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EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS

Published online by Cambridge University Press:  25 July 2018

CLAUDIANOR O. ALVES  and
CÉSAR E. TORRES LEDESMA
CLAUDIANOR O. ALVES
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900 Campina Grande, PB, Brazil E-mail: coalves@mat.ufcg.edu.br
CÉSAR E. TORRES LEDESMA
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú, Peru E-mail: ctl_576@yahoo.es
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Abstract

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In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation

$$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$
where ϵ is a positive parameter, 0 < α < 1, $1<p<\frac{n+2\alpha}{n-2\alpha}$, n > 2α; (−Δ)ρα is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ, Q, K are competing functions.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 2 , May 2019 , pp. 441 - 460
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

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