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We study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear couplingβ  &gt;  0{\beta&gt;0}   , we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constantsμ  β   ,   ν  β{\mu_{\beta},\nu_{\beta}}    by Lagrange multiplier rules. Then, asβ  →  ∞{\beta\to\infty}   , we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.

Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems