Singular anisotropic elliptic equations with gradient-dependent lower order terms

Abstract

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of ${\mathbb{R}}^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{ \begin{array}{ll} \mathcal A u+ Φ(u,\nabla u)=Ψ(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } Ω,\\ u=0 & \mbox{on } \partialΩ. \end{array} \right. \end{equation} Here $\mathcal{A}u=-{\sum }_{j=1}^N|{\partial }_{j}u{|}^{p_j-2}{\partial }_{j}u$ is the anisotropic $\overrightarrow{p}$-Laplace operator, while $\mathfrak{B}$ is an operator from $W_0^{1,\overrightarrow{p}}(\Omega )$ into $W^{-1,{\overrightarrow{p}}^{\prime }}(\Omega )$ satisfying suitable, but general, structural assumptions. $\Phi$ and $\Psi$ are gradient-dependent nonlinearities whose models are the following: \begin{equation*} \label{phi}Φ(u,\nabla u):=\left(\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}+1\right)|u|^{m-2}u, \quad Ψ(u,\nabla u):=\frac{1}{u}\sum_{j=1}^N |u|^{θ_j} |\partial_j u|^{q_j}. \end{equation*} We suppose throughout that, for every $1\le j\le N$, \begin{equation*}\label{ass} \mathfrak{a}_j\geq 0, \quad θ_j>0, \quad 0\leq q_j<p_j, \quad 1<p_j,m\quad \mbox{and}\quad p<N, \end{equation*} and we distinguish two cases: 1) for every $1\le j\le N$, we have ${\theta }_j\ge 1$; 2) there exists $1\le j\le N$ such that ${\theta }_j<1$. In this last situation, we look for non-negative solutions of \eqref{eq0}.