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

This paper studies positive solutions to the stationary nonlinear Schrödinger equation with a partial harmonic confinement: $$-\mathrm{\Delta }u+(x_1^2+\cdots +x_d^2)u+\lambda u=g(u),u\in H^1({\mathbb{R}}^N),$$ where $1\le d\le N-1$, and the nonlinearity $g(u)\ge 0$ satisfies standard regularity and subcritical growth conditions. While previous works only studied the case $\lambda \ge 0$, here by applying a spectral analysis of the Schrödinger operator $H=-\mathrm{\Delta }+\mathrm{\mid }y{\mathrm{\mid }}^2,y=(x_1,\cdots ,x_d)$, together with the variational method and moving planes method, we prove that \begin{itemize} \item A necessary condition for the existence of positive solutions is $\lambda \ge -d$, and there are no positive solutions for the zero-mass case $\lambda =-d$ as long as $N-d\le 4$. \item For $\lambda >-d$, this equation admits at least one positive solution. Moreover, any positive solution must be radially symmetric and decreasing with respect to the first $d$ (confined) variables $y=(x_1,\cdots ,x_d)$, and is also radially symmetric and decreasing, up to a translation, with respect to the remaining $N-d$ (unconfined) variables. \end{itemize} \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.