Electron. J. Differential Equations, Vol. 2017 (2017), No. 76, pp. 1-10.

Existence of standing waves for Schrodinger equations involving the fractional Laplacian

Everaldo S. de Medeiros, Jose Anderson Cardoso, Manasses de Souza

Abstract:
We study a class of fractional Schrodinger equations of the form
$$ \varepsilon^{2\alpha}(-\Delta)^\alpha u+ V(x)u = f(x,u) \quad\text{in } \mathbb{R}^N, $$
where $\varepsilon$ is a positive parameter, $0 < \alpha < 1$, $2\alpha < N$, $(-\Delta)^\alpha$ is the fractional Laplacian, $V:\mathbb{R}^{N}\to \mathbb{R}$ is a potential which may be bounded or unbounded and the nonlinearity $f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}$ is superlinear and behaves like $|u|^{p-2}u$ at infinity for some $2<p< 2^*_\alpha:=2N/(N-2\alpha)$. Here we use a variational approach based on the Caffarelli and Silvestre's extension developed in [3] to obtain a nontrivial solution for $\varepsilon$ sufficiently small.

Submitted September 16, 2016. Published March 20, 2017.
Math Subject Classifications: 35J20, 35J60, 35R11.
Key Words: Variational methods; critical points; fractional Laplacian.

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Everaldo S. de Medeiros
Departamento de Matemática
Universidade Federal da Paraíba, 58051-900
João Pessoa, PB, Brazil
email: everaldomedeiros1@gmail.com
Jose Anderson Cardoso
Departamento de Matemática
Universidade Federal de Sergipe, 49000-100
São Cristóvão, Brazil
email: anderson@mat.ufs.br
Manassés de Souza
Departamento de Matemática
Universidade Federal da Paraíba, 58051-900
João Pessoa, PB, Brazil
email: manassesxavier@gmail.com

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