Nonradial normalized solutions for nonlinear scalar field equations

Abstract

We study the following nonlinear scalar field equationHere f ∈ C(R, R), m > 0 is a given constant and µ ∈ R arises as a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity f , we show the existence of one nonradial solution for any N ≥ 4, and obtain multiple (sometimes infinitely many) nonradial solutions when N = 4 or N ≥ 6. In particular, all these solutions are sign-changing. MSC: 35J60, 58E05

Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation

“…It seems to be the first existence result of arbitrarily finitely many positive energy solutions in the investigation of L 2 constrained mass subcritical problems. In particular, it complements the recent related works [12,14] where multiple radial solutions of (P m ) were obtained at negative energy levels.…”

“…This case is commonly called mass subcritical and has been under extensive studies for decades. Among many possible choices, we refer the reader to [6,7,8,12,14,16,20,27,29] and to the references therein. In contrast however, to these previous works, we are herein interested in searching for constrained critical points at positive energy levels.…”

“…We end this section by observing that by an appropriate adaptation of the arguments for Theorem 1.5, a nonradial variant of that multiplicity result can be established when N = 4 or N ≥ 6. One should note here that the working space for proving Theorem 5.5 is the same as the one used in [14, Theorem 1.2] and that the special map in [14,Lemma 3.4] can be used to show the nonemptiness of a certain substitute to the family of odd continuous maps Γ m,k . The details of the proof are left to the interested reader.…”

On global minimizers for a mass constrained problem

“…This likely explains why the study of problem (Inf m ) is still the object of an intense activity. Among many others possible choices, we refer to [4,5,6,10,12,13,15,21,25,26] and to the references therein.…”

A global branch approach to normalized solutions for the Schrödinger equation

“…The study of the mass subcritical case, which can be traced back to the work of Stuart [37,38], had seen a major advance with the introduction of the Compactness by Concentration approach of Lions [27,28]. Nowadays, this case is rather well-understood and we refer to [20,22,33,36] for recent contributions.…”