Title: Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity Show Chinese title

Title: 非线性薛定谔方程临界组合幂非线性渐近轮廓

Abstract: We study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ 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 $\lambda\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. 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 $N\ge 5$. Show Chinese abstract

Abstract: 我们研究非线性薛定谔方程 $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ 的正基态解的渐近行为，其中 $N\ge 3$ 是一个整数，$2^*=\frac{2N}{N-2}$ 是 Sobolev 临界指数，$2<q<2^*$ 和$\lambda>0$是一个参数。已知当 $\lambda\to 0$ 时，经过尺度变换后，该方程的基态解收敛到临界 Emden-Fowler 方程 $-\Delta u=u^{2^*-1}$的一个特定解。 我们建立了一个精确的渐近特性，该特性以非平凡的方式依赖于空间维数 $N=3$, $N=4$或 $N\ge 5$。