Existence of Positive Solutions for the Nonlinear Kirchhoff Type Equations 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}

In this paper, we will study the following nonlinear Kirchhoff-type equation: -a+λ∫R3|∇u|2dx▵u+V(x)u=f(u),x∈R3,u∈H1(R3),u>0,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\left( a+\lambda \int _{{\mathbb {R}}^3}|\nabla u|^{2}dx\right) \triangle u+V(x)u=f(u), &{}x\in {\mathbb {R}}^3,\\ \displaystyle u\in H^1({\mathbb {R}}^3),\ u>0,&{}x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$\end{document}where a,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,\lambda $$\end{document} are positive constants, f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C({\mathbb {R}}, {\mathbb {R}})$$\end{document} is subcritical near infinity and superlinear near zero, and f also satisfies general Ambrosetti-Rabinowitz condition. Besides, V is a positive continuous potential satisfying 0<V0≤V(x)≤lim inf|x|→∞V(x)=V∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<V_0\le V(x)\le \liminf _{|x|\rightarrow \infty }V(x)=V_\infty $$\end{document}. Under the conditions of f and V, it is not easy to obtain bounded Palais–Smale sequences, so there we use a cut-off technique to overcome. Moreover, in order to conquer the difficulties of the lack of compactness due to the unboundedness of the domain 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} and V is a non-constant function, we use a new global compactness lemma to show the convergence of Palais–Smale sequences. Therefore, by the above methods, we prove the existence of positive solutions to the above equation for all λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} small.