zbMATH — the first resource for mathematics

Classification of solutions of higher order conformally invariant equations. (English) Zbl 0940.35082
For a positive integer \(p\) the authors investigate solutions of higher order conformally invariant nonlinear equations on \(\mathbb{R}^n\) of the following type: \[ (-\Delta)^p u = (n-1)! e^{nu} \] such that \(\int_{\mathbb{R}^n} e^{nu} < \infty\), where \(n=2p\), and \[ (-\Delta)^p u = u^{\frac{n+2p}{n-2p}} \] such that \(u>0\), where \(n>2p\).
Their main result is a characterization of the solutions as a certain class of radially symmetric functions. Their approach uses the method of the moving frame. To that purpose they first have to examine the asymptotic behaviour of the solution and derive a maximum principle.
Furthermore, it is shown that the exponent \(\frac{n+2p}{n-2p}\) of the nonlinearity in the second equation is critical, i.e. smaller exponents admit only the trivial solution. Conversely, the existence of a strictly positive solution implies for a large class of non-linearities \(f(u)\) on the right hand side that \(f\) is of power type with the given exponent.
Reviewer: W.Hoh (Bielefeld)

35J60 Nonlinear elliptic equations
58J05 Elliptic equations on manifolds, general theory
35G20 Nonlinear higher-order PDEs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text: DOI