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$\{\begin{array}{c}-\Delta u_1={\mu }_1{u}_1^p+\beta u_1^{\frac{p-1}{2}}{u}_2^{\frac{p+1}{2}}\text{ in }\Omega ,\\ -\Delta u_2={\mu }_2{u}_2^p+\beta u_2^{\frac{p-1}{2}}{u}_1^{\frac{p+1}{2}}\text{ in }\Omega ,\\ u_1,u_2>0\text{in }\Omega ;u_1=u_2=0\text{ on }\partial \Omega ,\end{array}$ where $p>1,{\mu }_1,{\mu }_2,\beta >0$ and Ω is a smooth bounded domain in $R^2$. We prove the a priori estimate for positive solutions. Moreover, under the natural condition that holds automatically for all positive solutions in star-shaped domains$p\underset{\Omega }{\int }|\nabla u_{1,p}{|}^2+|\nabla u_{2,p}{|}^{2}dx\le C,$ 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 ${\Vert u_{k,p}\Vert }_{L^{\infty }\left(\Omega \right)}\to \sqrt{e}$ 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 $n\in N_{\ge 2}$, and so on. In particular, we show that the “local mass” contributed by each concentration point must be one of $\left\{\right(8\pi ,8\pi ),(8\pi ,0),(0,8\pi \left)\right\}$.