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 methodsMSC
35R11 35A15 35B38 35P301 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
References
 Valdinoci, E: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. 49, 3344 (2009) MathSciNetMATHGoogle Scholar
 Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521573 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Servadei, R, Valdinoci, E: Mountain pass solutions for nonlocal elliptic operators. J. Math. Anal. Appl. 389, 887898 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Servadei, R, Valdinoci, E: Variational methods for nonlocal operators of elliptic type. Discrete Contin. Dyn. Syst., Ser. B 33, 21052137 (2013) MathSciNetMATHGoogle Scholar
 Fiscella, A: Saddle point solutions for nonlocal elliptic operators. Topol. Methods Nonlinear Anal. 44, 527538 (2014) MathSciNetView ArticleGoogle Scholar
 Zhang, BL, Bisci, GM, Servadei, R: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 22472264 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Ferrara, M, Bisci, GM, Zhang, BL: Existence of weak solutions for nonlocal fractional problems via Morse theory. Discrete Contin. Dyn. Syst., Ser. B 19, 24932499 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Chen, W, Deng, S: The Nehari manifold for nonlocal elliptic operators involving concaveconvex nonlinearities. Z. Angew. Math. Phys. 66, 13871400 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Bisci, GM, Rădulescu, VD: Multiplicity results for elliptic fractional equations with subcritical term. Nonlinear Differ. Equ. Appl. 22, 721739 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Bisci, GM, Pansera, BA: Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14, 591601 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Brändle, C, Colorado, E, de Pablo, A: A concaveconvex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb., Sect. A 143, 3971 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Fiscella, A, Servadei, R, Valdinoci, E: A resonance problem for nonlocal elliptic operators. Z. Anal. Anwend. 32, 411431 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Bisci, GM: Fractional equations with bounded primitive. Appl. Math. Lett. 27, 5358 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Bisci, GM: Sequences of weak solutions for fractional equations. Math. Res. Lett. 21, 113 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Caffarelli, LA, Silvestre, L: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 12451260 (2007) MathSciNetView ArticleMATHGoogle Scholar
 RosOton, X, Serra, J: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275302 (2014) MathSciNetView ArticleMATHGoogle Scholar
 RosOton, X, Serra, J: Fractional Laplacian: Pohozaev identity and nonexistence results. C. R. Math. Acad. Sci. Paris 350, 505508 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Cabré, X, Sire, Y: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, 2353 (2014) MathSciNetView ArticleMATHGoogle Scholar
 RosOton, X, Serra, J: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213, 587628 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Secchi, S: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^{N}\). J. Math. Phys. 54, 031501 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Chang, X, Wang, ZQ: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479494 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Dipierro, S, Palatucci, G, Valdinoci, E: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 1, 201216 (2013) MathSciNetMATHGoogle Scholar
 Autuoria, G, Pucci, P: Elliptic problems involving the fractional Laplacian in \(\mathbb{R}^{N}\). J. Differ. Equ. 255, 23042362 (2013) MathSciNetGoogle Scholar
 Felmer, P, Quaas, A, Tan, JG: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb., Sect. A 142, 12371262 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Bisci, GM, Rădulescu, VD: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. 54, 29853008 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Teng, K: Multiple solutions for a class of fractional Schrödinger equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 21, 7686 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Liu, ZL, Wang, ZQ: On Clark’s theorem and its applications to partially sublinear problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32, 10151037 (2015) View ArticleMATHGoogle Scholar
 Wang, ZQ: Nonlinear boundary value problems with concave nonlinearities near the origin. Nonlinear Differ. Equ. Appl. 8, 1533 (2001) MathSciNetView ArticleMATHGoogle Scholar
 Iannizzotto, A, Liu, SB, Perera, K, Squassina, M: Existence results for fractional pLaplacian problem via Morse theory. Adv. Calc. Var. 9, 101125 (2016) MathSciNetView ArticleMATHGoogle Scholar