NONLINEAR ELLIPTIC-EQUATIONS ON COMPACT RIEMANNIAN-MANIFOLDS AND ASYMPTOTICS OF EMDEN EQUATIONS

We study the asymptotics and the global solutions of the following Emden equations: -DELTA-u = lambda-e(u) in a 3-dim domain (lambda > 0) or -DELTA-u = u(q) + l\x\-2u (q > 1) in an N-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with l = 0 and q = N + 1)/(N - 3)(N > 3), we point out how the asymptotics are described via the Moebius group on S(N-1). For a conformally invariant equation -DELTA-u = epsilon\u\4/(N-2)u + l\x\-2u(epsilon = +/- 1) we prove the existence of a new type of solution of the form u(x) = \x\(2-N)/2-OMEGA(GAMMA(Ln\x\)(x/\x\)) where omega is defined on S(N-1) and GAMMA is-an-element-of C infinity (R; O(N)). Finally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenbock formula.