MULTIPLE SOLUTIONS TO A WEAKLY COUPLED PURELY CRITICAL ELLIPTIC SYSTEM IN BOUNDED DOMAINS

摘要

We study the weakly coupled critical elliptic system{-Delta u = mu(1)vertical bar u vertical bar(2)*( -2)u + lambda alpha vertical bar u vertical bar(alpha-2)vertical bar(beta)v vertical bar in Omega,-Delta v = mu(2)vertical bar v vertical bar(2)*( -2)v + lambda beta vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2) v in Omega,u = v = 0 on partial derivative Omega,where Omega is a bounded smooth domain in R-N, N = 3, 2*: = R-N, N = 3, 2* := 2N/N-2 is the critical Sobolev exponent, mu(1, )mu(2 0, alpha, beta 1, alpha + beta = 2)*. and lambda is an element of R.We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on Omega, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as lambda - -infinity.We also obtain existence of infinitely many solutions to this system in Omega = R-N.