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 $\mathrm{\Omega }$ 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 $1\le j\le N$ 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 $W_0^{1,\overrightarrow{p}}(\mathrm{\Omega })$ 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.

AMS Subject Classification: Primary 35J25; secondary 35B45, 35J60.