NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering

Abstract

We investigate a class of nonlinear equations of Schr\"odinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +\Delta u &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u \quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, \end{align*} where $N \geq 1$, $b_1, b_2>0$ and $p_1,p_2>2$. First, we establish the existence/nonexistence, symmetry, decay, uniqueness, non-degeneracy and instability of ground states. Then, we prove the scattering versus blowup below the ground state energy threshold. Our approach relies on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We also obtain an upper bound of the blow-up rate. The novelty here is that the equation does not enjoy any scaling invariance due to the presence of competing nonlinearities and the singular weights prevent the invariance by translation in the space variable. To the best of authors knowledge, this is the first time when inhomegeneous NLS equation with a focusing leading order nonlinearity and a defocusing perturbation is investigated.