Zhijie Chen

Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schr\"odinger equations

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