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

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.