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

TL;DR

This paper proves the existence of multiple positive solutions for a p-Laplace problem with subcritical growth, using Morse theory to relate solution multiplicity to topological invariants of the domain.

Contribution

It establishes a lower bound on the number of solutions for the p-Laplace problem via Morse theory, linking solution multiplicity to the Poincaré polynomial of the domain.

Findings

Existence of at least 2P_1(Ω)-1 solutions for small ε

Solutions are characterized using Morse theory

The multiplicity relates to topological invariants of Ω

Abstract

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega \end{array} \right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ 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 $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least $2{\mathcal P}_1(\Omega)-1$ solutions, possibly counted with their multiplicities, where ${\mathcal P}_t(\Omega)$ is the Poincar\'e 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 $(P_\varepsilon)$.