Multiple positive solutions for Kirchhoff-type problems in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document} involving critical Sobolev exponents

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent -a+b∫R3|∇u|2dxΔu+u=λf(x)uq-1+g(x)u5,x∈R3,u∈H1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll}-\left(a+b\displaystyle\int\limits_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\right)\Delta u+u=\lambda f(x)u^{q-1}+g(x)u^5,\quad x\in \mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array} \right.$$\end{document}where a, b >  0, 4 <  q <  6, and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document} is a positive parameter. Under certain assumptions on f(x) and g(x) and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document} is small enough, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g. The Nehari manifold and Ljusternik–Schnirelmann category are the main tools in our study. Moreover, using the Mountain Pass Theorem, we give an existence result about λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document} large.