Existence of solutions for degenerated problems in $L^1$ having lower order terms with natural growth
Lahsen Aharouch, Elhoussine Azroul, Abdelmoujib Benkirane · Portugaliae Mathematica · 2008
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 W_0^{1,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 L^1(Ω) .