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 on a bounded open subset Ω⊂RN (N≥2), where f∈L1(Ω) is arbitrary. Our models are Au=−∑Nj=1∂j(|∂ju|pj−2∂ju) and Φ(u,∇u)=(1+∑Nj=1aj|∂ju|pj)|u|m−2u, with m,pj>1,aj≥0 for 1≤j≤N and ∑Nk=1(1/pk)>1. The main novelty is the inclusion of a possibly singular gradient-dependent term Ψ(u,∇u)=∑Nj=1|u|θj−2u|∂ju|qj, where θj>0 and 0≤qj1 and 2) there exists 1≤j≤N such that θj≤1. In the latter situation, assuming that f≥0 a.e. in Ω, we obtain non-negative solutions for our problem