Concentration phenomena of positive solutions to weakly coupled Schrödinger systems with large exponents in dimension two

Abstract

We study the weakly coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -Δu_1 = μ_1 u_1^{p} +βu_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } Ω,\\ -Δu_2 = μ_2 u_2^{p} +βu_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} \text{ in } Ω,\\ u_1,u_2>0\quad\text{in }\;Ω;\quad u_1=u_2=0 \quad\text { on } \;\partialΩ, \end{cases} \end{equation*} where $p>1,{\mu }_1,{\mu }_2,\beta >0$ and $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^2$. Under the natural condition that holds automatically for all positive solutions in star-shaped domains \begin{align*} p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2 dx \leq C, \end{align*} we give a complete description of the concentration phenomena of positive solutions $(u_{1,p},u_{2,p})$ as $p\to +\infty$, including the $L^{\infty }$-norm quantization $\Vert u_{k,p}{\Vert }_{L^{\infty }(\Omega )}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p{\int }_{\Omega }|\nabla u_{1,p}{|}^2+|\nabla u_{2,p}{|}^{2}dx\to 8n\pi e$ with $n\in {\mathbb{N}}_{\ge 2}$, and so on. In particular, we show that the ``local mass'' contributed by each concentration point must be one of $\{(8\pi ,8\pi ),(8\pi ,0),(0,8\pi )\}$.

40 pages