Gelfand type quasilinear elliptic problems with quadratic gradient terms

In this paper, for 0 < m(1) <= m(x) <= m(2) and positive parameters lambda and p, we study the existence of positive solution for the quasilinear model problem { -Delta u + m(x) vertical bar del u vertical bar(2)/1 + u = lambda(1 + u)(P) in Omega, u = 0 on partial derivative Omega. We prove that the maximal set of lambda for which the problem has at least one positive solution is an interval (0, lambda*], with lambda* > 0, and there exists a minimal regular positive solution for every lambda is an element of (0, lambda*). We also prove, under suitable conditions depending on the dimension N and the parameters p, m(1), m(2), that for lambda = lambda* there exists a minimal regular positive solution. Moreover we characterize minimal solutions as those solutions satisfying a stability condition in the case m(1) = m(2). (C) 2013 Elsevier Masson SAS. All rights reserved.