Multiplicity of solutions for a p-Kirchhoff equation.

In this paper, we consider the following p-Kirchhoff equation: with Dirichlet boundary conditions, where Ω is a bounded domain in $\mathbb{R}^{N}$ . Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity $f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u$ ( $1< q< p< r< p^{*}$ ), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem. [ABSTRACT FROM AUTHOR]