Existence and estimates of some Mountain-Pass type solutions

Issue		ESAIM: COCV Volume 15, Number 3, July-September 2009


Page(s)		499 - 508
DOI		10.1051/cocv/2009015
Published online		02 July 2009

ESAIM: COCV 15 (2009) 499-508
DOI: 10.1051/cocv/2009015

Existence and $L_\infty$ estimates of some Mountain-Pass type solutions

José Maria Gomes

CMAF-Centro de Matemática e Aplicações Fundamentais, 2 Avenida Professor Gama Pinto, 1649-003 Lisboa, Portugal. jgomes@math.ist.utl.pt

Received September 26, 2006. Revised March 29, 2007. Published online July 2nd, 2009.

Abstract
We prove the existence of a positive solution to the BVP

$\begin{displaymath}(\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0, \end{displaymath}$

imposing some conditions on $\Phi$ and f. In particular, we assume $\Phi(t)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_\infty$ bound for the solution is provided by the $L_\infty$ norm of any test function with negative energy.

Mathematics Subject Classification. 34B18, 34C11

Key words: Second order singular differential equation, variational methods, Mountain Pass Theorem

© EDP Sciences, SMAI 2009

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