# Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem

## TL;DR

This paper proves the existence of two distinct solutions for a nonlocal variable exponent elliptic PDE within a Nehari manifold, using the Banach fixed point theorem, and shows these solutions are bounded in $L^{ty}(\,Omega)$.

## Contribution

It introduces a novel approach to establish multiple solutions for a nonlocal PDE with variable exponents using the Banach fixed point theorem within a Nehari manifold.

## Key findings

- Existence of two distinct nontrivial solutions.
- Solutions are bounded in $L^{\,infty}(\,Omega)$.
- Conditions on exponents ensure solution multiplicity.

## Abstract

In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\infty}(\Omega)$. \begin{align*} \begin{split} -\Delta_{p(x,y)}^{s(x,y)}u &= \beta|u|^{\alpha(x)-2}u+\lambda f(x,u)\,\,\mbox{in}\,\,\Omega,\\ u &= 0\,\, \mbox{in}\,\, \mathbb{R}^{N}\setminus\Omega \end{split} \end{align*} Here, $\lambda, \beta > 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $\Omega\subset\mathbb{R}^N (N\geq 2)$ is smooth and bounded. The relation between the exponents are assumed in the order $2 < \alpha^{-}\leq\alpha(x)\leq\alpha^{+} < p^{-}\leq p(x,y)\leq p^{+} < q^{+} < r^{+} < r^{+2} < p_{s}^{*}(x)$. Also, $\alpha(x)\leq p(x,x)\;\forall\;x\in\overline{\Omega}$ and $s(x,y)p(x,y) < N \;\forall\;(x,y)\in\overline{\Omega}\times\overline{\Omega}$.