Quasilinear elliptic equations with natural growth. (English) Zbl 1212.35078

Summary: In this paper we deal with the problem \(-\operatorname {div}(a(x,u)\nabla u)+g(x,u,\nabla u)=\lambda h(x)u+f\) in \(\Omega \), \(u=0\) on \(\partial \Omega \). The main goal of the work is to get hypotheses on \(a\), \(g\) and \(h\) such that the previous problem has a solution for all \(\lambda >0\) and \(f\in L^1(\Omega )\). In particular, we focus our attention in the model equation with \(a(x,u)=1+| u| ^m\), \(g(x,u,\nabla u)=\frac m2| u| ^{m-2}u| \nabla u| ^2\) and \(h(x)=| x| ^{-2}\).


35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems