Multiplicity of positive solutions for a class of critical Sobolev exponent problems involving Kirchhoff-type nonlocal term

Abstract

We study a class of critical Sobolev exponent problems involving Kirchhoff-type nonlocal term

$$\left\{\begin{array}{cc}-\left(a+b{\int }_{\Omega }|\nabla u{|}^{2}dx\right)\Delta u=u^5+g(x,u)+\lambda u^{q-1},\hfill & x\in \Omega ,\hfill \\ u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}\right.$$

where $\Omega \subset {\mathbb{R}}^3$ is a bounded domain, $a,b,\lambda >0,1<q<2,$ and $g\in C(\overline{\Omega }\times \mathbb{R})$ satisfies some conditions. By the variational method, we obtain two positive solutions. Moreover, we prove that there exists a positive ground state solution by the Nehari method.