Multiple normalized solutions for a competing system of Schrödinger equations

Abstract

We prove the existence of infinitely many solutions λ 1 , λ 2 ∈ R, u, v ∈ H 1 (R 3 ), for the nonlinear Schrödinger systemwhere a, µ > 0 and β ≤ −µ are prescribed. Our solutions satisfy u = v so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions. 2010 Mathematics Subject Classification. 35J50, 35J15, 35J60.1 1 Indeed, in [16] the authors exploit a uniform-in-β Palais-Smale condition to derive the convergence of the whole minimax structure to a limit problem.2 In [6] we showed that P β is a C 1 manifold since this was enough for our purpose, the extra regularity is straightforward.

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

“…The proof of Lemma 3.19 first relies on an equivariant version of Lemma 3.16, whose proof is almost identical to the one of Lemma 3.16. Then the lemma follows just as [5, Theorem 3.2] follows from [5,Proposition 3.9].…”

“…In view of Lemma 3.12 we can define Aiming to prove Theorem 1.2 we shall establish the existence of a Palais-Smale sequence (u n ) ⊂ Λ + (c) (respectively (u n ) ⊂ Λ − (c)) for F restricted to S(c). Our arguments are inspired from [5].…”

“…for any A ∈ G and any η ∈ C( Proof. We are only sketchy here and refer to [5] for the proofs of closely related results. The proof of Lemma 3.19 first relies on an equivariant version of Lemma 3.16, whose proof is almost identical to the one of Lemma 3.16.…”

“…At this point we conclude using Lemma 2.8 that (u k n ) n converges to a u k which is a critical point of F on S(c). Now to show that if two (or more) values of β k coincide, than F has infinitely many critical points at level c k , one can either proceed in the usual way, or adapt [5,Lemma 6.4] to the present setting.…”

Normalized solutions for nonlinear Schrödinger systems on bounded domains

“…Apart when global minimization can be applied, see [35], as far as we know the first result in the literature is due to Jeanjean [24], for the superlinear, Sobolev-subcritical NLS single equation on R N with a non-homogeneous nonlinearity. In recent years, other papers appeared, dealing with the NLS equation or system, always in the Sobolev subcritical regime, either on R N [5,21,7,9,4,22,6] or on a bounded domain [31,32,33,15,34,10]. These two settings are rather different in nature: each one requires a specific approach, and the results are in general not comparable.…”