Multiple solutions to a singular Lane-Emden-Fowler equation with convection term

Aranda, Carlos C. ; Dozo, Enrique Lami

Electronic journal of differential equations, Tome 2008 (2008)

Résumé

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.

Détail

Classification : 35J25, 35J60
Keywords: bifurcation, weighted principal eigenvalues and eigenfunctions