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 \[\mathcal A u+\Phi(x,u,\nabla u)=\Psi(u,\nabla u)+\mathfrak Bu +f \] on a bounded open subset \(\Omega\subset \mathbb R^N\) \((N\geq 2)\), where \(f\in L^1(\Omega)\) is arbitrary. Our models are \( \mathcal Au=-\sum_{j=1}^N \partial_j (|\partial_j u|^{p_j-2}\partial_j u)\) and \(\Phi(u,\nabla u)=\left(1+\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}\right)|u|^{m-2}u\), with \(m,p_j>1\), \(\mathfrak{a}_j\geq 0\) for \(1\leq j\leq N\) and \(\sum_{k=1}^N (1/p_k)>1\). The main novelty is the inclusion of a possibly singular gradient-dependent term \(\Psi(u,\nabla u)=\sum_{j=1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j}\), where \(\theta_j>0\) and \(0\leq q_j 1\) and 2) there exists \(1\leq j\leq N\) such that \(\theta_j\leq 1\). In the latter situation, assuming that \(f \ge 0\) a.e. in \(\Omega\), 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.