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

• Published in: arXiv.org 2020-05
• Main Authors: Ghosh, Sekhar, Choudhuri, Debajyoti, Ratan Kr Giri
• Format: Article
• Language: English
• Subjects: Nonlinearity; Operators (mathematics); Partial differential equations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2006.00260

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.