## Taiwanese Journal of Mathematics

### Multiplicity and Concentration of Solutions for Fractional Schrödinger Equations

#### Abstract

In this paper, we study the following fractional Schrödinger equations$(-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) + \mu \xi(x) |u|^{p-2}u, \quad x \in \mathbb{R}^{N},$where $\lambda \gt 0$ is a parameter, $V \in C(\mathbb{R}^{N})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two different nontrivial solutions. Moreover, the concentration of these solutions is also explored on the set $V^{-1}(0)$ as $\lambda \to \infty$. As an application, we also give the similar results and concentration phenomenons for the above problem with concave and convex nonlinearities.

#### Article information

Source
Taiwanese J. Math., Volume 21, Number 1 (2017), 187-210.

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874563

Digital Object Identifier
doi:10.11650/tjm.21.2017.7147

Mathematical Reviews number (MathSciNet)
MR3613980

Zentralblatt MATH identifier
1361.35194

#### Citation

Gao, Zu; Tang, Xianhua; Zhang, Wen. Multiplicity and Concentration of Solutions for Fractional Schrödinger Equations. Taiwanese J. Math. 21 (2017), no. 1, 187--210. doi:10.11650/tjm.21.2017.7147. https://projecteuclid.org/euclid.twjm/1498874563

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