Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations

Abstract

In this paper, we consider the existence of positive solutions to mixed local and nonlocal singular quasilinear singular elliptic equations $ \begin{align*} \left\{\begin{array}{rl} -\Delta_{\vec{p}}u(x)+\left(-\Delta\right)_{p}^{s}u(x) = \frac{f(x)}{u(x)^{\delta}}, &x\in\Omega, \\ u(x)>0, \; \; \; \; \; \; &x\in\Omega, \\ u(x) = 0, \; \; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{array} \right. \end{align*} $ where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^{N}(N > 2) $, $ -\Delta_{\vec{p}}u $ is an anisotropic $ p $-Laplace operator, $ \vec{p} = (p_{1}, p_{2}, ..., p_{N}) $ with $ 2\leq p_{1}\leq p_{2}\leq\cdot\cdot\cdot\leq p_{N} $, $ \left(-\Delta \right)_{p}^{s} $ is the fractional $ p $-Laplace operator. The major results shows the interplay between the summability of the datum $ f(x) $ and the power exponent $ \delta $ in singular nonlinearities.

Keywords

• mixed local and nonlocal
• anisotropic p-laplace equation
• singular elliptic operator