Multiple solutions for a fractional elliptic problem with critical growth

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Miyagaki, O. H.¹  Motreanu, D.^2,3  Pereira, F. R.¹ 

[1] Univ Fed Juiz de Fora, Dept Matemat ICE, BR-36036330 Juiz De Fora, Brazil

[2] Univ Perpignan, Dept Math, F-66860 Perpignan, France

[3] Yulin Normal Univ, Dept Math, Yulin 537000, Guangxi, Peoples R China

关键词: Nonlocal elliptic equations;    Variational methods;    Fractional critical growth;    Asymmetric nonlinearities;    Palais-Smale condition;   

DOI  :  10.1016/j.jde.2020.04.010

来源: Elsevier

【 摘 要 】

The paper focuses on a nonlocal Dirichlet problem with asymmetric nonlinearities. The equation is driven by the fractional Laplacian (-Delta)(s) for s is an element of (0, 1) and exhibits a sublinear term containing a parameter lambda, a linear term interfering with the spectrum of (-Delta)(s) and a superlinear term with fractional critical growth. The corresponding local problem governed by the standard Laplacian operator was investigated by F. O. de Paiva and A. E. Presoto. It can be recovered by letting s up arrow 1. The statement given here in the nonlocal setting is also related to extensively studied topics for local elliptic operators as the Brezis-Nirenberg problem and asymmetric nonlinearities. We go beyond the case of the standard Laplacian taking advantage of recent contributions on nonlocal fractional equations. Our main result establishes the existence of at least three nontrivial solutions, with one nonnegative and one nonpositive, provided the parameter lambda > 0 is sufficiently small. In order to overcome the difficulties in the nonlocal setting we develop new arguments that are substantially different from those used in previous works. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free