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

Daniele Barbera, Filippo Boni, Simone Dovetta, Lorenzo Tentarelli

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 $δ$-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.

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 -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.

Paper Structure

This paper contains 5 sections, 18 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:nomass']}
  3. Proofs of Theorems \ref{['thm:exsol']}--\ref{['thm:nonun']}
  4. Proof of Theorem \ref{['thm:propE']}
  5. Ground states: proof of Theorems \ref{['thm:gs1']}--\ref{['thm:gs2']}--\ref{['thm:gs3']}

Key Result

Theorem 1.1

Let $p,q>2$. For every $\lambda\in\mathbb{R}$, if $u\in H^1(\mathbb{R})$ is a nontrivial solution of nls-nomass, then $u$ is even and (up to a change of sign) positive and radially decreasing on $\mathbb{R}$. Moreover,

Figures (1)

  • Figure 1: The subsets of the $pq$-plane identified by the straight lines $p=2$, $p=6$, $q=2$, $q=4$, and $q=\frac{p}{2}+1$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 31 more