Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 70, pp. 1-14.
Title: Structure of ground state solutions for singular elliptic
equations with a quadratic gradient term
Authors: Antonio Luiz Melo (Univ. of Brasilia, Planaltina, Brazil)
Carlos Alberto Santos (Univ. of Brasilia, Brazil)
Abstract:
We establish results on existence, non-existence, and
asymptotic behavior of ground state solutions for the
singular nonlinear elliptic problem
$$\displaylines{
-\Delta u = g(u)| \nabla u |^2 + \lambda\psi(x) f(u)
\quad\hbox{in } \mathbb{R}^N,\\
u > 0 \quad\hbox{in } \mathbb{R}^N,\quad
\lim_{|x| \to \infty} u(x)=0,
}$$
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.
Submitted April 26, 2010. Published May 31, 2011.
Math Subject Classifications: 35J25, 35J20, 35J67.
Key Words: Singular elliptic equations; gradient term;
ground state solution.