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Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
Mathematische Zeitschrift, Volume: 304, Issue: 13
Swansea University Author: Vitaly Moroz
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We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solu...
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We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden-Fowler equation −Δu=u2∗−1. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3, N=4 or N≥5. We also discuss a connection of these results with a mass constrained problem associated to (Pλ). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L2 norm of the groundstates.
Nonlinear Schrödinger equation, critical Sobolev exponent, concentration compactness, asymptotic behaviour
Faculty of Science and Engineering
S.M. was supported by National Natural Science Foundation of China (Grant Nos.11571187, 11771182).