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

TL;DR

This paper investigates the existence and multiplicity of weak solutions for a fractional elliptic problem with critical exponential growth and a nonlocal Neumann boundary condition, expanding understanding of fractional PDEs with nonlocal boundaries.

Contribution

It establishes new results on the multiplicity of solutions for a class of fractional elliptic equations with critical exponential growth and nonlocal Neumann conditions.

Findings

Proves existence of multiple weak solutions.

Handles critical exponential growth in fractional PDEs.

Introduces methods for nonlocal Neumann boundary conditions.

Abstract

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R \setminus (a,b)\\ \mathcal{N}_{1/2}u(x) &= 0\;\;\mbox{in}\;\;(a,b), \end{aligned} \right. \end{equation} where $a,b\in \R$ with $a<b$, $(-\Delta)^{\frac{1}{2}}$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_{1/2}u(x) = \frac{1}{\pi} \int_{\R\setminus (a,b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy,\;\;x\in [a,b]. $$