Positive ground states for nonlinear Schrödinger–Kirchhoff equations with periodic potential or potential well in mathbf{R}^{3}

Abstract

This work is devoted to the nonlinear Schrödinger–Kirchhoff-type equation −(a+b∫R3|∇u|2dx)Δu+V(x)u=f(x,u),in R3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - \\biggl( a+b \\int _{\\mathbb{R}^{3}} \\vert \ abla u \\vert ^{2} \\,\ ext{d}x \\biggr) \\Delta u+V(x)u=f(x,u), \\quad \ ext{in } \\mathbb{R}^{3}, $$\\end{document} where a>0, bgeq 0, the nonlinearity f(x,cdot ) is 3-superlinear and the potential V is either periodic or exhibits a finite potential well. By the mountain pass theorem, Lions’ concentration-compactness principle, and the energy comparison argument, we obtain the existence of positive ground state for this problem without proving the Palais–Smale compactness condition.