Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation -Δu+u=u2∗-1+λuq-1inRN,(Pλ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta u+u=u^{2^*-1}+\\lambda u^{q-1} \\quad \ extrm{in}\\, {\\mathbb {R}}^N,\\qquad \\qquad \\qquad \\qquad \\qquad {(P_\\lambda )} \\end{aligned}$$\\end{document}where Nge 3 is an integer, 2^{*}=frac{2N}{N-2} is the Sobolev critical exponent, 2<q<2^* and lambda >0 is a parameter. It is known that as lambda rightarrow 0, after a rescaling the ground state solutions of (P_lambda ) converge to a particular solution of the critical Emden-Fowler equation -Delta u=u^{2^*-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 ge 5. We also discuss a connection of these results with a mass constrained problem associated to (P_{lambda }). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L^{2} norm of the groundstates.