Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity - Vitaly Moroz (Mathematische Zeitschrift, 2023)
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.