Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition

Claudianor O. Alves; César T. Ledesma

doi:10.3934/cpaa.2021058

Article Contents

2021, Volume 20, Issue 5: 2065-2100. Doi: 10.3934/cpaa.2021058

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Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition

Claudianor O. Alves^1, and
César T. Ledesma^2, ,

1.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil
2.
Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Peru

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: January 2021

Early access: April 2021

Published: May 2021

C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7 and C. E. Torres Ledesma was partially supported by INC Matemática 88887.136371/2017

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Abstract

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem

$ \begin{equation} \begin{cases} (-\Delta)^{\frac{1}{2}}u + u = Q(x)f(u)\;\;\mbox{in}\;\;\mathbb{R} \setminus (a, b)\\ \mathcal{N}_{1/2}u(x) = 0\;\;\qquad \qquad \quad \mbox{in}\;\;(a, b), \end{cases} \end{equation} \ \ \ \ \ \ \ \ (0.1) $

where $ a, b\in \mathbb{R} $ with $ a<b $, $ (-\Delta)^{\frac{1}{2}} $ denotes the fractional Laplacian operator and $ \mathcal{N}_{1/2} $ is the nonlocal operator that describes the Neumann boundary condition, which is given by

$ \mathcal{N}_{1/2}u(x) = \frac{1}{\pi} \int_{\mathbb{R}\setminus (a, b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy, \;\;x\in [a, b]. $

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Mathematics Subject Classification: Primary: 35A15; Secondary: 35J60, 34B10.

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