Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

TL;DR

This paper proves the existence of infinitely many solutions to a variable exponent fractional elliptic PDE with singular nonlinearity, using variational methods and Moser iteration for uniform bounds.

Contribution

It establishes the existence of infinitely many solutions to a nonlocal elliptic PDE with variable exponent and singularity, a novel result in this context.

Findings

Existence of infinitely many solutions proven.

Uniform boundedness of solutions established.

Solutions exist under certain growth conditions on f.

Abstract

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s} u&=\frac{\lambda}{|u|^{\gamma(x)-1}u}+f(x,u)~\text{in}~\Omega,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber \end{align} where $\Omega\subset\mathbb{R}^N,\, N\geq2$ is a smooth, bounded domain, $\lambda>0$, $s\in (0,1)$, $\gamma(x)\in(0,1)$ for all $x\in\bar{\Omega}$, $N>sp(x,y)$ for all $(x,y)\in\bar{\Omega}\times\bar{\Omega}$ and $(-\Delta)_{p(\cdot)}^{s}$ is the fractional $p(\cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{\infty}(\bar{\Omega})$ estimate of the solution(s) by the Moser iteration technique.