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+ \Phi(u,\nabla u)=\Psi(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } \Omega,\\ u=0 & \mbox{on } \partial\Omega. \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 $W^{-1,\overrightarrow{p}'}(\Omega)$ satisfying suitable, but general, structural assumptions. $\Phi$ and $\Psi$ are gradient-dependent nonlinearities whose models are the following: \begin{equation*} \label{phi}\Phi(u,\nabla u):=\left(\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}+1\right)|u|^{m-2}u, \quad \Psi(u,\nabla u):=\frac{1}{u}\sum_{j=1}^N |u|^{\theta_j} |\partial_j u|^{q_j}. \end{equation*} We suppose throughout that, for every $1\leq j\leq N$, \begin{equation*}\label{ass} \mathfrak{a}_j\geq 0, \quad \theta_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 $1\leq j\leq N$, we have $\theta_j\geq 1$; 2) there exists $1\leq j\leq N$ such that $\theta_j<1$. In this last situation, we look for non-negative solutions of \eqref{eq0}.