 Research
 Open Access
Infinitely many solutions for fractional Laplacian problems with local growth conditions
 Anran Li^{1}Email author and
 Chongqing Wei^{1}
https://doi.org/10.1186/s1366201609637
© Li and Wei 2016
 Received: 18 April 2016
 Accepted: 4 September 2016
 Published: 23 September 2016
Abstract
Keywords
 fractional Laplacian equation
 local growth condition
 Clark’s theorem
 variational methods
MSC
 35R11
 35A15
 35B38
 35P30
1 Introduction and main results
 (f_{1}):

f is a Carathéodory function defined on \(\Omega \times(\delta,\delta)\) for some \(\delta>0\) which can be chosen small;
 (f_{2}):

there exists a positive constant \(q_{1}\in(\frac {4}{2_{s}^{*}},2)\) such that$$\lim_{t\rightarrow0}\frac{F(x,t)}{t^{q_{1}}}=0, \quad\mbox{uniformly for a.e. } x\in\Omega; $$
 (f_{3}):

there exists a positive constant \(q_{2}\in(\frac {4}{2_{s}^{*}},2)\) such that$$\lim_{t\rightarrow0}\frac{F(x,t)}{t^{q_{2}}}=+\infty,\quad \mbox{uniformly for a.e. } x\in\Omega; $$
 (f_{4}):

\(f(x,t)\) is odd in t, for a.e. \(x\in\Omega\), \(t\in (\delta,\delta)\).
Theorem 1.1
Let (f_{1})(f_{4}) hold, then problem (P) enjoys a sequence of nontrivial solutions \(\{u_{m}\}\) with \(u_{m}_{\infty}\rightarrow0\) as \(m\rightarrow\infty\).
Remark 1.1
In this article, the nonlinearity f just satisfies some sublinear growth condition near the origin, While without any assumptions near infinity. In order to prove our results via variational approach, inspired by the methods of [27, 28], first, we need to modify and extend f to an appropriate f̃ and to show for the associated modified functional the existence of solutions. Second, in order to obtain solutions for the original problem (P), some \(L^{\infty}\)estimates for the solutions of the modified problem are absolutely necessary. However, as far as we have known there is few result about the \(L^{p}\)estimate for fractional Laplacian problem as the class Laplacian problem. Similar bounds were obtained before only in some special cases, for a semilinear fractional Laplacian equation with reaction term independent of u, or for the eigenvalue problem of some fractional elliptic operators. Recently, in [29], the authors provided a method to give a priori \(L^{\infty}\) bounds for the weak solutions of problems similar to (P). Inspired by this method, we are able to get a suitable estimate of \(L^{\infty}\) norm of the weak solutions (for more details please check Lemma 3.2 of our article). Finally, by the Sobolev embedding theorem and Lemma 2.2, we can get infinity many solutions for the original problem (P).
Remark 1.2
The key step of our article is to get a suitable estimate of \(L^{\infty}\) norm of the weak solutions. In condition (f_{2}), the assumption \(q_{1}> \frac{4}{2_{s}^{*}}\) will be applied to give a \(L^{\infty}\)estimate for the weak solutions.
Throughout the article, the letter C will denote various positive constants whose values may change from line to line but are not essential to the analysis of the problem. We denote the usual norm of \(L^{q}(\Omega)\) by \(\cdot _{q}\) for \(1\leqslant q\leqslant\infty\). Moreover, let \(0< s<1\) be real numbers, and the fractional critical exponent be defined as \(2_{s}^{*}= \frac{2N}{N2s}\).
The paper is organized as follows. In Section 2, we introduce some preliminary notions and notations and set the functional framework of the our problem. In Section 3, we will prove our main result Theorem 1.1.
2 Preliminary
In this preliminary section, for the reader’s convenience, we collect some basic results that will be used in the forthcoming section.
Definition 2.1
Let E be a Banach space, we say that a functional \(\Phi\in C^{1}(E, \mathbb{R})\) satisfies PalaisSmale condition at the level \(c\in\mathbb{R}\) ((PS)_{ c } in short) if any sequence \(\{u_{n}\}\subset E \) satisfying \(\Phi(u_{n}) \to c, \Phi'(u_{n}) \to0\) as \(n\to\infty\), has a convergent subsequence. Φ satisfies (PS) condition if Φ satisfies (PS)_{ c } condition at any \(c\in\mathbb{R}\).
The following Sobolev type embedding theorem holds.
Lemma 2.1
([2])
The embedding \(X(\Omega)\hookrightarrow L^{q}(\Omega)\) is continuous for all \(q\in [1,2_{s}^{*}]\), and compact for \(q\in[1,2_{s}^{*})\).
We also need the following new version of Clark’s theorem; see Theorem 1.1 in [27].
Lemma 2.2
 (1)
There exists a sequence of critical points \(\{u_{k}\}\) satisfying \(\Phi(u_{k})<0\) for all k and \(\u_{k}\\rightarrow0\) as \(k\rightarrow\infty\).
 (2)
There exists \(r>0\) such that for any \(a\in(0,r)\) there exists a critical point u such that \(\u\=a\) and \(\Phi(u)=0\).
3 Proof of main result
In order to get our main result by Lemma 2.2. First of all, we check that Φ̃ is coercive, i.e. \(\tilde{\Phi}(u)\rightarrow\infty\), as \(\u\\rightarrow\infty\), and Φ̃ satisfies the (PS) condition.
Lemma 3.1
The functional Φ̃ is bounded from below and satisfies (PS) condition.
Proof
Finally, in order to get the weak solutions of the original problem (P), we will prove that the above sequence of critical points \(\{u_{m}\}\) for Φ̃ enjoys the following property.
Lemma 3.2
Proof
Therefore, from the above discussion we can see that the original problem (P) also enjoys a sequence of nontrivial solutions \(\{ u_{m}\}\) satisfying \(u_{m}_{\infty}\rightarrow0\), as \(m\rightarrow\infty\). Thus the proof of Theorem 1.1 is complete. □
Declarations
Acknowledgements
This work was supported by NSFC11526126 and NSFC11571209.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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