NORMALIZED GROUND STATES TO SOBOLEV CRITICAL SCHRODINGER SYSTEMS WITH LINEAR AND NONLINEAR COUPLINGS

We consider the existence of normalized ground states to the following Sobolev critical Schrodinger systems with linear and nonlinear couplings. { -triangle v(1) + mu (1)v(1) - kappa v(2 )= |v(1)|(2 & lowast;-2)v(1) +xi alpha|v(1)|(alpha-2)|v(2)|beta(v1), in R-N, -triangle v(2) + mu 2v(2) - kappa v(1) = |v(2)|(2 & lowast;-2)v(2) +xi beta|v(1)|alpha|v(2)|beta-2(v2), in R-N, integral(RN) v(1)(2)dx= a(2)(1),integral(RN) v(2)(2)dx = a(2)(2), where N = 3 or 4, a1, a2 > 0, kappa is an element of R, xi>0, alpha>1, beta>1, alpha+beta is an element of (2, 2 & lowast;]\{2+ 4/N } and 2 & lowast; = 2N /N-2. Firstly, in the case alpha+ beta<2 + 4/N, by Ekeland's variational principle, we prove for any xi > 0, kappa > 0, the system admits a radially symmetric normalized solution provided a1, a2 are small; secondly, when 2 + 4N <alpha+ beta < 2 & lowast;, we also employ Ekeland's variational principle to prove that for any a1, a2 > 0, there are two positive constants xi(& lowast;& lowast;) and xi(& lowast;) with xi(& lowast;& lowast; )> xi(& lowast;) > 0 such that for any xi is an element of (xi(& lowast;), xi(& lowast;& lowast;)), the system possesses a radially symmetric normalized ground state solution whenever kappa is an element of (0, kappa<overline>) for some kappa<overline> > 0. In both cases above, we overcome the lack of compactness of the L2-norm by virtue of some delicate energy estimates. Finally, when alpha + beta = 2(& lowast;), we prove that the system does not have any nontrivial nonnegative solutions.