Displaying similar documents to “Trudinger–Moser inequality on the whole plane with the exact growth condition”

Semiclassical states of nonlinear Schrödinger equations with bounded potentials

Antonio Ambrosetti, Marino Badiale, Silvia Cingolani (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential V .

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

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Nonlinear Schrödinger equations (NLS) a with strongly singular potential a | x | - 2 on a bounded domain Ω are considered. If Ω = N and a > - ( N - 2 ) 2 / 4 , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here a = - ( N - 2 ) 2 / 4 is excluded because D ( P a ( N ) 1 / 2 ) is not equal to H 1 ( N ) , where P a ( N ) : = - Δ - ( N - 2 ) 2 / ( 4 | x | 2 ) is nonnegative and selfadjoint in L 2 ( N ) . On the other hand, if Ω is a smooth and bounded domain with 0 Ω , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov (2013)

Annales de l’institut Fourier

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In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the L 2 critical...

Non-generic blow-up solutions for the critical focusing NLS in 1-D

Joachim Krieger, Wilhelm Schlag (2009)

Journal of the European Mathematical Society

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We consider the L 2 -critical focusing non-linear Schrödinger equation in 1 + 1 -d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

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We investigate the propagation of the uniform spatial Gevrey G σ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Jean Bourgain, Aynur Bulut (2014)

Journal of the European Mathematical Society

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We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in 3 . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated...

Semiclassical states for weakly coupled nonlinear Schrödinger systems

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Remarks on global existence and compactness for L 2 solutions in the critical nonlinear schrödinger equation in 2D

Luis Vega Gonzalez (1998)

Journées équations aux dérivées partielles

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In the talk we shall present some recent results obtained with F. Merle about compactness of blow up solutions of the critical nonlinear Schrödinger equation for initial data in L 2 ( 𝐑 2 ) . They are based on and are complementary to some previous work of J. Bourgain about the concentration of the solution when it approaches to the blow up time.

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x N , u H ¹ ( N ) , where λ is a positive parameter, a and b are positive functions, while f : is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.