Existence and multiplicity results for supercritical nonlocal Kirchhoff problem

Abstract

We study the existence and multiplicity of solutions for the nonlocalperturbed Kirchhoff problem$$\displaylines{-\Big(a+b\int_\Omega |\nabla u|^2\,dx\Big)\Delta u=\lambda g(x,u)+f(x,u), \quad \text{in } \Omega,\\ u=0, \quad\text{on }\partial\Omega,}$$ where Ω is a bounded smooth domain in  \(\mathbb{R}^N\), \(N>4\),  \(a,b, \lambda > 0\), and  \(f,g:\Omega\times \mathbb{R}\to \mathbb{R}\)  are Caratheodory functions, with \(f\) subcritical, and \(g\) of arbitrary growth. This paper is motivated by a recent results by Faraci and Silva [4] where existence and multiplicity results were obtained when g is subcritical and f is a power-type function withcritical exponent.