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} (-Δ)_{p(\cdot)}^{s} u&=\fracλ{|u|^{γ(x)-1}u}+f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $\Omega \subset {\mathbb{R}}^N,N\ge 2$ 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.

21 pages

References in corpus (1)

• Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

Cited by in corpus (2)

• On critical variable-order Kirchhoff type problems with variable singular exponent
• Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents