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Ali Hyder

Publications and source records attributed to Ali Hyder.

33 records · Page 2Linked to original sources
arXiv2020

Concentration phenomena to a higher order Liouville equation

We study blow-up and quantization phenomena for a sequence of solutions $(u_k)$ to the prescribed $Q$-curvature problem $$ (-Δ)^nu_k= Q_ke^{2nu_k}\quad \text{in }Ω\subset\mathbb{R}^{2n},\quad \int_Ωe^{2nu_k}dx\leq C,$$ under natural assumptions on $Q_k$. It is well-known that, up to a subsequence, either $(u_k)$ is bounded in a suitable norm, or there exists $β_k\to\infty$ such that $ u_k=β_k(φ+o(1))$ in $Ω\setminus (S_1\cup S_φ)$ for some non-trivial non-positive $n$-harmonic function $φ$ and for a finite set $S_1$, where $S_φ$ is the zero set of $φ$. We prove quantization of the total curvature $\int_{\tildeΩ}Q_ke^{2nu_k}dx$ on the region $\tildeΩ\Subset(Ω\setminus S_φ)$. We also consider a non-local case in dimension three.

math.AP↗
arXiv2019

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-Δ)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).

math.AP↗
arXiv2019

On the general Toda system with multiple singular points

In this paper, we consider the following elliptic Toda system associated to a general simple Lie algebra with multiple singular sources \begin{equation*} \begin{cases} -Δw_i=\sum_{j=1}^na_{i,j}e^{2w_j}+2π\sum_{\ell=1}^mβ_{i,\ell}δ_{p_\ell} \quad&\mbox{in}\quad\mathbb{R}^2,\\ \\ w_i(x)=-2\log|x|+O(1)~\mbox{as}~|x|\to\infty,\quad &i=1,\cdots,n, \end{cases} \end{equation*} where $β_{i,\ell}\in[0,1)$. Under some suitable assumption on $β_{i,\ell}$ we establish the existence and non-existence results. This paper generalizes Luo-Tian's [19] and Hyder-Lin-Wei's [10] results to the general Toda system.

math.AP↗
arXiv2019

On $SU(3)$ Toda system with multiple singular sources

We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-Δw_1=2e^{2w_1}-e^{w_2}+2π\sum_{\ell=1}^mβ_{1,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -Δw_2=2e^{2w_2}-e^{w_1}+2π\sum_{\ell=1}^mβ_{2,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2 \\ w_i(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\, i=1,2, \end{array}\right.\end{align*} with $m\geq 3$ and $β_{i,\ell}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on $β_{i,\ell}$. This generalizes Luo-Tian's \cite{Luo-Tian} result for a singular Liouville equation in $\mathbb{R}^2$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^n$.

math.AP↗
arXiv2019

Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume

In this paper we study the following conformally invariant poly-harmonic equation $$Δ^mu=-u^\frac{3+2m}{3-2m}\quad\text{in }\mathbb{R}^3,\quad u>0,$$ with $m=2,3$. We prove the existence of positive smooth radial solutions with prescribed volume $\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$. We show that the set of all possible values of the volume is a bounded interval $(0,Λ^*]$ for $m=2$, and it is $(0,\infty)$ for $m=3$. This is in sharp contrast to $m=1$ case in which the volume $\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$ is a fixed value.

math.AP↗
arXiv2018

Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $Φ$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

math.AP↗
arXiv2018

The non-local mean-field equation on an interval

We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-Δ)^\frac{1}{2} u=ρ\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $ρ<2π$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.

math.AP↗
arXiv2018

Local and nonlocal singular Liouville equations in Euclidean spaces

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

math.AP↗
arXiv2016

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Let $m\ge 2$ be an integer. For any open domain $Ω\subset\mathbb{R}^{2m}$, non-positive function $φ\in C^\infty(Ω)$ such that $Δ^m φ\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(Ω)$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(Ω)$ solving the Liouville equation of order $2m$ $$(-Δ)^m u_k = V_ke^{2mu_k}\quad \text{in }Ω, \quad \limsup_{k\to\infty} \int_Ωe^{2mu_k}dx<\infty,$$ and blowing up exactly on the set $S_φ:=\{x\in Ω:φ(x)=0\}$, i.e. $$\lim_{k\to\infty} u_k(x)=+\infty \text{ for }x\in S_φ \text{ and }\lim_{k\to\infty} u_k(x)=-\infty \text{ for }x\in Ω\setminus S_φ,$$ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $Ω$ and to the case $Ω=\mathbb{R}^{2m}$. Several related problems remain open.

math.AP↗
arXiv2016

Conformally Euclidean metrics on $\mathbb{R}^n$ with arbitrary total $Q$-curvature

We study the existence of solution to the problem $$(-Δ)^\frac n2u=Qe^{nu}\quad\text{in }\mathbb{R}^{n},\quad κ:=\int_{\mathbb{R}^{n}}Qe^{nu}dx<\infty,$$ where $Q\geq 0$, $κ\in (0,\infty)$ and $n\geq 3$. Using ODE techniques Martinazzi for $n=6$ and Huang-Ye for $n=4m+2$ proved the existence of solution to the above problem with $Q\equiv const>0$ and for every $κ\in (0,\infty)$. We extend these results in every dimension $n\geq 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which $Q$ is non-constant, and under some decay assumptions on $Q$ we can also treat the cases $n=3$ and $4$.

math.AP↗
arXiv2015

Moser functions and fractional Moser-Trudinger type inequalities

We improve the sharpness of some fractional Moser-Trudinger type inequalities, particularly those studied by Lam-Lu and Martinazzi. As an application, improving upon works of Adimurthi and Lakkis, we prove the existence of weak solutions to the problem $(-Δ)^\frac{n}{2}u=λue^{bu^2} \,\text{ in }Ω,\, 0<λ<λ_1,\,b>0,$ with Dirichlet boundary condition, for any domain $Ω$ in $\mathbb{R}^n$ with finite measure. Here $λ_1$ is the first eigenvalue of $(-Δ)^\frac n2$ on $Ω$.

math.AP↗
arXiv2015

Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature

In this article we study the nonlocal equation \begin{align} (-Δ)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three we classify all solutions to the above equation in terms of their behavior at infinity.

math.AP↗
arXiv2015

Existence of entire solutions to a fractional Liouville equation in $\mathbb{R}^n$

We study the existence of solutions to the problem $$ (-Δ)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension $n\geq 3$ we show that to a certain extent the asymptotic behavior of $u$ and the constant $V$ can be prescribed simultaneously. Furthermore if $Q=-(n-1)!$ then $V$ can be chosen to be any positive number. This is in contrast to the case $n=3$, $Q=2$, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily $V\le |S^3|$, and to the case $n=4$, $Q=6$, where C-S. Lin showed that $V\le |S^4|$.

math.AP↗
arXiv2014

Conformal metrics on $R^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-Δ)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.

math.DG↗