Multiplicity of Positive Solutions for an Obstacle Problem in R

Abstract

In this paper we establish the existence of two positive solutions for the obstacle problem $$ \displaystyle \int_{\Re}\left[u'(v-u)'+(1+λV(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka $$ where $f$ is a continuous function verifying some technical conditions and $\Ka$ is the convex set given by $$ \Ka =\left\{v\in H^{1}(\Re); v \geq φ\right\}, $$ with $\phi \in H^1(\Re )$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle problem, Variational methods, Positive solutions.

To appear in Progress in Nonlinear Differential Equations and their Applications