An Improved Fountain Theorem and Its Application

Long-Jiang Gu; Huan-Song Zhou

doi:10.1515/ans-2016-6007

Article Open Access

An Improved Fountain Theorem and Its Application

Long-Jiang Gu and Huan-Song Zhou

Published/Copyright: December 21, 2016

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From the journal Advanced Nonlinear Studies Volume 17 Issue 4

Abstract

The main aim of the paper is to prove a fountain theorem without assuming the τ-upper semi-continuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

Keywords: Fountain Theorem; Strongly Indefinite Functional; Elliptic Equation; Sign-Changing Potentials; Infinitely Many Solutions

MSC 2010: 35J20; 35J60; 35Q55; 58E05

1 Introduction

Since the pioneering works of Bartsch and Willem [2, 4] (see also [21]), variant fountain theorems are established and which have been used to study the existence of infinitely many solutions for various elliptic problems, see, e.g., [2, 4, 3, 5, 6, 12, 22] and the references therein. In order to investigate infinitely many critical points of strongly indefinite functionals, Batkam and Colin [5] established a generalized fountain theorem based on the so-called τ-topology, introduced by Kryszewski and Szulkin [13]. In order to recall the fountain theorem that was proved in [5], we introduce some notations and definitions which are also often used in the following sections of the paper.

Let X be a separable Hilbert space and Y⊂X be a closed subspace of X endowed with inner product 〈⋅,⋅〉 and norm ∥⋅∥. Let

X = Y ⊕ Z , with Z = Y ⊥ , Y = ⊕ j = 0 ∞ ℝ ⁢ e j ¯ and Z = ⊕ j = 0 ∞ ℝ ⁢ f j ¯ ,

where {ej}j≥0 and {fj}j≥0 are orthonormal bases of Y and Z, respectively. Moreover, we define

(1.1) Y k := Y ⊕ ( ⊕ j = 0 k ℝ ⁢ f j ) and Z k := ⊕ j = k ∞ ℝ ⁢ f j ¯ ,

and let P:X→Y, Q:X→Z and Pk:X→Yk-1, Qk:X→Zk be the orthogonal projections. The τ-topology on X=Y⊕Z introduced in [13] is the topology associated to the following norm:

(1.2) ∥ u ∥ τ := max { ∑ j = 0 ∞ 1 2 j + 1 | 〈 P u , e j 〉 | , ∥ Q u ∥ } for u ∈ X .

By the above definition, we see that

(1.3) ∥ u ∥ τ ≤ max ⁡ { ∥ P ⁢ u ∥ , ∥ Q ⁢ u ∥ } ≤ ∥ u ∥ for ⁢ u ∈ X .

Furthermore, it follows from [13] and the appendix of [17] that, if {un}⊂X is bounded, then

(1.4) u n → 𝑛 u ⁢ in ⁢ τ ⁢ -topology ⇔ P ⁢ u n ⇀ 𝑛 P ⁢ u ⁢ and ⁢ Q ⁢ u n → 𝑛 Q ⁢ u .

Remark 1.1.

Let

e ~ j = { f j , j = 0 , 1 , … , k - 1 , e j - k , j ≥ k ,

and define the following norm:

(1.5) ∥ u ∥ τ k = max { ∑ j = 0 ∞ 1 2 j + 1 | 〈 P k u , e ~ j 〉 | , ∥ Q k u ∥ } .

Then ∥⋅∥τk and ∥⋅∥τ are equivalent for all k≥1. In fact, it is enough to show that ∥⋅∥τ1 and ∥⋅∥τ are equivalent. By (1.2),

∥ u ∥ τ 2 ≤ ( ∑ j = 0 ∞ 1 2 j + 1 | 〈 u , e j 〉 | ) 2 + ∑ j = 0 ∞ | 〈 u , f j 〉 | 2

= ( ∑ j = 0 ∞ 1 2 j + 1 | 〈 u , e j 〉 | ) 2 + | 〈 u , f 0 〉 | 2 + ∑ j = 1 ∞ | 〈 u , f j 〉 | 2

≤ 4 ( 1 2 | 〈 u , f 0 〉 | + ∑ j = 1 ∞ 1 2 j + 1 | 〈 u , e j - 1 〉 | ) 2 + 4 ∑ j = 1 ∞ | 〈 u , f j 〉 | 2

≤ 8 ⁢ ∥ u ∥ τ 1 2 .

On the other hand,

∥ u ∥ τ 1 ≤ 1 2 | 〈 u , f 0 〉 | + ∑ j = 0 ∞ 1 2 j + 2 | 〈 u , e j 〉 | + ( ∑ j = 1 ∞ | 〈 u , f j 〉 | 2 ) 1 2

≤ ∑ j = 0 ∞ 1 2 j + 1 | 〈 u , e j 〉 | + 2 1 2 ( | 〈 u , f 0 〉 | 2 + ∑ j = 1 ∞ | 〈 u , f j 〉 | 2 ) 1 2

≤ 2 3 2 ⁢ ∥ u ∥ τ .

Throughout the paper, for rk>0 and ρk>0, we always set

(1.6) B k := { u ∈ Y k : ∥ u ∥ < ρ k } and N k := { u ∈ Z k : ∥ u ∥ = r k } .

Since X is a Hilbert space, we have that if φ∈C1⁢(X,ℝ), then ∇⁡φ is given by the formula

〈 ∇ ⁡ φ ⁢ ( u ) , v 〉 = φ ′ ⁢ ( u ) ⁢ v for all ⁢ v ∈ X .

With the above notations, for an even functional, the fountain theorem proved in [5] can be stated as follows.

Theorem 1.1 ([5, Corollary 13]).

Let φ∈C1⁢(X,R) be an even functional satisfying the following:

∇⁡φ is weakly sequentially continuous, i.e., for any v∈X, φ′⁢(un)⁢v→𝑛φ′⁢(u)⁢v if un⇀𝑛u weakly in X.
φ is τ -upper semi-continuous, i.e., for any c∈ℝ, the set φc:={u∈X:φ⁢(u)≥c} is τ -closed.
For any c>0, φ satisfies the (PS)c condition, i.e., any sequence {un}⊂X with

φ ( u n ) → 𝑛 c 𝑎𝑛𝑑 φ ′ ( u n ) → 𝑛 0 in X ′ (the dual space of X )

has a convergent subsequence.

Additionally, if there exist ρk>rk>0 such that

(A1) d k := sup u ∈ Y k , ∥ u ∥ ≤ ρ k ⁡ φ ⁢ ( u ) < ∞ ,

(A2) a k := sup u ∈ Y k , ∥ u ∥ = ρ k ⁡ φ ⁢ ( u ) ≤ 0 ,

(A3) b k := inf u ∈ Z k , ∥ u ∥ = r k ⁡ φ ⁢ ( u ) → ∞ as ⁢ k → ∞ ,

then φ has an unbounded sequence of critical values.

The τ-upper semi-continuity was proposed in [13] for showing a generalized linking theorem. Similar to [13], this condition is also required in Theorem 1.1 (see also, e.g., [15]), which is mainly used to construct a suitable vector field. Theorem 1.1 can be used to deal with some strongly indefinite elliptic problems, but the τ-upper semi-continuity assumption requires that the primitive functions of the nonlinearities of the elliptic problems should be positive, see in [5, condition (f4)]. It is natural to ask what would happen if the nonlinear terms of an elliptic problem change sign and lose the positivity condition (f4). So, the main aim of the paper is to establish a variant fountain theorem without assuming the τ-upper semi-continuity, and then we may answer the above question, see Theorems 1.2 and 1.3. Our proofs are motivated by the papers [5, 7, 16]. We mention that if the τ-upper semi-continuity of φ is removed, then several steps in the proof of [5, Theorem 1.1] seem not working such as

We cannot construct the pseudogradient vector field as in [5] since the set φ-1⁢(-∞,c) may not be τ-open now. This difficulty is overcome in this paper by using some ideas from [7].
To the authors’ knowledge, the intersection lemma used in [5] is no longer applicable since the descending flow in our paper has different behavior from that of [5]. In this paper, we use the intersection lemma given in [16] instead.
We cannot make an explicit mini-max characterization on the critical values of φ because of the lack of τ-upper semi-continuity. So, it is hopeless to try getting infinitely many different critical points of φ by comparing their critical values as in [5] or [2]. In this paper, we get infinitely many different critical points {un} of φ by comparing their norm ∥un∥ and proving ∥un∥→𝑛+∞.

Now, we give our improved fountain theorem:

Theorem 1.2.

Let φ∈C1⁢(X,R) be an even functional satisfying the (PS)c condition (i.e., any sequence {un}⊂X with supn⁡φ⁢(un)≤c and φ′⁢(un)→𝑛0 has a convergent subsequence) and let ∇⁡φ be weakly sequentially continuous. For any k∈N, if there exists ρk>rk>0 such that, in addition to the above assumptions (A1) and (A3), we have

(A2’) a k := sup u ∈ Y k , ∥ u ∥ = ρ k ⁡ φ ⁢ ( u ) < inf u ∈ Z k , ∥ u ∥ ≤ r k ⁡ φ ⁢ ( u ) ,

(A4) sup ∥ u ∥ τ < δ ⁡ φ ⁢ ( u ) ≤ C δ < ∞ for any ⁢ δ > 0 ,

then φ has a sequence of critical points {ukm} such that limm→∞∥ukm∥→∞.

Remark 1.2.

In Theorem 1.2, the τ-upper semi-continuity is not assumed. But, we replaced condition (A2) in Theorem 1.1 by (A2’) and added a new assumption (A4). However, the conditions (A2’) and (A4) are easily verified in the applications, see, e.g., the proof of Theorem 1.3.

With Theorem 1.2 we may study the existence of infinitely many solutions for the following Schrödinger equation with strongly indefinite linear part and sign-changing nonlinear term:

(1.7) - Δ ⁢ u + V ⁢ ( x ) ⁢ u = g ⁢ ( x ) ⁢ | u | q - 2 ⁢ u + h ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , u ∈ H 1 ⁢ ( ℝ N ) , N ≥ 3 ,

where

1 < q < p p - 1 < 2 < p < 2 ∗ and 2 ∗ = 2 ⁢ N N - 2 ,

and V⁢(x), g⁢(x) and h⁢(x) are functions satisfying the following conditions:

V⁢(x)∈∩⁡C⁢(ℝN,ℝ)L∞, and 0 lies in a spectrum gap of the operator -Δ+V.
g∈∩⁡Lq0⁢(ℝN)L∞⁢(ℝN) with q0=2⁢N2⁢N-q⁢N+2⁢q.
h∈∩⁡Lp0⁢(ℝN)L∞⁢(ℝN) with p0=2⁢N2⁢N-p⁢N+2⁢p and h⁢(x)>0 a.e. in ℝN.

Since 0 lies in a gap of the spectrum of -Δ+V, problem (1.7) may be strongly indefinite. The nonlinearity in (1.7) has a super-linear part and a sub-linear part, which is usually called concave-convex nonlinearity. Some well-known results corresponding to concave-convex nonlinearities can be found in [1, 4] and the references therein. By (H2), we see that the weight function g⁢(x) may change sign, so, the variational functional of (1.7) does not satisfy the τ-upper semi-continuous assumption.

We mention that there are some papers on the existence of solutions for Schrödinger equations with both sign-changing potential V⁢(x) and indefinite nonlinearities, see, e.g., [9, 8, 10, 14, 19, 20], etc. But the problems discussed in [9, 8, 10, 19, 20] do not have strongly indefinite structure, and in [14] the potential V⁢(x) has to be periodic and the weight functions in the nonlinear term must satisfy some additional conditions. It appears that there are no results for problem (1.7) under conditions (H1)–(H3). Here we prove the following theorem.

Theorem 1.3.

Assume that conditions (H1)–(H3) hold. Then problem (1.7) has a sequence of nontrival solutions {un}⊂H1⁢(RN) with ∥un∥H1→∞ as n→∞.

2 Proof of Our Fountain Theorem

In this section, we are going to prove our fountain theorem, that is, Theorem 1.2. For doing this, some lemmas are required.

Lemma 2.1.

Let Yk and Zk be defined in (1.1). Let also φ∈C1⁢(X,R) be an even functional such that ∇⁡φ is weakly sequentially continuous. Assume that there exist k∈N and ρk>rk>0 such that conditions (A1) and (A2’) are satisfied and

(2.1) b k := inf u ∈ Z k , ∥ u ∥ = r k ⁡ φ ⁢ ( u ) > sup ∥ u ∥ τ < δ ⁡ φ ⁢ ( u ) for some ⁢ δ > 0 .

Then there exists a sequence {unk}⊂φdk+1:={u∈X:φ⁢(u)≤dk+1} such that

inf n ∥ u n k ∥ τ ≥ δ 2 𝑎𝑛𝑑 φ ′ ( u n k ) → 𝑛 0 in X ′ (the dual space of X ).

In order to prove Lemma 2.1, we need the following deformation lemma.

Lemma 2.2.

Under the assumptions of Lemma 2.1, let

(2.2) E = ∩ ⁡ φ d k + 1 { u ∈ X : ∥ u ∥ τ ≥ δ 2 } ,

with φdk+1:={u∈X:φ⁢(u)≤dk+1}, and δ given in (2.1). If there exists ϵ∈(0,12), with

(2.3) 0 < ϵ < b k - max ⁡ { a k , sup ∥ u ∥ τ < δ ⁡ φ ⁢ ( u ) } ,

such that ∥φ′⁢(u)∥>ϵ for any u∈E, then there exist T>0 and a map η⁢(t,u)∈C⁢([0,T]×Bk,X), with Bk given by (1.6), such that the following hold:

η⁢(0,u)=u and η⁢(t,-u)=-η⁢(t,u) for any u∈Bk and t∈[0,T].
φ⁢(η⁢(t,u)) is non-increasing in t∈[0,T] for fixed u∈Bk.
η is τ -continuous (i.e., η⁢(tm,um)→𝑚η⁢(t,u) in τ -topology, if tm→𝑚t and um→𝑚u in τ -topology) and η⁢(t,⋅):Bk→η⁢(t,Bk) is a τ -homeomorphism for any t∈[0,T].
η⁢(T,Bk)⊂φbk-ϵ.
For any (t,u)∈[0,T]×Bk, there exists a neighborhood W(t,u) of (t,u) in the |⋅|×τ-topology such that {v-η⁢(s,v):(s,v)∈∩⁡W(t,u)([0,T]×Bk)} is contained in a finite-dimensional subspace of X.

Proof.

For ϵ>0 given in (2.3), let

(2.4) B R = { u ∈ X : ∥ u ∥ ≤ R } , where ⁢ R = 2 ⁢ ( d k - b k + 2 ⁢ ϵ ) ϵ + ρ k + δ .

Firstly, we claim that there exists a vector field χ:φdk+1→X such that

χ is odd with ∥χ⁢(u)∥≤2ϵ and 〈∇⁡φ⁢(u),χ⁢(u)〉≤0 for any u∈φdk+ϵ.
χ⁢(u) is locally Lipschitz continuous and τ-locally Lipschitz τ-continuous on φdk+ϵ.
〈∇⁡φ⁢(u),χ⁢(u)〉<-1 for any u∈∩⁡φ-1⁢[bk-ϵ,dk+ϵ)BR.
For any u∈𝒲 (𝒲 is given by (2.9)), there exists a τ-open neighborhood Uu∈𝒩 of u such that χ⁢(Uu) is contained in a finite-dimensional subspace of X.

In fact, by our assumption, ∥φ′⁢(u)∥>ϵ for any u∈E, and we may define

ω ⁢ ( u ) = 2 ⁢ ∇ ⁡ φ ⁢ ( u ) ∥ ∇ ⁡ φ ⁢ ( u ) ∥ 2 for ⁢ u ∈ ∩ ⁡ E B R .

Then there exists a τ-neighborhood Vu⊂X of u such that

(2.5) 〈 ∇ ⁡ φ ⁢ ( v ) , ω ⁢ ( u ) 〉 > 1 for any ⁢ v ∈ ∩ ⁡ V u B R .

Otherwise, if such Vu does not exist, then there exists a sequence {vn}⊂BR such that

v n → 𝜏 u and lim n → ∞ ⁡ 〈 ∇ ⁡ φ ⁢ ( v n ) , ω ⁢ ( u ) 〉 ≤ 1 .

By (1.4) we have vn⇀u weakly in X, and this leads to a contradiction, since ∇⁡φ is weakly continuous and 〈∇⁡φ⁢(u),ω⁢(u)〉=2.

Note that BR is τ-closed [7], thus X∖BR is τ-open, and 𝒩=∪⁡{Vu:u∈∩⁡EBR}{X∖BR} forms a τ-open covering of E.

Since 𝒩 is metric, hence paracompact, there exists a local finite τ-open covering ℳ={Mi:i∈Λ} (where Λ is an index set) of E finer than 𝒩. If Mi⊂Vui for some ui∈E, then we choose ωi=ω⁢(ui), and if Mi⊂X∖BR, then we choose ωi=0. Let {λi⁢(u):i∈Λ} be a τ-Lipschitz continuous partition of unity subordinated to ℳ and let

ξ ⁢ ( u ) = ∑ i ∈ Λ λ i ⁢ ( u ) ⁢ ω i , u ∈ 𝒩 .

Since the τ-open covering ℳ of 𝒩 is local finite, each u∈𝒩 belongs to finitely many sets Mi. Therefore, for every u∈𝒩, the sum ξ⁢(u) is only a finite sum. It follows that, for any u∈𝒩, there exists a τ-open neighborhood Uu∈𝒩 of u such that ξ⁢(Uu) is contained in a finite-dimensional subspace of X. Then, by the equivalence of norms in a finite-dimensional vector space, we know that there exists C>0 such that

(2.6) ∥ ξ ⁢ ( v ) - ξ ⁢ ( w ) ∥ ≤ C ⁢ ∥ ξ ⁢ ( v ) - ξ ⁢ ( w ) ∥ τ for all ⁢ v , w ∈ U u .

On the other hand, by the τ-Lipschitz continuity of λi and (1.3), we have that there exists a constant Lu>0 such that

(2.7) ∥ ξ ⁢ ( v ) - ξ ⁢ ( w ) ∥ τ ≤ L u ⁢ ∥ v - w ∥ τ ≤ L u ⁢ ∥ v - w ∥ for all ⁢ v , w ∈ U u .

Then, from (2.6) and (2.7) we know that ξ⁢(u) is locally Lipschitz continuous and τ-locally Lipschitz τ-continuous. Moreover, by (2.5) and the property of λi, we also have that

〈 ∇ ⁡ φ ⁢ ( u ) , ξ ⁢ ( u ) 〉 > 1 and ∥ ξ ⁢ ( u ) ∥ < 2 ϵ for any ⁢ u ∈ ∩ ⁡ E B R .

Since φ is even, 𝒩 is symmetric, and we define ξ~⁢(u):=ξ⁢(u)-ξ⁢(-u)2 for u∈𝒩, with ξ~⁢(u) being odd. For δ>0 given by (2.1), let θ∈C∞⁢(ℝ,[0,1]) be such that

θ ⁢ ( t ) = { 0 , 0 ≤ t ≤ 2 ⁢ δ 3 , 1 , t ≥ δ .

Define the vector field χ:𝒩→X by

(2.8) χ ⁢ ( u ) = { - θ ⁢ ( ∥ u ∥ τ ) ⁢ ξ ~ ⁢ ( u ) , u ∈ 𝒩 , 0 , ∥ u ∥ τ ≤ 2 3 ⁢ δ .

It is easy to see that χ is an odd vector field which is well defined on

(2.9) 𝒲 := ∪ ⁡ 𝒩 { u ∈ X : ∥ u ∥ τ < δ } ,

satisfying

(2.10) ∥ χ ⁢ ( u ) ∥ ≤ 2 ϵ and 〈 ∇ ⁡ φ ⁢ ( u ) , χ ⁢ ( u ) 〉 ≤ 0 for any ⁢ u ∈ 𝒲 .

Since 0<ϵ<12, we have that 𝒲 covers ∪⁡φdk+ϵ(X∖BR), and this shows (a). By the construction of χ⁢(u), we know that χ⁢(u) is locally Lipschitz continuous and τ-locally Lipschitz τ-continuous on φdk+ϵ, and thus (b) is proved.

Moreover, by the choice of ϵ in (2.3), we have sup∥u∥τ≤δ⁡φ⁢(u)<bk-ϵ, i.e., {u∈X:∥u∥τ≤δ}⊂φbk-ϵ. So, by (2.8),

(2.11) 〈 ∇ ⁡ φ ⁢ ( u ) , χ ⁢ ( u ) 〉 < - 1 for any ⁢ u ∈ ∩ ⁡ φ - 1 ⁢ [ b k - ϵ , d k + ϵ ] B R ,

which implies (c). Then, by the definition of χ⁢(u), i.e., (2.8), and the properties of ξ⁢(u), we see that (d) holds. So, the claim is proved.

Next, we turn to proving (i)–(v) of the lemma. For this purpose, we construct a map η through the following Cauchy problem:

(2.12) d ⁢ η d ⁢ t = χ ⁢ ( η ) , η ⁢ ( 0 , u ) = u ∈ 𝒲 .

By the standard theory of ordinary differential equations in Banach spaces, we know that the initial problem has a unique solution η⁢(t,u) on [0,∞). Furthermore, a similar argument as the one in the proof of [21, Lemma 6.8] yields that η is τ-continuous. Moreover, η⁢(t,⋅):Bk→η⁢(t,Bk) is a τ-homeomorphism for any t∈[0,T]. So, part (iii) is proved.

Let Bk and BR be given by (1.6) and (2.4). Taking T=dk-bk+2⁢ϵ, we have

{ η ⁢ ( t , u ) : 0 ≤ t ≤ T , u ∈ B k } ⊂ B R .

Indeed, from (2.12) it follows that

η ⁢ ( t , u ) = u + ∫ 0 t χ ⁢ ( η ⁢ ( s , u ) ) ⁢ 𝑑 s for ⁢ u ∈ B k .

By the definition of dk (see condition (A1)), we know that Bk⊂𝒲. Then, by (2.3), (2.4) and (2.10), for any u∈Bk and t∈[0,T], we have

∥ η ( t , u ) ∥ ≤ ∥ u ∥ + ∫ 0 t ∥ χ ( η ( s , u ) ) ∥ d s ≤ ∥ u ∥ + ∫ 0 t 2 ϵ d s ≤ ∥ u ∥ + 2 ⁢ T ϵ ≤ R .

So, (i) is obvious by the oddness of χ⁢(u). By (a), we have

d d ⁢ t ⁢ φ ⁢ ( η ⁢ ( t , u ) ) = 〈 ∇ ⁡ φ ⁢ ( u ) , χ ⁢ ( u ) 〉 ≤ 0 for any ⁢ u ∈ B k .

So, φ⁢(η⁢(t,u)) is non-increasing in t∈[0,T] for fixed u∈Bk, and (ii) is proved.

Now, we claim that η⁢(T,Bk)⊂φbk-ϵ. Otherwise, there exists u∈Bk such that

(2.13) φ ⁢ ( η ⁢ ( T , u ) ) > b k - ϵ .

Since η⁢(t,u) is non-increasing along t, we have

η ⁢ ( t , u ) ∈ ∩ ⁡ φ - 1 ⁢ [ b k - ϵ , d k + ϵ ) B R for any ⁢ t ∈ [ 0 , T ] .

Then, using (2.11), we see that

φ ( η ( T , u ) = φ ( η ( 0 , u ) + ∫ 0 T 〈 φ ′ ( η ( s , u ) ) , χ ( η ( s , u ) ) 〉 d s ≤ φ ( η ( 0 , u ) + ∫ 0 T - 1 d s ≤ d k + ϵ - T = b k - ϵ ,

which contradicts (2.13), and thus (iv) is proved.

Finally, by (d) and (iii), similar to the proof of [21, Lemma 6.8], we see that (v) also holds. ∎

For the τk-norm defined in (1.5), the same as in [16], we introduce the following definition.

Definition 2.1.

Let Bk and Nk be defined in (1.6). For any T>0, the mapping γ:[0,T]×Bk→X is a τk-admissible homotopy if the following hold:

γ is τk-continuous in the sense that γ⁢(tm,um)→𝑚γ⁢(t,u) in τk-topology if tm→𝑚t, and um→𝑚u in τk-topology.
For any (t,u)∈[0,T]×Bk, there exist a neighborhood W(t,u) of (t,u) in the |⋅|×τk-topology such that {v-γ⁢(s,v):(s,v)∈∩⁡W(t,u)([0,T]×Bk)} is contained in a finite-dimensional subspace of X.

We remark that such γ does exist since the identity mapping Id⁢(t,u)≡u is a τk-admissible homotopy.

Let Yk and Zk be defined in (1.1), Bk and Nk be defined in (1.6). The following intersection lemma is proved in [16], where the genus of (∩⁡γ⁢(t,Bk)Nk) is also estimated (see also [11]).

Lemma 2.3 ([16, Proposition 7]).

Let φ∈C1⁢(X,R) be an even functional and let γ:[0,T]×Bk→X be a τk-admissible homotopy with the following properties:

γ⁢(0,u)=u for any u∈Bk,
γ⁢(t,-u)=-γ⁢(t,u),
φ⁢(γ⁢(t,u)) is non-increasing in t∈[0,T] for fixed u∈Bk,
for any t∈[0,T], γ⁢(t,⋅):Bk→γ⁢(t,Bk) is a τk-homeomorphism.

sup u ∈ Y k , ∥ u ∥ = ρ k ⁡ φ ⁢ ( u ) < inf u ∈ Z k , ∥ u ∥ ≤ r k ⁡ φ ⁢ ( u ) ,

with 0<rk<ρk , then

∩ ⁡ γ ⁢ ( t , B k ) N k ≠ ∅ for any ⁢ t ∈ [ 0 , T ] ,

where Bk is given by (1.6).

Now we can prove Lemma 2.1.

Proof of Lemma 2.1.

By contradiction, if the conclusion of Lemma 2.1 is false, then there exists ϵ>0 such that

∥ φ ′ ⁢ ( u ) ∥ > ϵ for any ⁢ u ∈ E ,

where E is defined by (2.2). By Lemma 2.2 we know that there exists a map η⁢(t,u)∈C⁢([0,T]×Bk,X) satisfying Lemma 2.2 (i)–(v). By Remark 1.1, the τ-topology and τk-topology are equivalent, thus it is easy to see that η⁢(t,u) satisfies the assumptions of Lemma 2.3, hence ∩⁡η⁢(T,Bk)Nk≠∅, and the definition of bk implies that supu∈Bk⁡φ⁢(η⁢(T,u))≥bk. However, Lemma 2.2 (iv) shows that

sup u ∈ B k ⁡ φ ⁢ ( η ⁢ ( T , u ) ) ≤ b k - ϵ ,

which leads to a contradiction. So, the proof is complete. ∎

Proof of Theorem 1.2.

Taking δ1>0, by (A4) we know that sup∥u∥τ<δ1⁡φ⁢(u)≤Cδ1 for some Cδ1>0. Then condition (A3) implies that there exists k1∈𝐍 sufficiently large such that

b k 1 > sup ∥ u ∥ τ < δ 1 ⁡ φ ⁢ ( u ) .

By Lemma 2.1, we know that there exists a sequence {unk1} such that

φ ′ ( u n k 1 ) → 𝑛 0 in X ′ , sup n φ ( u n k 1 ) < d k 1 + 1 , inf n ∥ u n k 1 ∥ τ ≥ δ 1 2 .

Since φ satisfies the (PS)c condition, {unk1} has a subsequence which is convergent to a critical point uk1 of φ with ∥uk1∥≥∥uk1∥τ≥δ12.

Now, we take δ2>2⁢∥uk1∥ and, similar to the above, there exists k2>k1 large enough such that

b k 2 > sup ∥ u ∥ τ < δ 2 ⁡ φ ⁢ ( u ) ,

and we can find the second critical point uk2 with ∥uk2∥≥δ22>∥uk1∥. Clearly, uk2≠uk1.

Repeating the above procedures, we get a sequence of critical points {ukm} with limm→∞∥ukm∥→∞. So, the theorem is proved. ∎

3 An Application

The aim of this section is to apply Theorem 1.2 to prove the existence of infinitely many solutions of problem (1.7). In this section, X=H1⁢(ℝN), N≥3, with the norm

∥ u ∥ H 1 = ( ∫ ℝ N ( | u | 2 + | ∇ ⁡ u | 2 ) ⁢ 𝑑 x ) 1 2 .

Also, Lp⁢(a⁢(x),ℝN) is the Lebesgue space with positive weight a⁢(x), endowed with the norm

| u | L a ⁢ ( x ) p := ( ∫ ℝ N a ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p ,

and this norm is simply denoted by |u|Lp if a⁢(x)≡1. For r>0, we set B⁢(x,r)={x∈ℝN:|x|<r}.

The variational functional of (1.7) is defined by

(3.1) φ ⁢ ( u ) := 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x - 1 q ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u | q ⁢ 𝑑 x - 1 p ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x

for u∈H1⁢(ℝN). By (H1)–(H3), we have φ⁢(u)∈C1⁢(H1⁢(ℝN)) and

φ ′ ⁢ ( u ) ⁢ ϕ = ∫ ℝ N ( ∇ ⁡ u ⁢ ∇ ⁡ ϕ + V ⁢ ( x ) ⁢ u ⁢ ϕ ) ⁢ 𝑑 x - ∫ ℝ N g ⁢ ( x ) ⁢ | u | q - 2 ⁢ u ⁢ ϕ ⁢ 𝑑 x - ∫ ℝ N h ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ⁢ ϕ ⁢ 𝑑 x

for any ϕ∈H1⁢(ℝN). Moreover, φ′ is weakly sequentially continuous by [21, Theorem A.2].

Let L:=-Δ+V⁢(x) be the Schrödinger operator acting on L2⁢(ℝN) with domain 𝒟⁢(L)=H2⁢(ℝN). Since L is self-adjoint and 0 lies in a gap of its spectrum, by the standard spectral theory, we know that the space H1⁢(ℝN) can be decomposed as H1⁢(ℝN)=Y⊕Z such that the quadratic form

u ∈ H 1 ⁢ ( ℝ N ) → ∫ ℝ N ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x

is negative and positive definite on Yand Z respectively, and both Y and Z may be infinite-dimensional. Let X=𝒟⁢(|L|12) be equipped with the following inner product and norm:

〈 u , v 〉 1 := 〈 | L | 1 2 ⁢ u , | L | 1 2 ⁢ v 〉 L 2 , ∥ u ∥ := 〈 | L | 1 2 ⁢ u , | L | 1 2 ⁢ u 〉 L 2 1 2 ,

where 〈⋅,⋅〉L2 is the usual inner product in L2⁢(ℝN). By condition (H1), similar to the appendix of [9], we know that X=H1⁢(ℝN) and the norms ∥⋅∥ and ∥⋅∥H1 are equivalent. Moreover, Y and Z are also orthogonal with respect to 〈⋅,⋅〉1.

Let P:X→Y and Q:X→Z be the orthogonal projections. Then (3.1) can be written as

φ ⁢ ( u ) := 1 2 ⁢ ( - ∥ P ⁢ u ∥ 2 + ∥ Q ⁢ u ∥ 2 ) - 1 q ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u | q ⁢ 𝑑 x - 1 p ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x .

Now, we set

Y k := Y ⊕ ( ⊕ j = 0 k ℝ ⁢ f j ) and Z k := ⊕ j = k ∞ ℝ ⁢ f j ¯ ,

where {fj}j≥0 is an orthonormal basis of (Z,∥⋅∥).

Before proving Theorem 1.3, we give some useful lemmas. The first lemma is the following embedding result which has been used in many papers (see, e.g., [18]). Here we give a simple proof for completeness.

Lemma 3.1.

If 1<q<2∗ and a⁢(x)∈∩⁡Lq0⁢(RN)L∞⁢(RN) with a⁢(x)≥0 a.e. in RN, where q0=2⁢N2⁢N-q⁢N+2⁢q. Then H1⁢(RN)↪Lq⁢(a⁢(x),RN) is compact.

Proof.

For u∈H1⁢(ℝn), by the Hölder and Sobolev inequalities, we see that

∫ ℝ N a ( x ) | u | q d x ≤ | a ( x ) | L q 0 ⋅ ( ∫ ℝ N | u | 2 ⁢ N N - 2 d x ) q ⁢ N - 2 ⁢ q 2 ⁢ N = | a ( x ) | L q 0 ⋅ | u | L 2 ∗ ⁢ ( ℝ N ) q ≤ C ∥ u ∥ q ,

that is, |u|La⁢(x)q≤C⁢∥u∥, which means that H1⁢(ℝN)↪Lq⁢(a⁢(x),ℝN). Let {un} be a bounded sequence in H1⁢(ℝN). Passing to a subsequence, we may assume that, for some u∈H1⁢(ℝN),

u n ⇀ 𝑛 u weakly in ⁢ H 1 ⁢ ( ℝ N ) , u n → 𝑛 u in ⁢ L loc q ⁢ ( ℝ N ) ⁢ for ⁢ 1 < q < 2 * .

To show that H1⁢(ℝN)↪Lq⁢(a⁢(x),ℝN) is compact, we need only to show that un strongly converges to u in Lq⁢(a⁢(x),ℝN) for q∈(1,2*). In fact, for any R>0,

∫ ℝ N a ⁢ ( x ) ⁢ | u n - u | q ⁢ 𝑑 x = ∫ ℝ N ∖ B ⁢ ( 0 , R ) a ⁢ ( x ) ⁢ | u n - u | q ⁢ 𝑑 x + ∫ B ⁢ ( 0 , R ) a ⁢ ( x ) ⁢ | u n - u | q ⁢ 𝑑 x

≤ ( ∫ ℝ N ∖ B ⁢ ( 0 , R ) | a ( x ) | q 0 d x ) 1 q 0 | u n - u | L 2 ∗ ⁢ ( ℝ N ) + | a | L ∞ ⁢ ( ℝ N ) ∫ B ⁢ ( 0 , R ) | u n - u | q d x → 0 ,

by letting n→+∞ and then R→+∞. ∎

Lemma 3.2.

Under condition (H3), let

β k := sup u ∈ Z k , ∥ u ∥ = 1 | u | L h ⁢ ( x ) p for any k ∈ 𝐍 .

Then βk→0 as k→∞.

Proof.

It is clear that 0<βk+1≤βk, and thus βk→β≥0 as k→∞. For every k≥0, there exists uk∈Zk with ∥uk∥=1 and |uk|Lh⁢(x)p≥βk2. By the definition of Zk, we have uk⇀𝑘0 in H1⁢(ℝN). Thus, Lemma 3.1 implies that uk→𝑛0 strongly in Lg⁢(x)q, so, β=0. ∎

Lemma 3.3.

If (H1)–(H3) hold, then φ satisfies the (PS)c condition in H1⁢(RN) for any c<+∞.

Proof.

Let {un}⊂H1⁢(ℝN) be any sequence satisfying

sup n ⁡ φ ⁢ ( u n ) ≤ c and φ ′ ⁢ ( u n ) → 𝑛 0 .

We claim that {un} is bounded in H1⁢(ℝN). Indeed, for n large, we have

c + 1 + ∥ u n ∥ ≥ φ ⁢ ( u n ) - 1 2 ⁢ 〈 ∇ ⁡ φ ⁢ ( u n ) , u n 〉 = ( 1 2 - 1 p ) ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u n | p ⁢ 𝑑 x + ( 1 2 - 1 q ) ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u n | q ⁢ 𝑑 x ,

that is,

(3.2) ∫ ℝ N h ( x ) | u n | p d x ≤ C + ∥ u n ∥ + C ∫ ℝ N | g ( x ) | | u n | q d x ≤ C + ∥ u n ∥ + C ∥ u n ∥ q .

Let un=yn+zn, with yn∈Y,zn∈Z. For n large,

∥ z n ∥ ≥ 〈 φ ′ ⁢ ( u n ) , z n 〉 = ∥ z n ∥ 2 - ∫ ℝ N g ⁢ ( x ) ⁢ | u | q - 2 ⁢ u ⁢ z n ⁢ 𝑑 x - ∫ ℝ N h ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ⁢ z n ⁢ 𝑑 x ,

thus

∥ z n ∥ 2 ≤ ∥ z n ∥ + ∫ ℝ N g ⁢ ( x ) ⁢ | u | q - 1 ⁢ z n ⁢ 𝑑 x + ∫ ℝ N h ⁢ ( x ) ⁢ | u | p - 1 ⁢ z n ⁢ 𝑑 x .

By the Hölder inequality and the Sobolev embeddings, we see that, for some C>0,

∥ z n ∥ 2 ≤ ∥ z n ∥ + | g ( x ) q - 1 q u n q - 1 | L q q - 1 | g ( x ) 1 q z n q - 1 | L q q - 1 + | h ( x ) p - 1 p u n p - 1 | L p p - 1 | h ( x ) 1 p z n p - 1 | L p p - 1

= ∥ z n ∥ + | u n | L g ⁢ ( x ) q q - 1 ⁢ | z n | L g ⁢ ( x ) q + ( ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) p - 1 p ⁢ | z n | L h ⁢ ( x ) p

≤ ∥ z n ∥ + C ⁢ ∥ u n ∥ q - 1 ⁢ ∥ z n ∥ + C ⁢ ( 1 + ∥ u n ∥ + ∥ u n ∥ q ) p - 1 p ⁢ ∥ z n ∥

≤ ∥ u n ∥ + C ⁢ ∥ u n ∥ q + C ⁢ ( 1 + ∥ u n ∥ + ∥ u n ∥ q ) p - 1 p ⁢ ∥ u n ∥ ,

where in the third line we used (3.2). Similarly, from ∥yn∥≥-〈φ′⁢(un),yn〉, it follows that

∥ y n ∥ 2 ≤ ∥ y n ∥ + | g ( x ) q - 1 q u n q - 1 | L q q - 1 | g ( x ) 1 q y n q - 1 | L q q - 1 + | h ( x ) p - 1 p u n p - 1 | L p p - 1 | h ( x ) 1 p y n p - 1 | L p p - 1

≤ ∥ u n ∥ + C ⁢ ∥ u n ∥ q + C ⁢ ( 1 + ∥ u n ∥ + ∥ u n ∥ q ) p - 1 p ⁢ ∥ u n ∥ .

Since ∥un∥2=∥yn∥2+∥zn∥2, the above conclusions show that

∥ u n ∥ 2 ≤ 2 ⁢ ∥ u n ∥ + C ⁢ ∥ u n ∥ q + C ⁢ ( 1 + ∥ u n ∥ + ∥ u n ∥ q ) p - 1 p ⁢ ∥ u n ∥ .

Thus, by q<pp-1, we know that {un} is bounded in H1⁢(ℝN).

By the boundedness of {un}, we may assume, up to a subsequence, that

y n ⇀ 𝑛 y in ⁢ H 1 ⁢ ( ℝ N ) , z n ⇀ 𝑛 z in ⁢ H 1 ⁢ ( ℝ N ) .

Let u=y+z. Then we get that 〈∇⁡φ⁢(un)-∇⁡φ⁢(u),yn-y〉→0 as n→∞, and

〈 φ ′ ⁢ ( u n ) - φ ′ ⁢ ( u ) , y n - y 〉

= - ∥ y n - y ∥ 2 + ∫ ℝ N g ⁢ ( x ) ⁢ ( | u | q - 2 ⁢ u - | u n | q - 2 ⁢ u n ) ⁢ ( y n - y ) ⁢ 𝑑 x + ∫ ℝ N h ⁢ ( x ) ⁢ ( | u | p - 2 ⁢ u - | u n | p - 2 ⁢ u n ) ⁢ ( y n - y ) ⁢ 𝑑 x .

Using the Hölder inequality, we see that yn→𝑛y in H1⁢(ℝN). Similarly, zn→𝑛z in H1⁢(ℝN). Hence, un→𝑛u in H1⁢(ℝN). So, we proved that φ satisfies the (PS)c condition for any c<+∞. ∎

Lemma 3.4.

Under the conditions (H1)–(H3), for any δ>0, there exists Cδ<∞ such that sup∥u∥τ<δ⁡φ⁢(u)<Cδ.

Proof.

Since u∈H1⁢(ℝN)=Y⊕Z with Z=Y⊥, we may set u=y+z for some y∈Y and z∈Z. Then

φ ⁢ ( u ) = 1 2 ⁢ ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) - 1 q ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u | q ⁢ 𝑑 x - 1 p ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x

≤ 1 2 ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) + 1 q ∫ ℝ N | g ( x ) | | u | q d x

≤ - 1 2 ⁢ ∥ y ∥ 2 + C ⁢ ∥ y ∥ q + 1 2 ⁢ ∥ z ∥ 2 + C ⁢ ∥ z ∥ q .

Since q<2, -12⁢∥y∥2+C⁢∥y∥q is bounded from above. By (1.2), we have ∥z∥≤∥u∥τ≤δ, so, there exists Cδ<∞ such that sup∥u∥τ<δ⁡φ⁢(u)<Cδ. ∎

Proof of Theorem 1.3.

By Theorem 1.2 and Lemmas 3.3–3.4, in order to prove Theorem 1.3, we need only to verify conditions (A1), (A2’) and (A3).

Clearly, (A1) is true, since φ maps a bounded set into a bounded set.

Next, we prove (A2’) by showing that ak→-∞ as ρk→∞. Let u=y+z with y∈Y and z∈Z. For any u∈Yk, by Lemma 3.1, we see that

φ ⁢ ( u ) = 1 2 ⁢ ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) - 1 q ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u | q ⁢ 𝑑 x - 1 p ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x

≤ 1 2 ⁢ ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) + 1 q ⁢ | u | L | g ⁢ ( x ) | q q - 1 p ⁢ | u | L h ⁢ ( x ) p p

≤ 1 2 ⁢ ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) + 2 q - 1 q ⁢ ( ∥ y ∥ L | g ⁢ ( x ) | q q + ∥ z ∥ L | g ⁢ ( x ) | q q ) - 1 p ⁢ | u | L h ⁢ ( x ) p p

≤ 1 2 ⁢ ( - ∥ y ∥ 2 + ∥ z ∥ 2 ) + C q ⁢ ( ∥ y ∥ q + ∥ z ∥ q ) - 1 p ⁢ | u | L h ⁢ ( x ) p p .

Since H1⁢(ℝN)↪Lp⁢(h⁢(x),ℝN), we denote by Ek the closure of Yk in Lp⁢(h⁢(x),ℝN). Then there exists a continuous projection from Ek to ⊕j=0kfj. Thus, there exists a constant C>0 such that

| z | L h ⁢ ( x ) p p ≤ C ⁢ | u | L h ⁢ ( x ) p p .

Note that all norms are equivalent in a finite-dimensional vector space, so, for any z∈⊕j=0kfj, there exists C>0 such that

∥ z ∥ p ≤ C ⁢ | z | L h ⁢ ( x ) p p ,

thus

φ ⁢ ( u ) ≤ ( - 1 2 ⁢ ∥ y ∥ 2 + C ⁢ ∥ y ∥ q ) + ( 1 2 ⁢ ∥ z ∥ 2 + C ⁢ ∥ z ∥ q - C ⁢ ∥ z ∥ p ) .

So,

a k := sup u ∈ Y k , ∥ u ∥ = ρ k ⁡ φ ⁢ ( u ) → - ∞ as ⁢ ρ k → ∞ .

Finally, for any u∈Zk with ∥u∥=rk, let u=y+z with y∈Y and z∈Z. Then y=0 and z=u. Furthermore,

φ ⁢ ( u ) = 1 2 ⁢ ∥ u ∥ 2 - 1 q ⁢ ∫ ℝ N g ⁢ ( x ) ⁢ | u | q ⁢ 𝑑 x - 1 p ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x

≥ 1 2 ∥ u ∥ 2 - 1 q ∫ ℝ N | g ( x ) | | u | q d x - 1 p ∫ ℝ N h ( x ) | u | p d x

≥ ( 1 4 ⁢ ∥ u ∥ 2 - C ⁢ ∥ u ∥ q ) + ( 1 4 ⁢ ∥ u ∥ 2 - 1 p ⁢ β k p ⁢ ∥ u ∥ p ) .

Choosing

r k = ( p 4 ) 1 p - 2 ⁢ 1 β k p p - 2 ,

we have

φ ⁢ ( u ) ≥ 1 4 ⁢ ∥ u ∥ 2 - C ⁢ ∥ u ∥ q = 1 4 ⁢ | r k | 2 - C ⁢ | r k | q .

Since βk→0 as k→∞, we have rk→𝑘+∞. Then

b k := inf u ∈ Z k , ∥ u ∥ = r k ⁡ φ ⁢ ( u ) → + ∞ as ⁢ k → ∞ .

Thus, (A3) is also proved. ∎

Communicated by Yiming Long

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11471331

Award Identifier / Grant number: 11501555

Funding statement: This work was supported by the NSFC under grants 11471331 and 11501555.

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Received: 2016-09-20

Revised: 2016-11-08

Accepted: 2016-11-09

Published Online: 2016-12-21

Published in Print: 2017-10-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2016-6007

Keywords for this article

Fountain Theorem; Strongly Indefinite Functional; Elliptic Equation; Sign-Changing Potentials; Infinitely Many Solutions

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