Gelfand type quasilinear elliptic problems with quadratic gradient terms

Abstract

In this paper, for $0<m_1\le m(x)\le m_2$ and positive parameters $\lambda$ and $p$, we study the existence of positive solution for the quasilinear model problem

We prove that the maximal set of $\lambda$ for which the problem has at least one positive solution is an interval $(0,{\lambda }^{\ast }]$, with ${\lambda }^{\ast }>0$, and there exists a minimal regular positive solution for every $\lambda \in (0,{\lambda }^{\ast })$. We also prove, under suitable conditions depending on the dimension $N$ and the parameters $p$, $m_1$, $m_2$, that for $\lambda ={\lambda }^{\ast }$ there exists a minimal regular positive solution. Moreover we characterize minimal solutions as those solutions satisfying a stability condition in the case $m_1=m_2$.