Singular anisotropic elliptic equations with gradient-dependent lower order terms

Abstract

We prove the existence of weak solutions for a general class of Dirichlet anisotropic elliptic problems of the form Au+Φ(x,u,∇u)=Ψ(u,∇u)+Bu+f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}{\\mathcal {A}} u+\\Phi (x,u,\ abla u)=\\Psi (u,\ abla u)+\\mathfrak Bu +f \\end{aligned}$$\\end{document}on a bounded open subset Omega subset {mathbb {R}}^N(Nge 2), where fin L^1(Omega ) is arbitrary. Our models are mathcal Au=-sum _{j=1}^N partial _j (|partial _j u|^{p_j-2}partial _j u) and Phi (u,nabla u)=left( 1+sum _{j=1}^N {mathfrak {a}}_j |partial _j u|^{p_j}right) |u|^{m-2}u, with m,p_j>1,{mathfrak {a}}_jge 0 for 1le jle N and sum _{k=1}^N (1/p_k)>1. The main novelty is the inclusion of a possibly singular gradient-dependent term Psi (u,nabla u)=sum _{j=1}^N |u|^{theta _j-2}u, |partial _j u|^{q_j}, where theta _j>0 and 0le q_j<p_j for 1le jle N. Under suitable conditions, we prove the existence of solutions by distinguishing two cases: 1) for every 1le jle N, we have theta _j> 1 and 2) there exists 1le jle N such that theta _jle 1. In the latter situation, assuming that f ge 0 a.e. in Omega , we obtain non-negative solutions for our problem.