Existence of ground state solution for a class of one-dimensional Kirchhoff-type equations with asymptotically cubic nonlinearities

摘要

Abstract This paper is concerned with the following Kirchhoff-type equation: -1+∫R|u′|2dxu′′+p(x)u=l(x)u3+f(x,u),x∈R,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} -left( 1+int _{mathbb {R}} |u'|^2dxright) u''+p(x)u=l(x)u^3+f(x,u) ,quad xin mathbb {R}, end{aligned}$$end{document}where p,l∈CRdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p,l in Cleft( mathbb {R}right) $$end{document} and f∈C(R×R,R)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$fin C(mathbb {R}times mathbb {R},mathbb {R})$$end{document}. Under some suitable assumptions on p, l and f, we establish the existence of nontrivial ground state solutions to the above equation. The results of this paper extend those in [Cheng, Li and Tang, Applicable Anal. 98 (2019), 1255-1266] for the one-dimensional case.