Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions

Abstract

We investigate normalized solutions for doubly nonlinear Schrödinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $\hat{I}´$-type at the origin. We provide a complete characterization of existence and uniqueness for normalized solutions and for energy ground states for every value of the nonlinearity powers. We show that the interplay between a defocusing standard and a focusing point nonlinearity gives rise to new phenomena with respect to those observed with single nonlinearities, standard combined nonlinearities, and combined focusing standard and pointwise nonlinearities.

25 pages, 1 figure. Keywords: doubly nonlinear Schrödinger, nonlinear point interactions, normalized solutions, ground states