Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

Cassani, Daniele; Tavares, Hugo; Zhang, Jianjun

doi:10.1016/j.jde.2020.01.036

Cited by 12 publications

(8 citation statements)

References 38 publications

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“…Recently [23,24] are concerned with existence and concentration results for a Coron-type problem in a bounded domain with one or multiple small holes in the case λ i = 0. In [5], the first author with D. Cassani and J. Zhang studied the existence of least energy positive solutions in the critical exponential case when N = 2.…”

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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“…In particular, N = 2 affects the notion of critical growth which is the maximal admissible growth for the nonlinearities to preserve the variational structure of the problem, for details see [11,13] for coupled Schrödinger systems and see [2,27,28] for a single Schrödinger equation for instance. In the case that ǫ = 1 in (1.2), the results on the existence and properties of solutions can be seen in [5,6,9,16,18,22] for N ≥ 3 and [7,11,12,15] for N = 2 for example.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Localized semiclassical states for Hamiltonian elliptic systems in dimension two

Zhang¹,

Yang²,

Zhang³

et al. 2022

Preprint

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In this paper, we consider the Hamiltonian elliptic system in dimension twowhere V ∈ C(R 2 ) has local minimum points, and f, g ∈ C 1 (R) are assumed to be either superlinear or asymptotically linear at infinity and of subcritical exponential growth in the sense of Trudinger-Moser inequality. Under only a local condition on V , we obtain a family of semiclassical states concentrating around local minimum points of V . In addition, in the case that f and g are superlinear at infinity, the decay and positivity of semiclassical states are also given. The proof is based on a reduction method, variational methods and penalization techniques.

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“…In literature, there are few contributions devoted to the study of coupled systems involving exponential nonlinearities. We refer to previous works [2][3][4][5][6][7] for more details. In that work, all systems are studied in dimension N = 2 and above all are led by the standard Laplacian −Δ.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of solution to singular Schrödinger systems involving the fractional p‐Laplacian with Trudinger–Moser nonlinearity in ℝN

Thin

2021

Math Methods in App Sciences

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In this paper, we study the existence of weak solution for singular fractional Schrödinger system in R N involving Trudinger-Moser nonlinearity as follows:where ), and H has exponential growth and does not satisfy the Ambrosetti-Rabinowitz condition. Note that our problem is the lack of compactness. First, we give a version of vanishing lemma due to Lions;using that result and a version of Mountain pass theorem without (PS) condition, we obtain the existence of nontrivial solution to the above system. When H satisfies the Ambrosetti-Rabinowitz condition, motivated by the work of Chen et al.,we study the existence of nontrivial solution to singular Schrödinger system involving the fractional (p 1 , p)-Laplace and Trudinger-Moser nonlinearity. In our best knowledge, this is the first time the above problems are studied.

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Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

Cited by 12 publications

References 38 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

Localized semiclassical states for Hamiltonian elliptic systems in dimension two

Existence of solution to singular Schrödinger systems involving the fractional p‐Laplacian with Trudinger–Moser nonlinearity in ℝN

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