Ground state solutions for fractional scalar field equations under a general critical nonlinearity

Claudianor O. Alves; Giovany M. Figueiredo; Gaetano Siciliano

doi:10.3934/cpaa.2019099

Article Contents

2019, Volume 18, Issue 5: 2199-2215. Doi: 10.3934/cpaa.2019099

This issue Previous Article Next Article On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

Ground state solutions for fractional scalar field equations under a general critical nonlinearity

1.
Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil
2.
Universidade de Brasilia, Campus Universitário Darcy Ribeiro, 70910-900, Brasília - DF - Brazil
3.
Universidade de São Paulo, Departamento de Matemática - IME, Rua do Matão 1010, 05508-090, São Paulo - SP - Brazil

^* Corresponding author
^* Corresponding author

Received: June 2017

Revised: November 2017

Published: April 2019

The first author is supported was partially supported by CNPq/Brazil Proc. 304036/2013-7; the second author was partially supported by CNPq, Capes and FAPDF, Brazil; the third author was partially supported by CNPq, Capes and FAPESP, Brazil

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Abstract

In this paper we study existence of ground state solution to the following problem

$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $

where $ (-\Delta)^{\alpha} $ is the fractional Laplacian, $ \alpha\in (0,1) $. We treat both cases $ N\geq2 $ and $ N = 1 $ with $ \alpha = 1/2 $. The function $ g $ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $ N\geq2 $, exponential if $ N = 1 $.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35A15, 35B33.

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References

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