Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

Abstract

We study the existence of solutions of the following nonlinear Schrödinger equation \begin{equation*} -Δu + \Big(V(x)-\fracμ{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where $V:{\mathbb{R}}^N\to \mathbb{R}$ and $f:{\mathrm{R}}^N\times \mathbb{R}\to \mathbb{R}$ are periodic in $x\in \mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-\Delta +V$ and $\mu <\frac{(N-2{)}^2}{4}$, $N\ge 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $\mu \ge 0$ we find a ground state solution as a minimizer of the energy functional on a natural constraint. If $\mu <0$ and $0$ lies below the spectrum of $-\Delta +V$, then ground state solutions do not exist.

References in corpus (2)

• Bifurcation into spectral gaps for a noncompact semilinear Schrödinger equation with nonconvex potential
• Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

Cited by in corpus (1)

• Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities