The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem
João R. Santos Júnior · arXiv (Cornell University) · 2013
Using minimax methods and Lusternik-Schnirelmann theory, we study multiple positive solutions for the Schrödinger - Kirchhoff equation $$ M\left(\dis\int_{Ω_λ}| abla u|^{2}dx+\dis\int_{Ω_λ}u^{2}dx\right)\left[-Δu + u \right]= f(u) $$ in $Ω_λ = λΩ$. The set $Ω\subset \mathbb{R}^3$ is a smooth bounded domain, $λ>0$ 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 $λ$, the number of solutions with the least number of closed and contractible in $\barΩ$ which cover $\barΩ$.