Ground states of the defocusing nonlinear Schrödinger equation with a point interaction in dimensions 2 and 3

Abstract

Abstract This paper is concerned with ground states of the defocusing nonlinear Schrödinger equation with a point interaction, $$ \textrm{i} \partial _t \psi = -\Delta _\alpha \psi + \psi |\psi |^{p - 2} \quad \text {in} \quad \mathbb {R} \times \mathbb {R}^N, $$ i ∂ t ψ = - Δ α ψ + ψ | ψ | p - 2 in R × R N , where $$- \Delta _\alpha $$ - Δ α denotes the Laplacian of point interaction centered at the origin with inverse s-wave scattering length $$- 2 (N - 1) \pi \alpha $$ - 2 ( N - 1 ) π α , and we suppose that either (i) $$N = 2$$ N = 2 , $$\alpha \in \mathbb {R}$$ α ∈ R and $$p > 2$$ p > 2 or (ii) $$N = 3$$ N = 3 , $$\alpha < 0$$ α < 0 and $$2< p < 3$$ 2 < p < 3 . At sufficiently small masses, (i) we prove that this equation admits ground states, (ii) we obtain some qualitative properties of ground states, and (iii) we obtain some results relating ground states with critical points of the associated action functional.