Electron. J. Diff. Eqns., Vol. 2008(2008), No. 05, pp. 1-21.

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

Carlos C. Aranda, Enrique Lami Dozo

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 2, 2008.
Math Subject Classifications: 35J25, 35J60.
Key Words: Bifurcation; weighted principal eigenvalues and eigenfunctions.

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Carlos C. Aranda
Mathematics Department
Universidad Nacional de Formosa, Argentina
email: carloscesar.aranda@gmail.com
Enrique Lami Dozo
CONICET-Universidad de Buenos Aires, Argentina.
and Univ. Libre de Bruxelles, Belgium
email: lamidozo@ulb.ac.be

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