Isolated singularities of polyharmonic operator in even dimension

Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions

Abstract: (1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.

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

Abstract: 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 .

Polyharmonic Boundary Value Problems

Abstract: Page and line numbers refer to the final version which appeared at Springer-Verlag. The preprint version, which can be found on our personal web pages, has different page and line numbers.

Concentration–compactness phenomena in the higher order Liouville's equation☆

Abstract: We investigate different concentration–compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in {R^2}, then that of a closed manifold and, finally, the particular case of the sphere {S^2m}. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in {R^2m}, blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.