Existence of solutions for Kirchhoff type problems with critical nonlinearity in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begi

AbstractIn this paper, we consider the existence and multiplicity of standing wave solutions of Kirchhoff type problems with critical nonlinearity in $${\mathbb{R}^N}$$RN : $$-\varepsilon^p \left(a + b \int\limits_{\mathbb{R}^N} \frac{1}{p}|\nabla u|^p{\rm d}x \right) \,{\rm div}(|\nabla u|^{p-2}\nabla u) + V(x)|u|^{p-2}u = K(x)|u|^{p^\ast-2}u + h(x,u),$$-εpa+b∫RN1p|∇u|pdxdiv(|∇u|p-2∇u)+V(x)|u|p-2u=K(x)|u|p*-2u+h(x,u), for all $${(t, x) \in \mathbb{R} \times \mathbb{R}^N}$$(t,x)∈R×RN, where V(x) is a nonnegative potential, and K(x) is a bounded positive function. Under suitable assumptions, we show that this equation has at least one solution provided that $${\varepsilon < \mathcal {E}}$$ε<E, for any $${m \in \mathbb{N}}$$m∈N, it has m pairs of solutions if $${\varepsilon < \mathcal {E}_m}$$ε<Em, where $${\mathcal {E}}$$E and $${\mathcal {E}_m}$$Em are sufficiently small positive numbers. Moreover, these solutions $${u_\varepsilon \rightarrow 0}$$uε→0 in $${W^{1,p}(\mathbb{R}^N)}$$W1,p(RN) as $${\varepsilon \rightarrow 0}$$ε→0.