Least energy sign-changing solutions for a class of nonlocal Kirchhoff-type problems

Abstract

In this paper, we consider the existence of least energy sign-changing solutions for a class of Kirchhoff-type problem \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\left( a+b\\int _{\\Omega }|\ abla u|^{2}dx\\right) \\Delta u =g(x,u), &\\quad x \\in \\Omega , \\\\ u=0, &\\quad x \\in \\partial \\Omega , \\end{array}\\right. \\qquad (K_b) \\end{aligned}$$\\end{document}-a+b∫Ω|∇u|2dxΔu=g(x,u),x∈Ω,u=0,x∈∂Ω,(Kb)where Omega is a bounded domain in {mathbb {R}}^N, N=1, 2, 3, with a smooth boundary partial Omega , a>0, b>0 and gin C^0(Omega times {mathbb {R}}, {mathbb {R}}). By using variational approach and some subtle analytical skills, the existence of the least energy sign-changing solutions of (K_b) is obtained successfully. Moreover, we prove that the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (K_b).