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.

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