Multiplicity of Positive Solutions for an Obstacle Problem in R

TL;DR

This paper proves the existence of two positive solutions for a specific obstacle problem involving variational inequalities in the real line, under certain conditions on the function and potential.

Contribution

It establishes the multiplicity of positive solutions for an obstacle problem in R using variational methods, which is a novel result for this class of problems.

Findings

Existence of two positive solutions proven.

Solutions are obtained via variational methods.

Results contribute to the theory of obstacle problems in unbounded domains.

Abstract

In this paper we establish the existence of two positive solutions for the obstacle problem $$ \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka $$ where $f$ is a continuous function verifying some technical conditions and $\Ka$ is the convex set given by $$ \Ka =\left\{v\in H^{1}(\Re); v \geq \varphi \right\}, $$ with $\varphi \in H^{1}(\Re)$ 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.