Positive and sign-changing solutions for the nonlinear Schrödinger systems with synchronization and separation

2024/07/14 by Wang, Qingfang, Wu, Wenju
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.10141

Abstract

In this paper, we consider the following nonlinear Schrödinger system: -Δ u+P(x)u=μ₁ u³+β uv², x ∈ R³, -Δ v+Q(x)v=μ₂ v³+β u²v, x ∈ R³, where P(x),Q(x) are positive radial potentials,~μ₁, μ₂>0, β ∈ R is a coupling constant. We constructed a new type of solutions which are different from the ones obtained in \citePW. This new family of solutions to system have a more complex concentration structure and are centered at the points lying on the top and the bottom circles of a cylinder with height h. Moreover, we examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type. In the attractive case, we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Moreover, we prove that there exist infinitely many sign-changing solutions whose energy can be arbitrarily large.

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