STABILITY OF GROUND STATES FOR LOGARITHMIC SCHRODINGER EQUATION WITH A δ′- INTERACTION

In this paper we study the one-dimensional logarithmic Schrodinger equation perturbed by an attractive delta'-interaction i partial derivative(t)u + partial derivative(2)(x)u + gamma delta' (x)u + u Log broken vertical bar u vertical bar(2) = 0, (x, t) is an element of R x 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'-interaction case, the set of the ground state is completely determined. More precisely: if 0 < gamma <= 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.