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Multiplicity results for a singular and quazilinear equation

Pages: 429 - 435, Issue Special, September 2007

 Abstract        Full Text (229.7K)              

J. Giacomoni - MIP-Ceremath, Manufacture des Tabacs-Bat C, 21, allée de Brienne, 31000 Toulouse, France (email)
K. Sreeandh - MIP-Ceremath, Manufacture des Tabacs-Bat C, 21, allée de Brienne, 31000 Toulouse, France (email)

Abstract: In this paper, we investigate the following quasilinear and singular problem:

-$\Delta_pu = \lambda/(u^\delta) + u^q$ in $\Omega$ $u|_(\partial\Omega) = 0 , u > 0$ in $\Omega$       (1)

where $\Omega$ is an open bounded domain with smooth boundary, 1 < $p$, $p - 1$ < $q$ and $\lambda$, $\delta$ > 0. We first prove that there exist weak solutions for $\lambda$ > 0 small in $W^(1,p)_0(\Omega) \cap C(bar(\Omega))$ if and only if $\delta < 2 + 1/(p - 1)$ . Investigating the radial symmetric case $(\Omega = B_R(0))$, we prove by a shooting method the global multiplicity of solutions to $(P)$ in $C(bar(\Omega))$ with $0 < \delta$, 1 < $p$ and $p - 1$ < $q$.

Keywords:  Quasilinear and singular equations, monotone methods, asymptotic methods for O.D.E.
Mathematics Subject Classification:  Primary: 35J35; Secondary: 35R05.

Received: September 2006;      Revised: June 2007;      Published: September 2007.