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Man Chun Leung

Publications and source records attributed to Man Chun Leung.

14 recordsLinked to original sources
arXiv2021

Construction of Blow-up Sequences for the Prescribed Scalar Curvature Equation on $S^n$. IV. Clustered Blow-ups

For the prescribed scalar curvature equation on $S^n$ ($n \ge 6$), we consider the situation where the number of bubbles tends to infinity in the Lyapunov-Schmidt (finite dimension) reduction method. In an outstanding paper by Wei and Yan, the special case where the bubbles are arranged "evenly" (close to a great circle) is considered. Here we are concerned with the generic scenario, where the bubbles are "planted" (arranged) in general position, in adjacent to the critical points of the prescribed function. The main interest of this article is to extract key information in the finite dimensional reduced functional, as well as in its first partial derivatives. Overall, there are two main contributions in the "expansions", namely, interaction between bubbles and curvature involvement. As each bubble becomes infinitesimally close to other bubbles in the neighborhood, the errors in the expansions are carefully estimated in terms of the geometric parameters. This lays the foundation for our next step on constructing blow-up sequences of solutions with dense blow-up points (clustered blow-up), where the (non-radial symmetric) prescribed scalar curvature function has non-isolated critical points.

math.AP↗
arXiv2017

Refined estimates for simple blow-ups of the scalar curvature equation on S^n

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

math.AP↗
arXiv2017

Conformal scalar curvature equation on S^n: functions with two close critical points (twin pseudo-peaks)

By using the Lyapunov-Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on S^n (n greater or equal to 3) when the prescribed function (after being projected to R^n) has two close critical points, which have the same value (positive), equal "flatness" (twin, flatness < n - 2), and exhibit maximal behavior in certain directions (pseudo-peaks). The proof relies on a balance between the two main contributions to the reduced functional - one from the critical points and the other from the interaction of the two bubbles.

math.AP↗
arXiv2002

Combining solutions of semilinear partial differential equations in R^n with critical exponent

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $Δu + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the points $ξ_1$ and $ξ_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ such that $u = u_1$ in $Ω_1$ and $u = u_2$ in $Ω_2$, where $Ω_1$ and $Ω_2$ are disjoint non-empty open domains in ${\R}^n$. $u$ satisfies the equation $Δu + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n.$ By the same result of Gidas, Ni and Nirenberg, $K \not\equiv 1$ in $R^n$. In this paper we discuss lower bounds on $\displaystyle{\sup_{\R^n} |K - 1|} .$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.

math.AP↗
arXiv2002

Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent

For an integer $n \ge 3$ and any positive number $ε$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le ε$, such that the equation $Δu + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n \setminus \{0 \}$ has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some cases K can be extended as a Lipschitz function on ${\R}^n.$ These provide counter-examples to a conjecture of C.-S. Lin when n > 4, and Taliaferro's conjecture.

math.AP↗
arXiv2000

Growth estimates on positive solutions of the equation $Δu + K u^{{n + 2}\over {n - 2}} = 0$ in ${\R}^n$

We construct unbounded positive $C^2$-solutions of the equation $Δu + K u^{(n + 2)/(n - 2)} = 0$ in ${\R}^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in ${\R}^n$, the conformal metric $g = u^{4/(n - 2)} g_o$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the $L^{2n/(n - 2)}$-norm of the solution and show that it has slow decay.

math.AP↗
arXiv1996

Asymptotic properties of energy of harmonic maps on asymptotically hyperbolic manifolds

Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in terms of a defining function for the boundary.

dg-ga↗
arXiv1996

Uniqueness of Positive Solutions of the Conformal Scalar Curvature Equation and Applications to Conformal Transformations

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We also study uniqueness of complete positive solutions and radial solutions.

dg-ga↗
arXiv1996

Rotationally Symmetric F-harmonic Maps Equations

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.

dg-ga↗
arXiv1996

Rotationally Symmetric $p$-harmonic maps

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investigated.

dg-ga↗
arXiv1995

On the $L^{n\over 2}$-norm of Scalar Curvature

Comparisons on $L^{n\over 2}$-norms of scalar curvatures between Riemannian metrics and standard metrics are obtained. The metrics are restricted to conformal classes or under certain curvature conditions.

dg-ga↗