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.