Structure of ground state solutions for singular elliptic equations with a quadratic gradient term

Melo, Antonio Luiz ; Santos, Carlos Alberto

Electronic Journal of Differential Equations, Tome 2011 (2011).

Résumé

Summary: We establish results on existence, non-existence, and asymptotic behavior of ground state solutions for the singular nonlinear elliptic problem $$\displaylines{ -\Delta u = g(u)| \nabla u |^2 + \lambda\psi(x) f(u) \quad\hbox{in } \mathbb{R}^N,\cr u > 0 \quad\hbox{in } \mathbb{R}^N,\quad \lim_{|x| \to \infty} u(x)=0, }$$ where $\lambda \in \mathbb{R}$ is a parameter, $\psi \geq 0 $, not identically zero, is a locally Holder continuous function; $g:(0,\infty) \to \mathbb{R}$ and $f:(0,\infty) \to (0,\infty)$ are continuous functions, (possibly) singular in 0; that is, $f(s)\to \infty$ and either $g(s)\to \infty$ or $g(s)\to -\infty$ as $s \to 0$. The main purpose of this article is to complement the main theorem in Porru and Vitolo [15], for the case $\Omega=\mathbb{R}^N$. No monotonicity condition is imposed on f or g.

Détail

Classification : 35J25, 35J20, 35J67
Keywords: singular elliptic equations, gradient term, ground state solution