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.

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

CASSANI, DANIELE;ZHANG, JIANJUN
2017-01-01

Abstract

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.
http://www.tandf.co.uk/journals/titles/03605302.asp
Concentration phenomena; critical growth; generalized Nehari manifold; Hamiltonian systems; Pohozaev–Trudinger–Moser inequality; positive solutions; Analysis; Applied Mathematics
Cassani, Daniele; Zhang, Jianjun
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2061988
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