Finite time blow-up of non-radial solutions for some inhomogeneous Schr\"{o}dinger equations

Abstract

This work studies the inhomogeneous Schr\"odinger equation $$ i\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $\lambda>-\frac{(N-2)^2}{4}$. The linear Schr\"odinger operator reads $\mathcal{K}_{s,\lambda}:= (-\Delta)^s +(2-s)\frac{\lambda}{|x|^2}$ and the focusing source term is local or non-local $$F(x,u)\in\{|x|^{-2\tau}|u|^{2(q-1)}u,|x|^{-\tau}|u|^{p-2}(J_\alpha *|\cdot|^{-\tau}|u|^p)u\}.$$ The Riesz potential is $J_\alpha(x)=C_{N,\alpha}|x|^{-(N-\alpha)}$, for certain $0<\alpha<N$. The singular decaying term $|x|^{-2\tau}$, for some $\tau>0$ gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely $1+\frac{2(1-\tau)}N<q<1+\frac{2(1-\tau)}{N-2s}$ and $1+\frac{2-2\tau+\alpha}{N}<p<1+\frac{2-2\tau+\alpha}{N-2s}$. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality.