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Critical and supercritical Hamiltonian systems of Schrödinger equations in dimension two Cassani, Daniele


We consider in the whole plane the Hamiltonian coupling of Schrödinger equations where the nonlinearities have critical or even supercritical growth in the sense of Moser. In the critical case, we prove that the (nonempty) set of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded 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 states solutions to singularly perturbed systems. Namely, in presence of an external potential which is bounded away from zero, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate. In the supercritical case we prove the existence of higher (though finite) energy solutions in a suitable Lorentz space framework.

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