Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term

Abstract

Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div[a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of R N , with N>2, f∈L m(Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m> N/2, and an unbounded entropy solution for N/2> m⩾2 N/( N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.