Bounded solutions for a quasilinear singular problem with nonlinear Robin boundary conditions

Abstract

The paper deals with the existence of bounded positive solutions for a quasilinear singular problem with nonlinear Robin conditions. The nonlinear term is singular with respect to the solution and has a quadratic growth with respect to its gradient. The result is obtained by approximation. We define an approximate problem which does not present the singularity and is bounded with respect to the gradient. In a first step we prove the existence of a bounded solution of this problem, using the Schauder fixed-point theorem. Then, we prove uniform a priori estimates for these solutions. Finally, by some equintegrability arguments, we pass to the limit in the approximate problem and show that the approximate solution converges to a solution of our problem. We also prove a strong-maximum property for the problem. In the last part of the paper, we consider the case of a nonsingular but more general quadratic growth with nonlinear Robin conditions.