A concentration behavior for semilinear elliptic systems with indefinite weight

Abstract

Consider the Schrodinger system $$\left\{ \begin{gathered} - \Delta u + V_{1,n} u = \alpha Q_n (x)|u|^{\alpha - 2} u|v|^\beta , \hfill \\ - \Delta v + V_{2,n} v = \beta Q_n (x)|u|^\alpha |v|^{\beta - 2} v, \hfill \\ u,v \in H_0^1 (\Omega ), \hfill \\ \end{gathered} \right.$$ where Ω ⊂ ℝN, α, β > 1, α + β 0} shrink to a point x0 ∈ Ω as n → +∞. We obtain the concentration phenomenon. Precisely, we first show that the system has a nontrivial solution (un, vn) corresponding to Qn, then we prove that the sequences (un) and (vn) concentrate at x0 with respect to the H1-norm. Moreover, if the sets {Qn > 0} shrink to finite points and (un, vn) is a ground state solution, then we must have that both un and vn concentrate at exactly one of these points. Surprisingly, the concentration of un and vn occurs at the same point. Hence, we generalize the results due to Ackermann and Szulkin [Arch. Rational Mech. Anal., 207, 1075–1089 (2013)].