Uniqueness of Positive Solutions for a Nonlinear Elliptic System

Ikoma, Norihisa

doi:10.1007/s00030-009-0017-x

Cited by 48 publications

(37 citation statements)

References 13 publications

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“…In fact, this conjecture was proved by Wei and Yao in [12] with the assumption i) β > max{ν 1 , ν 2 } or ii) β > 0 sufficiently small, see also Ikoma [6]. Employing the arguments used in [13], Nguyen, Tian, Deconinck and Sheils in [10] showed that the ground state is concerned with the best constant for the vector valued Gagliardo-Nirenberg inequality which in our case is…”

Section: Introductionmentioning

confidence: 63%

Concentration of blow-up solutions for the coupled Schrödinger equations inR2

Gao

Wang

2015

Journal of Mathematical Analysis and Applications

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We prove a concentration result for blow-up solutions of the coupled Schrödinger equations with non-spherically symmetric initial data in H 1 (R 2 ). Using the profile decomposition technique and assuming a conjecture of Sirakov is valid, which was proved by Wei and Yao and Ikoma for some cases, we could remove the radial restriction on the initial data and improve the concentration rate from Lü and Liu (2011) [9].

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Section: Introductionmentioning

confidence: 63%

Concentration of blow-up solutions for the coupled Schrödinger equations inR2

Gao

Wang

2015

Journal of Mathematical Analysis and Applications

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“…There are many works on the existence of non-trivial positive solutions of (1.5)- (1.8). See [2][3][4][5]11,12,[18][19][20]24,26,30,33,34]. Sign and size of β are important in the study of (1.5)-(1.8) and various situations are studied in the above papers.…”

Section: Introductionmentioning

confidence: 98%

A local mountain pass type result for a system of nonlinear Schrödinger equations

Ikoma

Tanaka

2010

Calc. Var.

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We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ 1 , μ 2 , β > 0 and V 1 (x), V 2 (x) : R N → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ∈ R N the least energy level m(P) for non-trivial vector solutions of the rescaled "limit" problem: ( * * ) We assume that there exists an open bounded setWe show that ( * ) possesses a family of non-trivial vector positive solutions {(v 1ε (x), v 2ε (x))} ε∈(0,ε 0 ] which concentrates-after extracting a subsequence ε n → 0-to a point P 0 ∈ Λ Communicated by P. Rabinowitz. 123 450 N. Ikoma, K. Tanaka with m(P 0 ) = inf P∈Λ m(P). Moreover (v 1ε (x), v 2ε (x)) converges to a least energy nontrivial vector solution of ( * * ) after a suitable rescaling. Mathematics Subject Classification (2000)35B25 · 35J65 · 58E05

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“…Even in this case, it is not easy to get an explicit expression for E(y). To make matters worse, the uniqueness of a vector solution is known only for very restricted cases, typically in case W 1 (y) = W 2 (y) (refer to [20,36]). To overcome the difficulty in the characterization of a concentration point and the restricted knowledge on the uniqueness of a vector solution of a limiting problem in the construction for one bump vector solutions of (3), we approach the problem (3) with a new variational view point regarding (4) as a perturbation of ∇ L = 0 in a neighborhood of a set of approximate solutions in R N × (H 1 ) 2 .…”

Section: Introductionmentioning

confidence: 99%

“…β,μ 1 ,μ 2 λ 1 ,λ 2 (γ (s, t)) ≤ max max (s,t)∈[1,R] 2 J β,μ 1 ,μ 2 λ 1 ,λ 2 (γ (s, t)), L λ 1 ,μ 1 , L λ 1 ,μ 1 ≤ max max (s,t)∈[0,R] 2 J β,μ 1 ,μ 2 λ 1 ,λ 2 (γ (s, t)), L λ 1 ,μ 1 , L λ 1 ,μ 1 = max (s,t)∈[0,R] 2 J β,μ 1 ,μ 2 λ 1 ,λ 2 (γ (s, t)) (20). …”

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confidence: 99%

Semi-classical standing waves for nonlinear Schrödinger systems

Byeon

2015

Calc. Var.

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For N ≤ 3 and β > 0, we consider the following singularly perturbed elliptic systemThere are an enormous number of results for localized solutions of singularly perturbed scalar problems using variational methods or finite dimensional reduction methods. However, there exist no general existence results of localized solutions for elliptic systems. We present some such results here. In the first, by a mini-max characterization for a limiting problem, for small ε > 0, we show the existence of one bump solutions with a common concentration point of u 1 , u 2 in a domain O when certain conditions for the limiting problem are satisfied. Typical examples of potentials W 1 , W 2 satisfying the condition are the following: (1) W 1 , W 2 have a common non-degenerate critical point in O which share the same stable, unstable directions; (2) for the outnormal n onWe also give some nonexistence results for some potentials W 1 , W 2 , not satisfying these conditions, but each W 1 , W 2 having a structurally stable critical point in O.Mathematics Subject Classification 35J47 · 35J50 · 35J60 Communicated by P. Rabinowitz. B Jaeyoung Byeon

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Uniqueness of Positive Solutions for a Nonlinear Elliptic System

Abstract: Abstract. In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system:We prove that for sufficiently small β > 0, the above system has a unique nontrivial positive solution.

Cited by 48 publications

References 13 publications

Concentration of blow-up solutions for the coupled Schrödinger equations inR2

Concentration of blow-up solutions for the coupled Schrödinger equations inR2

A local mountain pass type result for a system of nonlinear Schrödinger equations

Semi-classical standing waves for nonlinear Schrödinger systems

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