The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem

TL;DR

This paper investigates how the shape of a domain influences the number of positive solutions to a Kirchhoff-type elliptic PDE, using advanced variational methods to establish solution multiplicity for large domain scaling.

Contribution

It introduces a novel analysis linking domain topology to solution count in Kirchhoff problems, employing minimax and Lusternik-Schnirelmann techniques.

Findings

Number of solutions increases with domain size

Solutions correspond to topological features of the domain

Method applies to superlinear, subcritical nonlinearities

Abstract

Using minimax methods and Lusternik-Schnirelmann theory, we study multiple positive solutions for the Schr\"{o}dinger - Kirchhoff equation $$ M\left(\dis\int_{\Omega_{\lambda}}|\nabla u|^{2}dx+\dis\int_{\Omega_{\lambda}}u^{2}dx\right)\left[-\Delta u + u \right]= f(u) $$ in $\Omega_{\lambda} = \lambda\Omega$. The set $\Omega \subset \mathbb{R}^3$ is a smooth bounded domain, $\lambda>0$ is a parameter, $M$ is a general continuous function and $f$ is a superlinear continuous function with subcritical growth. Our main result relates, for large values of $\lambda$, the number of solutions with the least number of closed and contractible in $\bar{\Omega}$ which cover $\bar{\Omega}$.