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Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction

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  • In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive $δ^{\prime}$-interaction

    $i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$

    where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0 < γ≤ 2$, then there is a single ground state and it is an odd function; if $γ>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.

    Mathematics Subject Classification: 35Q51, 35Q55, 37K40, 34B37.


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  • Figure 1.  The graph of the curve $\mathcal{I}_{1}\cup \mathcal{I}_{2}$.

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