On an elliptic Kirchhoff-type problem depending on two parameters

Abstract

In this paper, on a bounded domain ${\Omega\subset {\bf R}^n}$, we consider a non-local problem of the type

$$\left\{\begin{array}{l}-K\left(\int_{\Omega}|\nabla u(x)|^2dx\right)\Delta u =\lambda f(x,u)+\mu g(x,u) \quad {\rm in}\,\,\Omega\\ u=0 \quad {\rm on}\,\,\partial\Omega.\end{array}\right.$$

Under rather general assumptions on K and f, we prove, in particular, that there exists λ* > 0 such that, for each λ > λ* and each Carathéodory function g with a sub-critical growth, the above problem has at least three weak solutions for every μ ≥ 0 small enough.