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Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. (English) Zbl 0755.35036
Summary: 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 \(-\Delta u=u^ q+\ell| x|^{-2}u\) \((q>1)\) 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 \(\ell=0\) and \(q=(N+1)/(N-3)\) \((N>3)\), we point out how the asymptotics are described via the Möbius group on \(S^{N-1}\). For a conformally invariant equation \(-\Delta u=\varepsilon| u|^{4/(N-2)}u+\ell| x|^{-2}u\) \((\varepsilon=\pm 1)\) 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 \(\Gamma\in C^ \infty\) \((\mathbb R;O(N))\). Finally, we extend and simplify the results of B. Gidas and J. Spruck [Commun. Pure Appl. Math. 34, 525–598 (1981; Zbl 0465.35003)] on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Lichnerowicz-Weitzenböck formula.

35J60 Nonlinear elliptic equations
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
35B40 Asymptotic behavior of solutions to PDEs
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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