PreprintAnisotropic elliptic equations with gradientdependent lower order terms and \(L^1\) dataBarbara Brandolini and Florica C. CîrsteaAbstractFor 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 divergenceform nonlinear anisotropic operator \(\mathcal A\), the prototype of which is \(\mathcal A u=\sum_{j=1}^N \partial_j(\partial_j u^{p_j2}\partial_j u)\) 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 pseudomonotone from \(W_0^{1,\overrightarrow{p}}(\Omega)\) into its dual, as well as a gradientdependent 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.
This paper is available as a pdf (440kB) file. It is also on the arXiv: arxiv.org/abs/arXiv:2001.02754.
