Journal ArticleDOI

# Isolated singularities of polyharmonic operator in even dimension

02 Jan 2016-Complex Variables and Elliptic Equations (Taylor & Francis)-Vol. 61, Iss: 1, pp 55-66

TL;DR: In this paper, the authors considered the problem of finding the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma), where a is a nonnegative measurable function in some Lebesgue space.

AbstractWe consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.

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TL;DR: In this article, a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4>>\s was established for the stereograph projection of the biharmonic operator.
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 .

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BookDOI
01 Jan 2010
TL;DR: The preprint version of this paper has different page and line numbers from the final version which appeared at Springer-Verlag as mentioned in this paper, which can be found on their personal web pages.
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.

442 citations

Journal ArticleDOI

145 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension and showed that on a locally conformally flat manifold of non-positive Euler characteristic, one always has compactness.
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.

53 citations