Nonlinear Elliptic Equation

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Shuyu Hsu - One of the best experts on this subject based on the ideXlab platform.

  • some properties of the yamabe soliton and the related Nonlinear Elliptic Equation
    Calculus of Variations and Partial Differential Equations, 2014
    Co-Authors: Shuyu Hsu
    Abstract:

    Firstly we prove the non-existence of positive radially symmetric solution of the Nonlinear Elliptic Equation \(\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot \nabla u=0\) in \(\mathbb{R }^{n}\) when \(n\ge 3\), \(0Elliptic Equation for other parameter range of \(\alpha \) and \(\beta \). Then these results are applied to prove some results on Yamabe solitons including the exact behaviour of the metric of the Yamabe soliton, its scalar curvature and sectional curvature, at infinity. A new proof of a result of Daskalopoulos and Sesum (The classification of locally conformally flat Yamabe solitons, http://arxiv.org/abs/1104.2242) on the positivity of the sectional curvature of Yamabe solitons is also presented.

  • exact decay rate of a Nonlinear Elliptic Equation related to the yamabe flow
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Shuyu Hsu
    Abstract:

    Let 0 2, $\alpha=(2\beta +\rho)/(1-m)$ and $\beta>m\rho/(n-2-mn)$ for some constant $\rho>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$, v>0, in $R^n$. When m=(n-2)/(n+2), the metric $g=v^{4/(n+2)}dx^2$ corresponds to a locally conformally flat Yamabe shrinking gradient soliton with positive sectional curvature. We prove that the solution $v$ of the above Nonlinear Elliptic Equation has the exact decay rate $\lim_{r\to\infty}r^2v(r)^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(\alpha (1-m)-2\beta)}$.

  • some properties of the yamabe soliton and the related Nonlinear Elliptic Equation
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Shuyu Hsu
    Abstract:

    We will prove the non-existence of positive radially symmetric solution of the Nonlinear Elliptic Equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla u=0$ in $R^n$ when $n\ge 3$, $0 \frac{\rho}{n-2}>0$, the scalar curvature $R(r)\to\rho$ as $r\to\infty$ if either $\beta>\frac{\rho}{n-2}>0$ or $\rho=0$ and $\alpha>0$ holds, and $\lim_{r\to\infty}R(r)=0$ if $\rho 0$. We give a simple different proof of a result of P.Daskalopoulos and N.Sesum \cite{DS2} on the positivity of the sectional curvature of rotational symmetric Yamabe solitons $g=v^{\frac{4}{n+2}}dx^2$ with $v$ satisfying the above Equation with $m=\frac{n-2}{n+2}$. We will also find the exact value of the sectional curvature of such Yamabe solitons at the origin and at infinity.

  • singular limit and exact decay rate of a Nonlinear Elliptic Equation
    Nonlinear Analysis-theory Methods & Applications, 2012
    Co-Authors: Shuyu Hsu
    Abstract:

    Abstract For any n ≥ 3 , 0 m ≤ ( n − 2 ) / n , and constants η > 0 , β > 0 , α ≤ β ( n − 2 ) / m , we prove the existence of radially symmetric solution of n − 1 m Δ v m + α v + β x ⋅ ∇ v = 0 , v > 0 , in R n , v ( 0 ) = η , without using the phase plane method. When 0 m ( n − 2 ) / n , n ≥ 3 , we prove that v satisfies lim | x | → ∞ | x | 2 v ( x ) 1 − m log | x | = 2 ( n − 1 ) ( n − 2 − n m ) β ( 1 − m ) if α = 2 β / ( 1 − m ) > 0 and lim | x | → ∞ | x | α / β v ( x ) = A for some constant A > 0 if 2 β / ( 1 − m ) > max ( α , 0 ) . For β > 0 or α = 0 , we prove that the radially symmetric solution v ( m ) of the above Elliptic Equation converges uniformly on every compact subset of R n to the solution of an Elliptic Equation as m → 0 .

  • singular limit and exact decay rate of a Nonlinear Elliptic Equation
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Shuyu Hsu
    Abstract:

    For any $n\ge 3$, $0 0$, $\beta>0$, $\alpha$, satisfying $\alpha\le\beta(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}\Delta v^m+\alpha v +\beta x\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=\eta$, without using the phase plane method. When $0 0$, we prove that the radially symmetric solution $v$ of the above Elliptic Equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{\beta(1-m)}$. In particular when $m=\frac{n-2}{n+2}$, $n\ge 3$, and $\alpha=2\beta/(1-m)>0$, the metric $g_{ij}=v^{\frac{4}{n+2}}dx^2$ is the steady soliton solution of the Yamabe flow on $\R^n$ and we obtain $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|}=\frac{(n-1)(n-2)}{\beta}$. When $0 \max (\alpha,0)$, we prove that $\lim_{|x|\to\infty}|x|^{\alpha/\beta}v(x)=A$ for some constant $A>0$. For $\beta>0$ or $\alpha=0$, we prove that the radially symmetric solution $v^{(m)}$ of the above Elliptic Elliptic Equation converges uniformly on every compact subset of $\R^n$ to the solution $u$ of the Equation $(n-1)\Delta\log u+\alpha u+\beta x\cdot\nabla u=0$, $u>0$, in $\R^n$, $u(0)=\eta$, as $m\to 0$.

Marino Badiale - One of the best experts on this subject based on the ideXlab platform.

Laurent Veron - One of the best experts on this subject based on the ideXlab platform.

Andrey Shishkov - One of the best experts on this subject based on the ideXlab platform.

Noureddine Zeddini - One of the best experts on this subject based on the ideXlab platform.

  • Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R ^2
    Potential Analysis, 2003
    Co-Authors: Noureddine Zeddini
    Abstract:

    For a bounded regular Jordan domain Ω in R ^2, we introduce and study a new class of functions K (Ω) related on its Green function G . We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular Nonlinear Elliptic Equation Δ u +ϕ( x , u )=0, in D ′(Ω), with u =0 on ∂Ω and u ∈ C ―(Ω), where ϕ is a nonnegative Borel measurable function in Ω×(0,∞) that belongs to a convex cone which contains, in particular, all functions ϕ( x , t )= q ( x ) t ^−γ,γ>0 with nonnegative functions q ∈ K (Ω). Some estimates on the solution are also given.

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