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

Abstract

<abstract><p>We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. 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 $ \Phi $ with an "anisotropic natural growth" in the gradient and a good sign condition.</p></abstract>