Some existence results for a nonlocal non-isotropic problem

Abstract

In this paper we deal with the following problem $\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{, \Omega }\left\vert \partial {i}u\right\vert ^{p{i}}dx\right) \partial {i}\left( \left\vert \partial {i}u\right\vert ^{p{i}-2}\partial {i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \ u\geq 0 & in\ \Omega, \ u=0 & on\ \partial \Omega, \end{cases}$ where $\Omega$ is a bounded regular domain in $\mathbb{R}^{N}$. We will assume without loss of generality that $1\leq p{1}\leq p{2}\leq \ldots\leq p_{N}$ and that $f$ and $g$ are non-negative functions belonging to a suitable Lebesgue space $L^{m}(\Omega)$, $1\lt q\lt \overline{p}^{\ast}$, $a \gt 0$, $b \gt 0$ and $0\lt\gamma \lt 1.$