Ground states of nonlinear Schrödinger systems with mixed couplings

Wei, Juncheng; Wu, Yuanze

doi:10.1016/j.matpur.2020.07.012

Cited by 50 publications

(20 citation statements)

References 55 publications

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“…In [39,40], by dividing the d components into m groups, the authors proved that system (1.1) has a least energy positive solution under appropriate assumptions on β ij . Other related and recent results in the subcritical case can be found in [14,15,16,18,29,31,47] and references, where we stress that the first two mentioned papers deal with a general p.…”

Section: Introductionmentioning

confidence: 99%

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Tavares

You

2020

Calc. Var.

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In this paper we investigate the existence of solutions to the following Schrödinger system in the critical caseHere, Ω ⊂ R 4 is a smooth bounded domain, d ≥ 2, −λ1(Ω) < λi < 0 and βii > 0 for every i, βij = βji for i = j, where λ1(Ω) is the first eigenvalue of −∆ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that βij ≥ 0 (cooperation) whenever components i and j belong to the same group, while βij < 0 or βij is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on βij , we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case Ω = R 4 and λ1 = . . . = λm = 0, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch.

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Section: Introductionmentioning

confidence: 99%

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Tavares

You

2020

Calc. Var.

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“…Furthermore, we mention the works of Chen and Zou 9 and Li and Tang, 10 they generalized and improved some results of Ambrosetti, Cerami and Ruiz 7 . Related studies with those presented above can be found in Ambrosetti, Colorado and Ruiz, 11 Chen and Zou, 12 Liu and Wang, 13 Maia, Ruviaro and Moura, 14 Wei and Wu, 15 and Zhang and Zhang 16 . We would like to point out that most of the above works considered the cases that the nonlinear terms have subcritical or critical‐type polynomial growth in the sense of Sobolev embedding.…”

Section: Introductionmentioning

confidence: 72%

Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

Wei

Lin

Tang

2021

Math Methods in App Sciences

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We consider the following two coupled nonlinear Schrödinger system: −normalΔu+u=f1false(x,ufalse)+λfalse(xfalse)v,x∈ℝ2,−normalΔv+v=f2false(x,vfalse)+λfalse(xfalse)u,x∈ℝ2, where the coupling parameter satisfies 0 < λ(x) ≤ λ0 < 1 and the reactions f1, f2 have critical exponential growth in the sense of Trudinger–Moser inequality. Using non‐Nehari manifold method together with the Lions's concentration compactness and the Trudinger‐Moser inequality, we show that the above system has a Nehari‐type ground state solution and a nontrivial solution. Our results improve and extend the previous results.

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“…As far as we know, there are some results about the existence and nonexistence of ground state solution. J. Wei and Y. Wu [28] gave an (almost) complete study on the existence and nonexistence of ground state solution with different Morse indices of (2.1) under different conditions by the idea of block decomposition and measure the total interaction between different blocks for 3-coupled system when the system admits mixed couplings.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…Inspired by the above-mentioned works, especially by [14,22,28], in this paper our goal is two-folds. One is to give a complete classification of ground state solution with different Morse indices for three-coupled Schrödinger system under suitable conditions.…”

Section: Introductionmentioning

confidence: 99%

Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrodinger System

sci¹

2021

ATA

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There have been intensive studies for the system on existence/non-existence and classification of ground state solutions when N = 2. However fewer results about the classification of ground state solution are available for N ≥ 3. In this paper, we first give a complete classification result on ground state solutions with Morse indices 1, 2 or 3 for three-coupled Schrödinger system. Then we generalize our results to Ncoupled Schrödinger system for ground state solutions with Morse indices 1 and N. We show that any positive ground state solutions with Morse index 1 or Morse index N must be the form of (d 1 w, d 2 w, • • • , d N w) under suitable conditions, where w is the unique positive ground state solution of certain equation. Finally, we generalize our results to fractional N-coupled Schrödinger system.

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Ground states of nonlinear Schrödinger systems with mixed couplings

Cited by 50 publications

References 55 publications

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear $N$-Coupled Schrodinger System

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