Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Recently, there have been much analytic work on the conformally invariant operators as well as its associated differential equations. A well known second order conformally invariant operator comes from the Yamabe problem or, more generally, the problem of prescribed scalar curvature. Given a smooth positive function K defined on a compact Riemannian manifold (M, g0) of dimension n ≥ 2, we ask whether there exists a metric g conformal to g0 such that K is the scalar curvature of the new metric g. Let g = eg0 for n = 2 or g = u 4 n−2 g0 for n ≥ 3, then the problem is reduced to find solutions of the following nonlinear elliptic equations:

Classification of solutions of higher order conformally invariant equations

where a > 0 is some constant and x ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [8]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([24]–[27]); see also related works of Viaclovsky ([40]–[41]) and Chang, Gursky and Yang ([13]–[14]). The method used in [21], as well as in much of the above cited work, is the method of moving planes. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Aleksandrov [1], Serrin [38], Gidas, Ni and Nirenberg [21]–[22], Berestycki and Nirenberg [2], and others. In [30], Li and Zhu gave a proof of the above mentioned theorem of Caffarelli, Gidas and Spruck using the method of moving spheres (i.e. the method of moving planes together with the conformal invariance), which fully exploits the conformal invariance of the problem and, as a result, captures the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. Significant simplifications to the proof in [30] have been made in Li and Zhang [29]. The method of moving spheres has been used in [24]–[27]. Liouville type theorems for various conformally invariant equations have received much attention; see, in addition to the above cited papers, [23], [17], [15], [33], [42] and [43].

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=6&iss=2&rank=1

Remark on some conformally invariant integral equations: the method of moving spheres

The proof of the Lane–Emden conjecture in four space dimensions

In this paper, we consider the following conformally invariant equations of fourth order¶
 
$ \cases {\Delta^2 u = 6 e^{4u} &in $\bf {R}^4,$ \cr e^{4u} \in L^1(\bf {R}^4),\cr}$
 (1)¶and¶
 
$ \cases {\Delta^2 u = u^{n+4 \over n-4}, \cr u>0 & in $ {\bf R}^n $ \qquad for $ n \ge5 $, \cr} $
 (2) where 
$ \Delta^2 $
 denotes the biharmonic operator in R

 n
 . By employing the method of moving planes, we are able to prove that all positive solutions of (2) are arised from the smooth conformal metrics on S n
 by the stereograph projection. For equation (1), we prove a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4
 .

https://www.ems-ph.org/journals/show_pdf.php?issn=0010-2571&vol=73&iss=2&rank=2

C.-S. Lin

Papers

A classification of solutions of a conformally invariant fourth order equation in Rn