Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 05, pp. 1-21.
Title: Multiple solutions to a singular Lane-Emden-Fowler equation
with convection term
Authors: Carlos C. Aranda (Univ. Nacional de Formosa, Argentina)
Enrique Lami Dozo (Univ. de Buenos Aires, Argentina)
Abstract:
This article concerns the existence of multiple
solutions for the problem
$$\displaylines{
-\Delta u = K(x)u^{-\alpha}+s(\mathcal{A}u^\beta+\mathcal{B}
|\nabla u|^\zeta)+f(x) \quad \hbox{in }\Omega\cr
u > 0 \quad \hbox{in }\Omega\cr
u = 0 \quad \hbox{on }\partial\Omega\,,
}$$
where $\Omega$ is a smooth, bounded domain in $\mathbb{R}^n$ with
$n\geq 2$, $\alpha$, $\beta$, $\zeta$, $\mathcal{A}$,
$\mathcal{B}$ and $s$ are real positive numbers, and $f(x)$ is a
positive real valued and measurable function. We start with the
case $s=0$ and $f=0$ by studying the structure of the range of
$-u^\alpha\Delta u$. Our method to build $K$'s which give at least
two solutions is based on positive and negative principal
eigenvalues with weight. For $s$ small positive and for values
of the parameters in finite intervals, we find
multiplicity via estimates on the bifurcation set.
Submitted August 12, 2007. Published January 02, 2008.
Math Subject Classifications: 35J25, 35J60.
Key Words: Bifurcation; weighted principal eigenvalues and eigenfunctions.