Multiple mixed interior and boundary peaks synchronized solutions for linearly coupled Schrödinger systems - Jin, Ke (Topological Methods in Nonlinear Analysis, 2023)
In the present paper we consider the problem: equation 0N_ε cases -ε^2Δ u+u=u^3+λ v& in Ω, -ε^2Δ v+v=v^3+λ u& in Ω, u> 0, v> 0& in Ω, un=vn=0& on Ω, cases equation where ε> 0, 0< λ< 1, ΩR^3 is smooth and bounded, and n denotes the outer normal vector defined on Ω, the boundary of Ω. By the Lyapunov-Schmidt reduction method and the maximum principle of elliptic equations, we construct synchronized solutions of (0) with mixed interior and boundary peaks for any 0< ε< ε_0 and λ(0,1){λ_0}, where λ_0(0,1) is given and ε_0> 0 is sufficiently small. As ε approaches 0, the interior peaks concentrate at sphere packing points in Ω and the boundary peaks concentrate at the critical points of the mean curvature function of the boundary.