Equivalent Moser type inequalities in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>and the zero mass case

Cassani, Daniele; Sani, Federica; Tarsi, Cristina

doi:10.1016/j.jfa.2014.09.022

Cited by 62 publications

(32 citation statements)

References 49 publications

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“…The scenario changes remarkably from the higher dimensional case N ≥ 3 to the planar case N = 2. In particular, N = 2 affects the notion of critical growth which is the maximal admissible growth for the nonlinearities in order to preserve the variational structure of the problem; we refer to [8][9][10] for a discussion on this topic and to [5,30] for a survey on systems of the form (1.1) in the case of bounded domains. As far as we are concerned with minimal energy solutions in the whole space, existence results have been first established in [31], see also [34], in the higher dimensional case and then recently extended to N = 2 in [16] , where the Trudinger-Moser critical case is covered, see also [4,6].…”

Section: Introductionmentioning

confidence: 99%

A prioriestimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

Cassani

Zhang

2017

Communications in Partial Differential Equations

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We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations (Formula presented.) where V0>0, f,g have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground-state solutions is compact in H1(ℝ2) × H1(ℝ2) up to translations. Moreover, for each (u,v)∈S, one has that u,v are uniformly bounded in L∞(ℝ2) and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground-state solutions to the singularly perturbed system (Formula presented.) where V∈(ℝ2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant →0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate

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Section: Introductionmentioning

confidence: 99%

A prioriestimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

Cassani

Zhang

2017

Communications in Partial Differential Equations

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“…We remark that in dimension two, the upper bound for the AT (α, β) was also obtained in [2] using the critical Trudinger-Moser inequality in [16].…”

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confidence: 87%

Best constants and existence of maximizers for weighted Trudinger–Moser inequalities

Dong

2016

Calc. Var.

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The main purpose of this paper is three-fold. First of all, we will establish a weighted Trudinger-Moser inequality on the entire space without requiring the functions under consideration being radially symmetric (see Theorem 1.1). We will also prove the existence of a maximizer of this sharp weighted inequality. Therefore, our result improves the earlier one where such type of inequality has only been proved for spherically symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in (Ann Inst H Poincaré Anal Non Linaire 31(2):297-314, 2014) (except in the case s = 0). Second, we will prove stronger weighted Trudinger-Moser inequalities for functions which are not required to be radially symmetric (see Theorems 1.2 and 1.3). Finally, the existence of a maximizer of these stronger inequalities is proved in Theorem 1.4. Mathematics Subject Classification

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“…We refer to [15] for the details. Next we recall an Adachi-Tanaka type inequality due to Cassani-Sani-Tarsi [9].…”

Section: 2mentioning

confidence: 99%

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

Cassani

Tavares

Zhang

2020

Journal of Differential Equations

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We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systemswhere Ω is a bounded smooth domain, λ 1 , λ 2 > −Λ 1 (the first eigenvalue of (−∆, H 1 0 (Ω)), µ 1 , µ 2 > 0 and β is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For 'small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities.

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Equivalent Moser type inequalities inR2and the zero mass case

Cited by 62 publications

References 49 publications

A prioriestimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

A prioriestimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

Best constants and existence of maximizers for weighted Trudinger–Moser inequalities

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

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