Existence of solutions for degenerated problems in <var>L</var>¹ having lower order terms with natural growth

Abstract

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation

Au + g(x,u,∇u) = f,

where A is a Leray–Lions operator from the weighted Sobolev space W01,p(Ω, w) into its dual W −1,p'(Ω, w*). While g(x,s,ξ) is a nonlinear term having natural growth with respect to ξ and no growth with respect to s, it satisfies a sign condition on s, i.e., g(x,s,ξ) · s ≥ 0 for every s∈ℝ. The right-hand side f belongs to L1(Ω).