SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Infinitely many weak solutions for a fractional Schrödinger equation

Wei Dong1*, Jiafa Xu2 and Zhongli Wei23

Author Affiliations

1 Department of Mathematics, Hebei University of Engineering, Handan 056038, Hebei, China

2 School of Mathematics, Shandong University, Jinan 250100, Shandong, China

3 Department of Mathematics, Shandong Jianzhu University, Jinan 250101, Shandong, China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:159  doi:10.1186/s13661-014-0159-6

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/159


Received:22 February 2014
Accepted:13 June 2014
Published:12 July 2014

© 2014 Dong et al.; licensee Springer

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper we are concerned with the fractional Schrödinger equation , , where , , stands for the fractional Laplacian of order α, V is a positive continuous potential, and f is a continuous subcritical nonlinearity. We obtain the existence of infinitely many weak solutions for the above problem by the fountain theorem in critical point theory.

Keywords:
fractional Laplacian; subcritical nonlinearity; fountain theorem; weak solution

Introduction

In this paper we consider the following fractional Schrödinger equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M1">View MathML</a>

(1)
where , , stands for the fractional Laplacian of order α, and the potential is a continuous function satisfying

(V) .

The nonlinearity is a continuous function, satisfying the subcritical condition.

(H1) There exist , and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M2">View MathML</a>

(2)
where is the fractional critical exponent.

Recently, there have appeared plenty of works on the fractional Schrödinger equations; for example, see [[1]–[11]] and the references therein. In [[1]], Shang and Zhang considered the critical fractional Schrödinger equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M3">View MathML</a>

(3)
where ε and λ are positive parameters, V and f satisfy (V) and (H1), respectively. They obtained the result that (1.2) has a nonnegative ground state solution and investigated the relation between the number of solutions and the topology of the set where V attains its minimum for all sufficiently large λ and small ε. In [[2]], Shang et al. considered the existence of nontrivial solutions for (1.2) with , where .

In [[8]], Hua and Yu studied the critical fractional Laplacian equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M4">View MathML</a>

(4)
where , , is a bounded domain. They obtained the result that the problem (1.3) possesses a nontrivial ground state solution for any .

In [[7]], Secchi investigated the existence of radially symmetric solutions for (1.1) replacing by , where g satisfies the following conditions.

(g1) is of class for some , and odd,

(g2) ,

(g3) ,

(g4) for some such that .

Inspired by the mentioned papers, we first establish a compact embedding lemma via a fractional Gagliardo-Nirenberg inequality. Then by virtue of the fountain theorem in critical point theory, we get two existence results of infinitely many weak solutions for (1.1).

Preliminary results

In this section we offer some preliminary results which enable us to obtain the main existence theorems. First, we collect some useful facts of the fractional order Sobolev spaces.

For any , the fractional Sobolev space is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M5">View MathML</a>

(5)
endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M6">View MathML</a>

(6)
where is the so-called Gagliardo semi-norm of u. Let be the Schwartz space of rapidly decaying functions in , for any and , and let be defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M7">View MathML</a>

(7)
The symbol P.V. stands for the Cauchy principal value, and is a dimensional constant that depends on N and α, precisely given by .

Indeed, the fractional Laplacian can be viewed as a pseudo-differential operator of symbol , as stated in the following.

Lemma 2.1

(see [[12]])

Letandbe the fractional Laplacian operator defined by (2.1). Then for any,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M8">View MathML</a>

(8)
whereis the Fourier transform, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M9">View MathML</a>

(9)

Now we can see that an alternative definition of the fractional Sobolev space via the Fourier transform is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M10">View MathML</a>

(10)
It can be proved that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M11">View MathML</a>

(11)
As a result, the norms on ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M12">View MathML</a>

(12)
are all equivalent.

In this paper, in view of the presence of potential , we consider its subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M13">View MathML</a>

(13)
We define the norm in E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M14">View MathML</a>

(14)
where . Moreover, by [[6]], E is a Hilbert space with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M15">View MathML</a>

(15)
Note that by (2.2) and (2.3), together with the condition (V), we know that is equivalent to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M16">View MathML</a>

(16)
The corresponding inner product is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M17">View MathML</a>

(17)
Throughout out this paper, we will use the norm in E.

Definition 2.2

We say that is a weak solution of (1.1), if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M18">View MathML</a>

(18)

Lemma 2.3

(see [[7]] and [[12]])

Eis continuously embedded intoforand compactly embedded intofor.

Lemma 2.4

Eis compactly embedded intoforwith.

Proof

By [[4]], we know E is compactly embedded into , i.e., if there exists a sequence and such that weakly in E, passing to a subsequence if necessary, we have strongly in . Therefore, we only consider . In order to do this, we need the following fractional Gagliardo-Nirenberg inequality, see [[13], Corollary 2.3]. Let , , and . Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M19">View MathML</a>

(19)
with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M20">View MathML</a>

(20)
Note that the dimension , we can take , and then whence as . Consequently, from (2.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M21">View MathML</a>

(21)
Furthermore, note that (2.4); we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M22">View MathML</a>

(22)
Then by (2.6) and , we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M23">View MathML</a>

(23)
Therefore, E is compactly embedded into for with , as required. This completes the proof. □

The functional associated with (1.1) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M24">View MathML</a>

(24)
where .

Now, we list our assumptions on f and F.

(H2) uniformly for .

(H3) There exist and such that and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M25">View MathML</a>

(25)

(H4) for .

(H5) There exist , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M26">View MathML</a>

(26)

(H6) for all .

Remark 2.5

(1) Let , for all and . Then (H1), (H2), (H4), and (H6) hold. Moreover, we easily have and , so (H3) is satisfied.

However, we can see that does not satisfy the Ambrosetti-Rabinowitz condition (see [[6], ()]):

(AR) there is a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M27">View MathML</a>

(27)

Indeed, is impossible for all and .

(2) Let and , for all and . Then (H1), (H2), (H4), and (H6) hold. Moreover, from , and (H5) holds.

Note that from Theorem 4 in [[14]] we have (H4) and (H5) imply (H2).

Lemma 2.6

(see [[15], Lemma 1])

Let (V) and (H1) hold. Thenand its derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M28">View MathML</a>

(28)
Moreover, the critical points ofJare weak solutions of (1.1).

To complete the proofs of our theorems, we need the following critical point theorems in [[16]–[19]].

Definition 2.7

Let be a real Banach space, . We say that J satisfies the () condition if any sequence such that and as has a convergent subsequence.

Let X be a Banach space equipped with the norm and , where for any . Set and .

Lemma 2.8

Letbe a real reflexive Banach space, satisfies the () condition for anyandJis even. If for each sufficiently large, there existsuch that the following conditions hold:

(i) as,

(ii) ,

then the functionalJhas an unbounded sequence of critical values, i.e., there exists a sequencesuch thatandas.

In the following, we will introduce a variant fountain theorem by Zou [[16]]. Let X and the subspace and be defined above. Consider the following -functional defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M29">View MathML</a>

(29)

Lemma 2.9

If the functionalsatisfies

(T1) maps bounded sets to bounded sets uniformly for, and, moreover, for all,

(T2) for all; moreover, oras,

(T3) there existsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M30">View MathML</a>

(30)
then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M31">View MathML</a>

(31)
whereand. Moreover, for a.e. , there exists a sequencesuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M32">View MathML</a>

(32)

Remark 2.10

As mentioned in [[6]], E is a Hilbert space. Let be an orthonormal basis of E and define , , and , . Clearly, with for all .

Existence of weak solutions for (1.1)

Theorem 3.1

Assume that (V), (H1)-(H4), and (H6) hold. Then (1.1) has infinitely many weak solutionssatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M33">View MathML</a>

(33)

Proof

We first prove that J satisfies the () condition for any . Let be a () sequence, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M34">View MathML</a>

(34)
which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M35">View MathML</a>

(35)

In what follows, we shall show that is bounded. Otherwise, up to a subsequence, is unbounded in E, and we may assume that as . We define the sequence by ,  . Clearly, and for any n. Going over, if necessary, to a subsequence, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M36">View MathML</a>

(36)
Suppose that in E. Dividing by in both sides of (2.7), noting that , we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M37">View MathML</a>

(37)
On the other hand, denote , by (H2), for all , and we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M38">View MathML</a>

(38)
If , using Fatou’s lemma, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M39">View MathML</a>

(39)
This contradicts (3.4). Hence, has zero measure, i.e., a.e. in . Let such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M40">View MathML</a>

(40)
Then we claim is bounded. If , ; if , . Therefore, is bounded when . If for n large enough

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M41">View MathML</a>

(41)
Consequently, by (H3), noting that (3.1) and (3.2) hold, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M42">View MathML</a>

(42)
where is a positive constant. But fixing any , we let . Note that from (H1) we see that there exist , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M43">View MathML</a>

(43)
Then by (3.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M44">View MathML</a>

(44)
Then for n large enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M45">View MathML</a>

(45)
This also contradicts (3.5).

Now the sequence is bounded, as required. Next, we verify that has a convergent subsequence. Without loss of generality, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M46">View MathML</a>

(46)
Combining this with (H1) and the Hölder inequality, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M47">View MathML</a>

(47)
Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M48">View MathML</a>

(48)
with the fact that when . Therefore, we prove that J satisfies the () condition for any .

Clearly, by (H6). It remains to prove that the conditions (i) and (ii) of Lemma 2.8 hold. Let with , where is defined in Remark 2.10. Then by Lemma 3.8 of [[19]], as for the fact that .

Now for with , we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M49">View MathML</a>

(49)
Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M50">View MathML</a>

(50)

Next we shall prove that, for any finite dimensional subspace , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M51">View MathML</a>

(51)
Suppose the contrary. For some sequence with , there is a such that for all . Put and then . Up to a subsequence, assume that weakly in E. Since , in E, a.e. on , and . Denote , then and for a.e. , . It follows from (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M52">View MathML</a>

(52)
But, for large n, on account of F being nonnegative, (H2) and Fatou’s Lemma enable us to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M53">View MathML</a>

(53)
as . This contradicts (3.10). Consequently, (3.9) holds, as required. Note that in Remark 2.10, and there exist positive constants such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M54">View MathML</a>

(54)
Combining this and (3.8), we can take , and thus . Until now, we have proved the functional J satisfies all the conditions of Lemma 2.8. Hence, J has an unbounded sequence of critical values, i.e., there exists a sequence such that and as . This completes the proof. □

We prove that there exists ( is determined in (H5)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M55">View MathML</a>

(55)
Indeed, by (3.6) we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M56">View MathML</a>

(56)
This, together with (H1), implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M57">View MathML</a>

(57)
Clearly, (3.12) holds true with . In the following theorem, we make the following assumption instead of (V):

(V′): , , where in (3.12), β in (H5).

Especially, by (V′), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M58">View MathML</a>

(58)

Now, we define a class of functionals on E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M59">View MathML</a>

(59)
It is easy to know that for all and the critical points of correspond to the weak solutions of problem (1.1). Note that , where J is the functional defined in (2.7).

Theorem 3.2

Assume that (V′), (H1), and (H4)-(H6) hold. Then (1.1) possesses infinitely many weak solutions.

Proof

We first prove that there exist a positive integer and two sequences as such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M60">View MathML</a>

(60)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M61">View MathML</a>

(61)
where and are defined in Remark 2.10.

Step 1. We claim that (3.14) is true.

By (3.6) and (H4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M62">View MathML</a>

(62)
Since with , and from Theorem 3.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M63">View MathML</a>

(63)
Let as . Then there exists such that , . Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M64">View MathML</a>

(64)

Step 2. We show that (3.15) is true.

We apply the method in Lemma 2.6 of [[20]] to verify the claim. First, we prove that there exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M65">View MathML</a>

(65)
There would otherwise exist a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M66">View MathML</a>

(66)
For each , let . Then , and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M67">View MathML</a>

(67)
Passing to a subsequence if necessary, we may assume in E for some since is of finite dimension. We easily find . Consequently, there exists a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M68">View MathML</a>

(68)
Indeed, if not, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M69">View MathML</a>

(69)
which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M70">View MathML</a>

(70)
This leads to , contradicting . In view of and the equivalence of any two norms on , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M71">View MathML</a>

(71)
For every , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M72">View MathML</a>

(72)
and , where is defined by (3.20). Then for n large enough, by (3.20), we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M73">View MathML</a>

(73)
Consequently, for n large enough, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M74">View MathML</a>

(74)
This contradicts (3.22). Therefore, (3.17) holds. For the ε given in (3.17), we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M75">View MathML</a>

(75)
Then by (3.17), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M76">View MathML</a>

(76)
As is well known, (H5) implies (H2), and hence for any , there is a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M77">View MathML</a>

(77)
where ε is determined in (3.17). Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M78">View MathML</a>

(78)
Now for any , if we take , so is large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M79">View MathML</a>

(79)

Step 3. Clearly, implies that maps bounded sets to bounded sets uniformly for . In view of (H6), for all . Thus the condition (T1) of Lemma 2.9 holds. Besides, the condition (T2) of Lemma 2.9 holds for the fact that as and since . Evidently, Step 1 and Step 2 imply that the condition (T3) of Lemma 2.9 also holds for all . Consequently, Lemma 2.9 implies that for any and a.e. , there exists a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M80">View MathML</a>

(80)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M81">View MathML</a>

(81)
Furthermore, we easily have , , where and as .

Claim 1. possesses a strong convergent subsequence in E, a.e. and . In fact, by the boundedness of , passing to a subsequence, as , we may assume in E. By the method of Theorem 3.1, we easily prove that strongly in E.

Thus, for each , we can choose such that for the sequence we have obtained a convergent subsequence, and passing again to a subsequence, we may assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M82">View MathML</a>

(82)
Thus we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M83">View MathML</a>

(83)

Claim 2. is bounded in E and has a convergent subsequence with the limit for all . For convenience, we set for all . Consequently, (3.12) and (H5) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/159/mathml/M84">View MathML</a>

(84)
Therefore, is bounded in E. By Claim 1, we see that has a convergent subsequence, which converges to an element for all .

Hence, passing to the limit in (3.24), we see and , and . Since as , we get infinitely many nontrivial critical points of . Therefore (1.1) possesses infinitely many nontrivial solutions by Lemma 2.9. This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

WD and JX obtained the results in a joint research. All the authors read and approved the final manuscript.

Acknowledgements

The authors are highly grateful for the referees’ careful reading of this paper and their comments. Supported by the NNSF-China (11371117), Shandong Provincial Natural Science Foundation (ZR2013AM009) and Hebei Provincial Natural Science Foundation (A2012402036).

References

  1. Shang, X, Zhang, J: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity. 27, 187–207 (2014). Publisher Full Text OpenURL

  2. Shang, X, Zhang, J, Yang, Y: On fractional Schrödinger equation in with critical growth. J. Math. Phys.. 54, (2013). Publisher Full Text OpenURL

  3. Cabré, X, Sire, Y: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 31, 23–53 (2014). Publisher Full Text OpenURL

  4. Chang, X: Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys.. 54, (2013). Publisher Full Text OpenURL

  5. Chang, X, Wang, Z: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 26, 479–494 (2013). Publisher Full Text OpenURL

  6. Secchi, S: Ground state solutions for nonlinear fractional Schrödinger equations in . J. Math. Phys.. 54, (2013). Publisher Full Text OpenURL

  7. [arXiv:1210.0755] webcite

  8. Hua, Y, Yu, X: On the ground state solution for a critical fractional Laplacian equation. Nonlinear Anal.. 87, 116–125 (2013). Publisher Full Text OpenURL

  9. Barrios, B, Colorado, E, de Pablo, A, Sánchez, U: On some critical problems for the fractional Laplacian operator. J. Differ. Equ.. 252, 6133–6162 (2012). Publisher Full Text OpenURL

  10. Autuori, G, Pucci, P: Elliptic problems involving the fractional Laplacian in . J. Differ. Equ.. 255, 2340–2362 (2013). Publisher Full Text OpenURL

  11. Dipierro, S, Pinamonti, A: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ.. 255, 85–119 (2013). Publisher Full Text OpenURL

  12. Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math.. 136, 521–573 (2012). Publisher Full Text OpenURL

  13. Hajaiej, H, Yu, X, Zhai, Z: Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms. J. Math. Anal. Appl.. 396, 569–577 (2012). Publisher Full Text OpenURL

  14. Wu, X: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in . Nonlinear Anal., Real World Appl.. 12, 1278–1287 (2011). Publisher Full Text OpenURL

  15. Ye, Y, Tang, C: Multiple solutions for Kirchhoff-type equations in . J. Math. Phys.. 54, (2013). Publisher Full Text OpenURL

  16. Zou, W: Variant fountain theorems and their applications. Manuscr. Math.. 104, 343–358 (2001). Publisher Full Text OpenURL

  17. Struwe, M: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, New York (2000)

  18. Ambrosetti, A, Rabinowitz, P: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text OpenURL

  19. Willem, M: Minimax Theorems, Birkhäuser, Boston (1996)

  20. Zhang, Q, Xu, B: Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential. J. Math. Anal. Appl.. 377, 834–840 (2011). Publisher Full Text OpenURL