A Concentration Behavior for Semi linear Elliptic Systems with Indefinite Weight

作者：Zhong Xue Xiu^*; Zou Wen Ming

来源：Acta Mathematica Sinica-English Series, 2014, 30(12): 2014-2026.

DOI：10.1007/s10114-014-3509-5

• Schrodinger systems
• concentration
• ground state solution

Consider the Schrodinger system
{-Delta u + V(1,n)u = alpha Q(n)(x)vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta), -Delta u + V(2,n)u = alpha Q(n)(x)vertical bar u vertical bar(beta-2)u vertical bar v vertical bar(beta-2)v, u, v is an element of H-0(1)(Omega),
where Omega subset of R-N, alpha,beta > 1, alpha + beta < 2* and the spectrum sigma(-Delta + V-i,V-n) subset of (0,+infinity), i = 1,2; Q(n) is a bounded function and is positive in a region contained in Si and negative outside. Moreover, the sets {Q(n) > 0} shrink to a point x(0) is an element of Omega as n -> +infinity. We obtain the concentration phenomenon. Precisely, we first show that the system has a nontrivial solution (u(n), v(n)) corresponding to Q(n), then we prove that the sequences (u(n)) and (v(n)) concentrate at x(0) with respect to the H-1-norm. Moreover, if the sets {Q(n) > 0} shrink to finite points and (u(n), v(n)) is a ground state solution, then we must have that both u(n), and v(n), concentrate at exactly one of these points. Surprisingly, the concentration of u(n), and v(n), occurs at the same point. Hence, we generalize the results due to Ackermann and Szulkin [Arch. Rational Mech. Anal., 207, 1075-1089 (2013)].

• 出版日期2014-12
• 单位清华大学