)Hλ(u)=λI(u)-J(u),u∈X,λ∈[K,1].
Assume

J(u)≥0, ∀u∈X;

|I(u)|+J(u)→+∞ as ∥u∥→+∞;

for all λ∈[K,1], Hλ is τ-sequentially upper semicontinuous; that is, if |||un-u|||→0, then(40)limsupn→∞Hλ(un)≤Hλ(u),

and Hλ′ is weakly sequentially continuous. Moreover, Hλ maps bounded sets to bounded sets;

there exist u0∈X-∖{0} with ∥u0∥=1 and R>r>0 such that, for all λ∈[K,1],(41)infNHλ>sup∂MHλ.

Then there exists E⊂[K,1] such that the Lebesgue measure of [K,1]∖E is zero and, for every λ∈E, there exist cλ and a bounded (PS)cλ-sequence for Hλ, where cλ satisfies
(42)supMHλ≥cλ≥infNHλ.

For 0<K<1 and λ∈[K,1], let
(43)Ψλ(u)=λ2∫RNV-(x)u2dx-(12∫RN(|∇u|2+V+(x)u2)dx+ψ(u)),u∈X.
Then
(44)Ψ1=-Φ
and it is easy to verify that a critical point u of Ψλ is a weak solution of
(45)-Δu+Vλ(x)u+f(x,u)=0,u∈X,
where
(46)Vλ=V+-λV-.

Lemma 9.

Suppose that (v) and (f1)–(f3) hold. Then, there exist 0<K*<1 and E⊂[K*,1] such that the Lebesgue measure of [K*,1]∖E is zero and, for every λ∈E, there exist cλ and a bounded (PS)cλ-sequence for Ψλ, where cλ satisfies
(47)+∞>supλ∈Ecλ≥infλ∈Ecλ>0.

Proof.

For u∈X, let
(48)I(u)=12∫RNV-(x)u2dx,J(u)=12∫RN(|∇u|2+V+(x)u2)dx+ψ(u).
Then, I and J satisfy assumptions (a) and (b) in Proposition 8, and, by (43), Ψλ(u)=λI(u)-J(u).

From (43) and (20), for any u∈X and λ∈[K,1], we have
(49)Ψλ(u)=λ-12∫RNV-(x)u2dx-(12∫RN(|∇u|2+V(x)u2)dx+∫RNF(x,u)dx)=12∥u-∥2-12∥u+∥2-1-λ2∫RNV-(x)u2dx-∫RNF(x,u)dx.
Let u*∈X and {un}⊂X be such that |||un-u*|||→0. It follows that un-→u*-, un+⇀u*+, and un⇀u*. In addition, up to a subsequence, we can assume that un→u* a.e. in RN. Then, we have
(50)∥un-∥2⟶∥u*-∥2,(51)liminfn→∞∫RNV-(x)un2dx≥∫RNV-(x)u*2dx(by Fatou,s lemma),liminfn→∞∥un+∥2≥∥u*+∥2.
By Remark 1, F(x,t)≥0 for all x and t. This together with Fatou’s lemma implies
(52)liminfn→∞∫RNF(x,un)dx≥∫RNF(x,u*)dx.
Then, by (49), we obtain
(53)limsupn→∞Ψλ(un)≤Ψλ(u*).
This implies that Ψλ is τ-sequentially upper semicontinuous.

If un⇀u* in X, then, for any fixed φ∈X, as n→∞,
(54)〈-Ψλ′(un),φ〉=∫RN(∇un∇φ+Vλunφ)dx+∫RNf(x,un)φdx⟶∫RN(∇u*∇φ+Vλu*φ)dx+∫RNf(x,u*)φdx=〈-Ψλ′(u*),φ〉.
This implies that Ψλ′ is weakly sequentially continuous. Moreover, it is easy to see that Ψλ maps bounded sets to bounded sets. Therefore, Ψλ satisfies assumption (c) in Proposition 8.

Finally, we will verify assumption (d) in Proposition 8 for Ψλ.

From assumption (f1) and f(x,t)/t→0 as t→0 uniformly for x∈RN, we deduce that, for any ϵ>0, there exists Cϵ>0 such that
(55)F(x,t)≤ϵt2+Cϵ|t|p,∀(x,t)∈RN×R.
From (49) and (55), we have, for u∈N,
(56)Ψλ(u)≥12∥u∥2-1-λ2∫RNV-(x)u2dx-ϵ∫RNu2dx-Cϵ∫RN|u|pdx.
Then by the Sobolev inequality ∥u∥Lp(RN)≤C∥u∥ and ∥u∥L2≤C∥u∥ (by (21) and (22)), we deduce that there exists a constant C>0 such that
(57)Ψλ(u)≥12∥u∥2-C(1-λ)maxRNV-(x)∥u∥2-ϵC∥u∥2-CCϵ∥u∥p.
Choose 0<K*<1 and ϵ>0 such that C(1-K*)maxRNV-(x)<1/4 and Cϵ=1/8. Then, for every λ∈[K*,1], we have
(58)Ψλ(u)≥18∥u∥2-CCϵ∥u∥p.
Let r>0 be such that rp-2CCϵ=1/16 and β=r2/16. Then, from (58), we deduce that, for N={u∈X-∣∥u∥=r},
(59)infNΨλ≥β,∀λ∈[K*,1].

We will prove that supK*≤λ≤1Ψλ(u)→-∞ as ∥u∥→∞ and u∈X+⊕R+u0. Arguing indirectly, assume that, for some sequences λn∈[K*,1] and un∈X+⊕R+u0 with ∥un∥→+∞, there is L>0 such that Ψλn(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+.

First, we consider the case w≠0. Dividing both sides of (49) by ∥un∥2, we get that
(60)-L∥un∥2≤Ψλn(un)∥un∥2=12∥wn-∥2-12∥wn+∥2-1-λn2∫RNV-(x)wn2dx-∫RNF(x,un)∥un∥2dx.

From (5) and the result (a) of Lemma 6, we deduce that
(61)liminf|t|→∞F(x,t)t2≥A2>12maxRNV-≥12μ-1,
where A is defined by (34). Note that, for x∈{x∈RN∣w≠0}, we have |un(x)|→+∞. This implies that, when n is large enough,
(62)∫{x∈RN∣w≠0}F(x,un)un2wn2dx≥A+μ-14∫{x∈RN∣w≠0}wn2dx.
By (10), we have, when n is large enough,
(63)∫RNF(x,un)∥un∥2dx=∫RNF(x,un)un2wn2dx≥∫{x∈RN∣w≠0}F(x,un)un2wn2dx.
Combining the above two inequalities yields
(64)liminfn→∞(∫RNF(x,un)∥un∥212∥wn-∥2-12∥wn+∥2-1-λn2∫RNV-(x)wn2dx-∫RNF(x,un)∥un∥2dx)≤liminfn→∞(∫RN12∥wn-∥2-12∥wn+∥2-A+μ-14∫{x∈RN∣w≠0}wn2dx∫RN)≤12∥w-∥2-12∥w+∥2-A+μ-14∫RNw2dx≤12∥w-∥2-12∥w+∥2-A+μ-14∥w-∥L22.
We used the inequalities
(65)limn→∞∥wn-∥2=∥w-∥2,liminfn→∞∥wn+∥2≥∥w+∥2,liminfn→∞∫{x∈RN∣w≠0}wn2dx≥∫RNw2dx
in the second inequality of (64).

Since w-=tu0 for some t∈R, by (36), we get that
(66)A+μ-14∥w-∥L22≥A+μ-14γ∥w-∥2.
Note that, by the choice of γ (see (35)), we have ((A+μ-1)/4γ)>1/2. Then by (64) and the fact that w≠0, we have that
(67)liminfn→∞(F(x,un)∥un∥2d12∥wn-∥2-12∥wn+∥2-1-λn2∫RNV-(x)wn2dx-∫RNF(x,un)∥un∥2dx)≤-(A+μ-14γ-12)∥w-∥2-12∥w+∥2<0.
It contradicts (60), since -L/∥un∥2→0 as n→∞.

Second, we consider the case w=0. In this case, limn→∞∥wn-∥=0. It follows that
(68)liminfn→∞∥wn+∥≥1,
since ∥wn∥=1 and wn=wn++wn-. Therefore, the right hand side of (60) is less than -1/4 when n is large enough. However, as n→∞, the left hand side of (60) converges to zero. It induces a contradiction.

Therefore, there exists R>r such that
(69)supλ∈[K*,1]sup∂MΨλ≤0.
This implies that Ψλ satisfies assumption (d) in Proposition 8 if λ∈[K*,1]. Finally, it is easy to see that
(70)supλ∈[K*,1]supMΨλ<+∞.
Then, the results of this lemma follow immediately from Proposition 8.

Lemma 10.

Suppose that (v) and (f1)–(f3) are satisfied. Let λ∈[K*,1] be fixed, where K* is the constant in Lemma 9. If {vn} is a bounded (PS)c-sequence for Ψλ with c≠0, then, for every n∈N, there exists an∈ZN such that, up to a subsequence, un:=vn(·+an) satisfies
(71)un⇀uλ≠0,Ψλ(uλ)≤c,Ψλ′(uλ)=0.

Proof.

The proof of this lemma is inspired by the proof of Lemma 3.7 in [19]. Because {vn} is a bounded sequence in X, up to a subsequence, either

limn→∞supy∈RN∫B1(y)|vn|2dx=0 or

there exist ϱ>0 and an∈ZN such that ∫B1(an)|vn|2dx≥ϱ.

If (a) occurs, using the Lions lemma (see, e.g., [21, Lemma 1.21]), a similar argument as for the proof of [19, Lemma 3.6] shows that
(72)limn→∞∫RNF(x,vn)dx=0,limn→∞∫RNf(x,vn)vn±dx=0.
It follows that
(73)limn→∞∫RN(2F(x,vn)-f(x,vn)vn)dx=0.
On the other hand, as {vn} is a (PS)c-sequence of Ψλ, we have 〈Ψλ′(vn),vn〉→0 and Ψλ(vn)→c≠0. It follows that
(74)∫RN(f(x,vn)vn-2F(x,vn))dx=2Ψλ(vn)-〈Ψλ′(vn),vn〉⟶2c≠0,n⟶∞.
This contradicts (73). Therefore, case (a) cannot occur.

If case (b) occurs, let un=vn(·+an). For every n,
(75)∫B1(0)|un|2dx≥ϱ.
Because V and F(x,t) are 1-periodic in every xj, {un} is still bounded in X,
(76)limn→∞Ψλ(un)≤c,Ψλ′(un)⇀0,n⟶∞.
Up to a subsequence, we assume that un⇀uλ in X as n→∞. Since un→uλ in Lloc2(RN), it follows from (75) that uλ≠0. Recall that Ψλ′(un) is weakly sequentially continuous. Therefore, Ψλ′(un)⇀Ψλ′(uλ) and, by (76), Ψλ′(uλ)=0.

Finally, by (f3) and Fatou’s lemma
(77)c=limn→∞(Ψλ(un)-12〈Ψλ′(un),un〉)=limn→∞∫RNF~(x,un)≥∫RNF~(x,uλ)=Ψλ(uλ).

Lemma 11.

There exist 0<K**<1 and η>0 such that, for any λ∈[K**,1], if u≠0 satisfies Ψλ′(u)=0, then ∥u∥≥η.

Proof.

We adapt the arguments of Yang [23, p. 2626] and Liu [12, Lemma 2.2]. Note that, by (f1) and (f2), for any ϵ>0, there exists Cϵ>0 such that
(78)|f(x,t)|≤ϵ|t|+Cϵ|t|p-1.
Let u≠0 be a critical point of Ψλ. Then u is a solution of
(79)-Δu+Vλu+f(x,u)=0,u∈X.
Multiplying both sides of this equation by u±, respectively, and then integrating into RN, we get that
(80)0=±∥u±∥2+(1-λ)∫RNV-(x)unu±dx+∫RNf(x,u)u±dx.
It follows that
(81)∥u±∥2=∓(1-λ)∫RNV-(x)uu±dx∓∫RNf(x,u)u±dx≤(1-λ)supRNV-∫RN|u|·|u±|dx+ϵ∫RN|u|·|u±|dx+Cϵ∫RN|u|p-1|u±|dx≤C1((1-λ)+ϵ)∥u∥·∥u±∥+C2∥u∥p-1∥u±∥,
where C1 and C2 are positive constants related to the Sobolev inequalities and supRNV-. From the above two inequalities, we obtain
(82)∥u∥2=∥u+∥2+∥u-∥2≤2C1((1-λ)+ϵ)∥u∥2+2C2∥u∥p.
Because p>2, this implies that ∥u∥≥η for some η>0 if ϵ>0 and 1-K**>0 are small enough and λ∈[K**,1]. The desired result follows.

Let K=max{K*,K**}, where K* and K** are the constants that appeared in Lemmas 9 and 11, respectively. Combining Lemmas 9–11, we obtain the following lemma.

Lemma 12.

Suppose (v) and (f1)–(f3) are satisfied. Then, there exist η>0, {λn}⊂[K,1], and {un}⊂X such that λn→1,
(83)supnΨλn(un)<+∞,∥un∥≥η,Ψλn′(un)=0.

3. A Priori Bound of Approximate Solutions and Proof of the Main Theorem

In this section, we give a priori bound for the sequence of approximate solutions {un} obtained in Lemma 12. We then give the proofs of Theorem 3.

Lemma 13.

Suppose (v) and (f1)–(f3) are satisfied. Let {un} be the sequence obtained in Lemma 12. Then, {un}⊂L∞(RN) and
(84)supn∥un∥L∞(RN)≤D.

Proof.

From Ψλn′(un)=0, we deduce that un is a weak solution of (45) with λ=λn; that is,
(85)-Δun+Vλn(x)un+f(x,un)=0inRN.
By assumption (f1) and the bootstrap argument of elliptic equations, we deduce that un∈L∞(RN).

Multiplying both sides of (85) by vn=(un-D)+:=max{un-D,0} and integrating into RN, we get that
(86)∫RN|∇vn|2dx+∫un≥D(Vλn(x)un+f(x,un))vndx=0.
Recall that Vλn=V+-λnV- and λn≤1. Then by (5), we get that
(87)∫un≥D(Vλn(x)un+f(x,un))vndx=∫un≥D(Vλn(x)+f(x,un)un)unvndx≥0.
This together with (86) yields vn=0. It follows that un(x)≤D on RN.

Similarly, multiplying both sides of (85) by wn=(un+D)-:=max{-(un+D),0} and integrating into RN, we can get that un≥-D on RN. Therefore, for all n,∥un∥L∞(RN)≤D.

Lemma 14.

Suppose that (v), (f1), (f2), (f3), and (f4) are satisfied. Let {un} be the sequence obtained in Lemma 12. Then
(88)0<infn∥un∥≤supn∥un∥<+∞.

Proof.

As Ψλn′(un)=0 and un≠0, Lemma 11 implies that infn∥un∥>0.

To prove supn∥un∥<+∞, we apply an indirect argument and assume by contradiction that ∥un∥→+∞.

Since Ψλn′(un)=0, by (81), we get that
(89)∥un±∥2=∓(1-λn)∫RNV-(x)unun±dx∓∫RNf(x,un)un±dx=∓∫RNf(x,un)un±dx+(1-λn)O(∥un∥2).
It follows that
(90)∥un∥2+∫RNf(x,un)(un+-un-)dx=∥un+∥2+∥un-∥2+∫RNf(x,un)(un+-un-)dx=(1-λn)O(∥un∥2).
Set wn=un/∥un∥. Then, by (90),
(91)∥un∥2(1+∫RNf(x,un)un(wn+-wn-)wndx)=(1-λn)O(∥un∥2).
Then, by λn→1 as n→∞, we have that
(92)∫RNf(x,un)un(wn+-wn-)wndx⟶-1,n⟶∞.

From Lemma 12,
(93)C0:=supnΨλn(un)<+∞.
Then, by Ψλn′(un)=0, we obtain
(94)2C0≥2Ψλn(un)-〈Ψλn′(un),un〉=2∫RNF~(x,un)dx.
From (f3), we have
(95)2C0≥2∫RNF~(x,un)dx≥2∫{x∣D≥|un(x)|≥κ}F~(x,un)dx,
where κ is the constant in (f4). As the continuous function F~ is 1-periodic in every xj variable, we deduce from (8) that there exists a constant C′>0 such that
(96)F~(x,t)≥C′t2,forevery(x,t)∈RN×Rwithκ≤|t|≤D.
Combining (95) and (96) leads to
(97)C0≥C′∫{x∣D≥|un(x)|≥κ}un2dx.
Dividing both sides of this inequality by ∥un∥2 and sending n→∞, we obtain
(98)limn→∞∫{x∣D≥|un(x)|≥κ}wn2dx=0.

From (7), (21), and (22), we have that
(99)∫{x∣|un(x)|<κ}|f(x,un)un(wn+-wn-)wn|dx≤ν∫{x∣|un(x)|<κ}|(wn+-wn-)wn|dx≤ν∫RN|(wn+-wn-)wn|dx≤ν∥wn∥L22≤νμ0∥wn∥2=νμ0<1,
where μ0 is the constant defined in (v).

Since f∈C(RN×R) and limt→0f(x,t)/t=0, we deduce that there exists C>0 such that, for every (x,t)∈RN×R with |t|≤D,
(100)|f(x,t)|≤C|t|.
This together with (98) gives
(101)∫{x∣D≥|un(x)|≥κ}|f(x,un)un(wn+-wn-)wn|dx≤C∫{x∣D≥|un(x)|≥κ}|(wn+-wn-)wn|dx≤C∥wn+-wn-∥L2(∫{x∣D≥|un(x)|≥κ}wn2dx)1/2≤2C∥wn∥L2(∫{x∣D≥|un(x)|≥κ}wn2dx)1/2⟶0,n⟶∞.
Combining (99) and (101) yields
(102)limsupn→∞∫RN|f(x,un)un(wn+-wn-)wn|dx≤limsupn→∞∫{x∣|un(x)|<κ}|f(x,un)un(wn+-wn-)wn|dx+limsupn→∞∫{x∣D≥|un(x)|≥κ}|f(x,un)un(wn+-wn-)wn|dx<1.
This contradicts (92). Therefore, {un} is bounded in X.

Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>.

Let {un} be the sequence obtained in Lemma 12. From Lemma 14, {un} is bounded in X. Therefore, up to a subsequence, either

limn→∞supy∈RN∫B1(y)|un|2dx=0 or

there exist ϱ>0 and yn∈ZN such that ∫B1(yn)|un|2dx≥ϱ.

According to (72), if case (a) occurs,
(103)limn→∞∫RNf(x,un)un±dx=0.
Then, by (81) and λn→1, we have
(104)∥un±∥2=∓(1-λn)∫RNV-(x)unun±dx∓∫RNf(x,un)un±dx≤C(1-λn)∥un∥L22+|∫RNf(x,un)un±dx|⟶0.
This contradicts infn∥un∥>0 (see (88)). Therefore, case (a) cannot occur. As case (b) therefore occurs, wn=un(·+yn) satisfies wn⇀u0≠0. From (14) and (43), we have that
(105)Ψλ(u)=-Φ(u)+λ-12∫RNV-u2dx,∀u∈X.
It follows that
(106)〈Ψλ′(u),φ〉=-〈Φ′(u),φ〉+(λ-1)∫RNV-uφdx,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀u,φ∈X.
By Ψλn′(un)=0 (see Lemma 12), we have Ψ′λn(wn)=0. From (106), we have that, for any φ∈X,
(107)〈Ψ′λn(wn),φ〉=-〈Φ′(wn),φ〉+(λn-1)×∫RNV-(x)wnφdx.
Together with Ψλn′(wn)=0 and λn→1, this yields
(108)〈Φ′(wn),φ〉⟶0,∀φ∈X.
Finally, by wn⇀u0≠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.
Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions on the paper. Shaowei Chen was supported by Science Foundation of Huaqiao University and Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY119).

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