Multiple positive solutions for a p-Laplace Benci-Cerami type problem (1<p<2), via Morse theory

Abstract

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p Δ_{p}u + u^{p-1} = f(u) & \hbox{in} \ Ω \newline u>0 & \hbox{in} \ Ω \newline u=0 & \hbox{on} \ \partial Ω \end{array} \right. \] where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with smooth boundary, $N\ge 2$, $1<p<2$, $\epsilon >0$ is a parameter and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists ${\epsilon }^{\ast }>0$ such that, for every $\epsilon \in (0,{\epsilon }^{\ast })$, $(P_{\epsilon })$ has at least $2{\mathcal{P}}_1(\Omega )-1$ solutions, possibly counted with their multiplicities, where ${\mathcal{P}}_t(\Omega )$ is the Poincaré polynomial of $\Omega$. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on $\Omega$, approximating $(P_{\epsilon })$.

to be published in "Communications in Contemporary Mathematics"