Multiple solutions for a class of $$N$$ N -Kirchhoff type equations via variational methods

Abstract

In this article, we consider the following \(N\)-Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} -M\left( \int \limits _{\Omega }|\nabla u|^N\,dx\right) \Delta _N u = \lambda f(x,u) +\mu g(x,u) \quad \text { in } \Omega ,\\ u =0 \quad \text { on } \partial \Omega , \end{array}\right. \end{aligned}$$ where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\), \(N \ge 2\), \(M: {\mathbb {R}}^+_0 \rightarrow {\mathbb {R}}\) is a continuous function, \(\Delta _Nu = {\mathrm {div}} (|\nabla u|^{N-2}\nabla u)\), \(f,g: \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) are two Caratheodory functions and \(\lambda , \mu \) are positive parameters. Using variational method, we show the existence of at least three weak solutions for the problem.