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

Abstract

AbstractWe study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+u=u^{2^*-1}+\lambda u^{q-1} \quad \textrm{in}\, {\mathbb {R}}^N,\qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}$$ - Δ u + u = u 2 ∗ - 1 + λ u q - 1 in R N , ( P λ ) where $$N\ge 3$$ N ≥ 3 is an integer, $$2^{*}=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 is the Sobolev critical exponent, $$2<q<2^*$$ 2 < q < 2 ∗ and $$\lambda >0$$ λ > 0 is a parameter. It is known that as $$\lambda \rightarrow 0$$ λ → 0 , after a rescaling the ground state solutions of $$(P_\lambda )$$ ( P λ ) converge to a particular solution of the critical Emden-Fowler equation $$-\Delta u=u^{2^*-1}$$ - Δ 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 = 3 , $$N=4$$ N = 4 or $$N \ge 5$$ N ≥ 5 . We also discuss a connection of these results with a mass constrained problem associated to $$(P_{\lambda })$$ ( P λ ) . Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the $$L^{2}$$ L 2 norm of the groundstates.