Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

César E. Torres  Ledesma

doi:10.3934/cpaa.2016.15.535

Article Contents

2016, Volume 15, Issue 2: 535-547. Doi: 10.3934/cpaa.2016.15.535

This issue Previous Article Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter Next Article Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities

Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

César E. Torres Ledesma^1,

1.
Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Perú

Received: June 30, 2015

Revised: November 30, 2015

Published: February 2016

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Abstract

In this paper we study the non-linear fractional Schrödinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u)\ in\ R^{n}, \ u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.

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Mathematics Subject Classification: Primary: 35J35; Secondary: 35J60.

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