## 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.

: Primary 35J75; secondary 35J60, 35Q35.

This paper is available as a pdf (544kB) file. It is also on the arXiv: arxiv.org/abs/arXiv:2001.02887.

 Friday, September 9, 2022