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 $$-\Delta u+u=u^{2^{\ast }-1}+\lambda u^{q-1}\mathrm{in}{\mathbb{R}}^N,$$ where $N\ge 3$ is an integer, $2^{\ast }=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^{\ast }$ and $\lambda >0$ is a parameter. It is known that as $\lambda \to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^{\ast }-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=3$, $N=4$ or $N\ge 5$.

25 pages

References in corpus (3)

• The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
• Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth
• Global dynamics above the ground state energy for the combined power type nonlinear Schrodinger equations with energy critical growth at low frequencies