## Anisotropic elliptic equations with gradient-dependent lower order terms and $$L^1$$ data

### Barbara Brandolini and Florica C. Cîrstea

#### Abstract

For every summable function $$f$$, we prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems in a bounded open subset $$\Omega$$ of $$\mathbb R^N$$. The principal part is a divergence-form nonlinear anisotropic operator $$\mathcal A$$, the prototype of which is $$\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u)$$ with $$p_j>1$$ for all $$1\leq j\leq N$$ and $$\sum_{j=1}^N (1/p_j)>1$$. As a novelty in this paper, our lower order terms involve a new class of operators $$\mathfrak B$$ such that $$\mathcal{A}-\mathfrak{B}$$ is bounded, coercive and pseudo-monotone from $$W_0^{1,\overrightarrow{p}}(\Omega)$$ into its dual, as well as a gradient-dependent nonlinearity with an "anisotropic natural growth" in the gradient and a good sign condition.

Keywords: Nonlinear anisotropic elliptic equations, Leray–Lions operators, summable data.

: Primary 35J25; secondary 35B45, 35J60.

This paper is available as a pdf (440kB) file. It is also on the arXiv: arxiv.org/abs/arXiv:2001.02754.

 Wednesday, August 25, 2021