Singular anisotropic elliptic equations with gradient-dependent lower order terms

Barbara Brandolini and Florica C. Cîrstea

Abstract

We prove the existence of weak solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$\mathcal{A}u+\mathrm{\Phi }(x,u,\mathrm{\nabla }u)=\mathrm{\Psi }(u,\mathrm{\nabla }u)+\mathfrak{B}u+f$$ on a bounded open subset $\mathrm{\Omega }\subset {\mathbb{R}}^N$ $(N\ge 2)$, where $f\in L^1(\mathrm{\Omega })$ is arbitrary. Our models are $\mathcal{A}u=-\sum _{j=1}^N{\partial }_j(|{\partial }_{j}u{|}^{p_j-2}{\partial }_{j}u)$ and $\mathrm{\Phi }(u,\mathrm{\nabla }u)=(1+\sum _{j=1}^N{\mathfrak{a}}_j|{\partial }_{j}u{|}^{p_j})|u{|}^{m-2}u$, with $m,p_j>1$, ${\mathfrak{a}}_j\ge 0$ for $1\le j\le N$ and $\sum _{k=1}^N(1/p_k)>1$. The main novelty is the inclusion of a possibly singular gradient-dependent term $\mathrm{\Psi }(u,\mathrm{\nabla }u)=\sum _{j=1}^N|u{|}^{{\theta }_j-2}u|{\partial }_{j}u{|}^{q_j}$, where ${\theta }_j>0$ and \(0\leq q_j 1\) and 2) there exists $1\le j\le N$ such that ${\theta }_j\le 1$. In the latter situation, assuming that $f\ge 0$ a.e. in $\mathrm{\Omega }$, we obtain non-negative solutions for our problem.

Keywords: Leray–Lions operators, anisotropic operators, boundary singularity, summable data.

AMS Subject Classification: Primary 35J75; secondary 35J60, 35Q35.