Electron. J. Diff. Equ., Vol. 2011 (2011), No. 70, pp. 1-14.

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

Antonio Luiz Melo, Carlos Alberto Santos

Abstract:
We establish results on existence, non-existence, and asymptotic behavior of ground state solutions for the singular nonlinear elliptic problem
$$\displaylines{ -\Delta u = g(u)| \nabla u |^2 + \lambda\psi(x) f(u) \quad\hbox{in } \mathbb{R}^N,\cr u > 0 \quad\hbox{in } \mathbb{R}^N,\quad \lim_{|x| \to \infty} u(x)=0, }$$
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.

Submitted April 26, 2010. Published May 31, 2011.
Math Subject Classifications: 35J25, 35J20, 35J67.
Key Words: Singular elliptic equations; gradient term; ground state solution.

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Antônio Luiz Melo
Department of Mathematics, University of Brasília
Planaltina, 70910-990, Brazil
email: almelo@unb.br
Carlos Alberto Santos
Department of Mathematics, University of Brasília
Brasília, 70910-900, Brazil
email: csantos@unb.br, capdsantos@gmail.com

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