On Solutions for Strongly Coupled Critical Elliptic Systems on Compact Riemannian Manifolds

In this paper, by using variational methods we investigate theexistence of solutions for the following system of elliptic equations??? - delta(g)u+a(x)u+b(x)v = (alpha)/(2 & lowast;)f(x)u|u|(alpha-2)|v|(beta) in M, -delta(g)v+b(x)u+c(x)v = (beta)/(2 & lowast;)f(x)v|v|(beta-2)|u|(alpha) in M,where (M,g) is a smooth closed Riemannian manifold of dimension n >= 3,delta gis the Laplace-Beltrami operator, a, b and care functions Holdercontinuous in M,f is a smooth function and alpha > 1,beta > 1aretworealnumbers such that alpha + beta = 2 & lowast;,where 2(& lowast; )= 2n/(n-2) denotes the critical Sobolev exponent. We get these results by assuming su?cient conditions on the function h=(alpha) /(2)& lowast; a+(2 root alpha beta)/(2)& lowast; b+(beta)/(2)& lowast;crelated to the linear geometric potential (n-2)/R-4(n-1) (g), where R-g is the scalar curvature associated to the metric g.