Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential

Abstract

We consider the following semilinear Schrodinger equation with inverse square potential \\begin{document}$\\begin{array}{l}\\left\\{ \\begin{align} & -\\vartriangle u+(V(x)-\\frac{\\mu }{|x{{|}^{2}}}u=f(x,u),\\ \\ \\ \\ \\ x\\in {{\\mathbb{R}}^{N}}, \\\\ & u\\in {{H}^{1}}({{\\mathbb{R}}^{N}}), \\\\ \\end{align} \\right.\\end{array}$ \\end{document} where $N≥ 3$, $f$ is asymptotically linear, $V$ is 1-periodic in each of $x_1, ..., x_N$ and $\\sup[σ(-\\triangle +V)\\cap (-∞, 0)]＜0＜{\\rm{inf}}[σ(-\\triangle +V)\\cap (0, ∞)]$. Under some mild assumptions on $V$ and $f$, we prove the existence and asymptotical behavior of ground state solutions of Nehari-Pankov type to the above problem.