Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations
Tóm tắt
We study the asymptotics and the global solutions of the following Emden equations: −Δu=λe
u
in a 3-dim domain (λ>0) or −Δu=u
q
+ℓ|x|−2
u (q>1) in anN-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 andq=(N+1)/(N−3) (N>3), we point out how the asymptotics are described via the Moebius group onS
N−1. For a conformally invariant equation −Δu=ɛ|u|4/(N−2)
u+ℓ|x|−2
u(ɛ=±1) we prove the existence of a new type of solution of the formu(x)=|x|(2−N)/2ω(Γ(Ln|x|)(x/|x|)) where ω is defined onS
N−1 and Γ∈C
∞ (ℝ;O(N)). Finnally, 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-Weitzenböck formula.
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