Multiplicity of solutions for a class of fractional elliptic problem with critical exponential growth and nonlocal Neumann condition
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 $$ \mathcal{N}_{1/2}u(x) = \frac{1}{\pi} \int_{\R\setminus (a,b)} \frac{u(x)  u(y)}{xy^{2}}dy,\;\;x\in [a,b]. $$
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.02422
 Bibcode:
 2019arXiv191002422A
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 arXiv admin note: substantial text overlap with arXiv:1812.04881