A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems

Abstract

The paper deals with the existence of normalized solutions to the systemfor any µ 1 , µ 2 , a 1 , a 2 > 0 and β < 0 prescribed. We present a new approach that is based on the introduction of a natural constraint associated to the problem. We also show that, as β → −∞, phase separation occurs for the solutions that we find. Our method can be adapted to scalar nonlinear Schrödinger equations with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.

Multiple normalized solutions for a competing system of Schrödinger equations

“…This problem possesses many physical motivations, e.g. it appears in models for nonlinear optics and Bose-Einstein condensation (we refer to [6] and the references therein for a more exhaustive discussion). Due to the physical background, it seems natural to search for normalized solutions (i.e.…”

“…λ 1 , λ 2 < 0 are prescribed, and the L 2 -constraints are neglected), and not much is known about the full problem (1.1). The only results available in the setting considered here are presented in [5,6], where for possibly non-symmetric systems we proved existence of one positive radial normalized solution, both for suitable choices of β > 0 [5], and for all β < 0 [6]. In this paper we consider the symmetric problem (1.1) with µ 1 = µ 2 and a 1 = a 2 and, exploiting the symmetry, we prove the existence of infinitely many solutions, which will be found as critical points of the energy functional J β : S → R, defined by…”

“…With regard to this, we mention that while for L 2subcritical problems many results are available (see e.g. [13,14,21,22] for equations and [7,11,15] for systems), the L 2 -critical or supercritical ones are much less understood, and we refer to [3,12] for equations and to [4,5,6] for systems. Let us focus now on system (1.1) in the symmetric case a 1 = a 2 and µ 1 = µ 2 .…”

Nonradial normalized solutions for nonlinear scalar field equations

“…One also refers to a mass critical case when the boundedness from below does depend on the value m > 0. In this paper we focus on mass subcritical cases and we refer to the papers [1,2,14] for results in the mass supercritical cases.…”

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

“…In addition, we refer the readers to the literature [30][31][32] about the normalized solutions for nonlinear Schrödinger systems.…”