Discontinuous Ground States for NLSE on $\mathbb{R}$ with a Fülöp-Tsutsui $\delta$ interaction

Abstract

We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a global minimizer of the action functional on the Nehari manifold and the defect considered is a Fülöp-Tsutsui $\delta$ type, namely a $\delta$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results from the Grillakis-Shatah-Strauss theory.

24 pages, 7 figures