# Existence of solutions for a class of fractional elliptic problems on exterior domains

## TL;DR

This paper proves the existence of solutions for a class of fractional elliptic problems on exterior domains using variational and topological methods, expanding the understanding of nonlocal elliptic equations with fractional Laplacians.

## Contribution

It introduces a general variational framework for fractional elliptic problems on exterior domains, applicable to a broad class of nonlocal equations.

## Key findings

- Existence of nontrivial solutions established.
- Applicable to fractional Laplacian problems with superlinear nonlinearity.
- Method can be adapted for other nonlocal elliptic problems.

## Abstract

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in} \quad \Omega \quad \mbox{and} \quad u \not\equiv 0, \\ u=0 \quad \mathbb{R}^N \setminus \Omega, \end{array} \right. \end{equation*} involving the fractional Laplacian operator $(-\Delta)^{s}$, where $s\in (0,1)$, $N> 2s$, $\Omega \subset \R^N$ is an exterior domain with (non-empty) smooth boundary $\partial \Omega$ and $p\in (2, 2_{s}^{*})$. The main technical approach is based on variational and topological methods. The variational analysis that we perform in this paper dealing with exterior domains is quite general and may be suitable for other goals too.