Stability of ground states for logarithmic Schr\"{o}dinger equation with a $\delta^{\prime}$-interaction

Abstract

In this paper we study the one-dimensional logarithmic Schr\"odinger equation perturbed by an attractive $\delta^{\prime}$-interaction \[ i\partial_{t}u+\partial^{2}_{x}u+ \gamma\delta^{\prime}(x)u+u\, \mbox{Log}\left|u\right|^{2}=0, \quad (x,t)\in\mathbb{R}\times\mathbb{R}, \] where $\gamma>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $\delta^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0<\gamma\leq 2$, then there is a single ground state and it is an odd function; if $\gamma>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.