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

Abstract

Using minimax methods and Lusternik-Schnirelmann theory, we study multiple positive solutions for the Schrödinger - Kirchhoff equation $$ M\left(\dis\int_{Ω_λ}|\nabla u|^{2}dx+\dis\int_{Ω_λ}u^{2}dx\right)\left[-Δu + u \right]= f(u) $$ 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 $\bar{\Omega }$.

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