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

Abstract

<abstract><p>In this paper, we consider the existence of positive solutions to mixed local and nonlocal singular quasilinear singular elliptic equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>