Solutions for a critical elliptic system with periodic boundary condition
Qingfang Wang, Wenju Wu, Mingxue Zhai
TL;DR
The paper analyzes a nonlinear, coupled critical Schrödinger system with periodic boundary conditions on a strip and proves the existence of bubbling solutions for large periods $L$ under weak conditions on the periodic coefficients $K_1$ and $K_2$. It introduces a Green’s-function framework for the periodic domain and a weighted Lyapunov–Schmidt reduction to overcome Sobolev embedding obstructions, establishing invertibility of the linearized operator via Fredholm theory. A finite-dimensional reduction identifies a precise balance of bubble parameters, yielding a true solution $(u,v)$ concentrated at a single bubble with $L$-periodicity, and enabling the construction of infinitely many periodic solutions by varying $L$. The results extend the bubbling phenomenon to periodic strips with coupled critical exponents and demonstrate how the Green’s function and weighted spaces facilitate analysis in the periodic setting.
Abstract
In this paper, we consider the following nonlinear critical Schrödinger system: \begin{eqnarray*}\begin{cases} -Δu=K_1(y)u^{2^*-1}+\frac{1}{2} u^{\frac{2^*}{2}-1}v^\frac{2^*}{2}, \,\,\,\,\,y\inΩ,\,\,\,\,\,u>0,\cr -Δv=K_2(y)v^{2^*-1}+\frac{1}{2} v^{\frac{2^*}{2}-1}u^\frac{2^*}{2}, \,\,\,\,\,y\inΩ,\,\,\,\,\,v>0,\cr u(y'+Le_j,y'')=u(y), \,\,\,\,\,\frac{\partial u(y'+Le_j,y'')}{\partial y_j}=\frac{\partial u(y)}{\partial y_j}, \,\,\,\,\,if\,\, y'=-\frac{L}{2}e_j,\,\,\,j=1, \ldots, k,\cr v(y'+Le_j,y'')=v(y), \,\,\,\,\,\frac{\partial v(y'+Le_j,y'')}{\partial y_j}=\frac{\partial v(y)}{\partial y_j}, \,\,\,\,\,if\,\, y'=-\frac{L}{2}e_j,\,\,\,j=1, \ldots, k,\cr u,v \to 0 \,\,as \,\,|y''|\to \infty, \end{cases} \end{eqnarray*} where $K_1(y),\,K_2(y)$ satisfy some periodic conditions and $Ω$ is a strip. Under some conditions which are weaker than Li, Wei and Xu(J. Reine Angew. Math. 743: 163-211, 2018), we prove that there exists a single bubbling solution for the above system. Moreover, as the appearance of the coupling terms, we construct different forms of solutions, which makes it more interesting. Since there are periodic boundary conditions, this expansion for the difference between the standard bubbles and the approximate bubble can not be obtained by using the comparison theorem as one usually does for Dirichlet boundary condition. To overcome this difficulty, we will use the Green's function of $-Δ$ in $Ω$ with periodic boundary conditions which helps us find the approximate bubble. Due to the lack of the Sobolev inequality, we will introduce a suitable weighted space to carry out the reduction.