Gelfand type quasilinear elliptic problems with quadratic gradient terms

Arcoya, David; Carmona, José; Martínez-Aparicio, Pedro J.

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 249-265

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

$$\{\begin{array}{cc}-\Delta u+m\left(x\right)\frac{{\left|\nabla u\right|}^2}{1+u}=\lambda {(1+u)}^p\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega .\hfill \end{array}$$

We prove that the maximal set of λ 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 $\lambda \in (0,{\lambda }^{⁎})$. 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 $m_1=m_2$.

Keywords: Gelfand problem, Quasilinear elliptic equations, Quadratic gradient, Stability condition, Extremal solutions

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