Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations

We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-Δu+|y|^2u+λu=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-Δ+ |y|^2$ is given by $[d,\infty)$. We prove that $λ\geq-d$ is necessary for positive