Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
type: Item
instance of: scholarly article
title: Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
zbMATH Open document ID: 1325.35300
author: Tianling Jin
author: Ali Maalaoui
author: Luca Martinazzi
author: Jingang Xiong
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-03-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.4299
review text: In this paper, the authors study the existence and asymptotics of solutions of the equation \[ (-\Delta)^\frac{3}{2}u = 2 e^{3u}\quad\text{on}\quad \mathbb R^3\tag{1} \] with \[ V:=\int_{\mathbb R^3} e^{3u}\, dx <\infty.\tag{2} \] Note that (1) can be considered as a prescribed \(Q\)-curvature equation, in the sense that if a smooth function \(u\) solves the equation \((-\Delta)^\frac{3}{2}u = K e^{3u}\) for some function \(K\), then the metric \( \tilde g:= e^{2u}g_0\) has \(Q\)-curvature \(K\) and the quantity \(V\) appearing in (2) is simply the volume of the conformal metric \(\tilde g\). Here, \(g_0\) is the flat Euclidean metric on \(\mathbb R^3\) and \((-\Delta)^\frac{3}{2}\) is interpreted as \((-\Delta)^\frac{1}{2} \circ (-\Delta)\). To define \((-\Delta)^{\frac{1}{2}}v\) for a function \(v\), it is required that \[ v \in L_{1/2}(\mathbb R^3):= \left\{v \in L^1_{\operatorname{loc}}: \int_{\mathbb R^3} \frac{| v(x)|}{1+| x|^4}dx < \infty\right\}, \] which makes \((-\Delta)^{\frac{1}{2}}v\) be a tempered distribution. Given a tempered distribution \(f\) on \(\mathbb R^3\), we say that \(u\) is a solution of \((-\Delta)^\frac{3}{2}u = f\) if \(u \in W^{2,1}_{\operatorname {loc}}(\mathbb R^3), \Delta u \in L_{1/2}(\mathbb R^3)\) and \(\int_{\mathbb R^3} (-\Delta u)(-\Delta)^{\frac{1}{2}} \varphi\), \(dx = \langle f\), \(\varphi\rangle\) for every \(\varphi \in \mathcal{S}(\mathbb R^3)\), where \(\mathcal{S}(\mathbb R^3)\) is the Schwarz space of rapidly decreasing smooth functions on \(\mathbb R^3\). One of the main results in this paper is the following. For every \(V \in (0, 2\pi^2)\), the problem (1)-(2) has at least one solution \(u \in C^\infty(\mathbb R^3)\) and \(u\) is of the form \(u = v+p\), where \(p\) is a polynomial of degree 0 or 2 and \[ v(x) = \frac{1}{2\pi^2}\int_{\mathbb R^3} \log\left(\frac{| y|}{| x-y|}\right)e^{3u(y)}\, dy. \] Using this decomposition property for the solutions, the authors investigate the behavior of the solution \(u\) at …
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 6418468
zbMATH Keywords: asymptotics for solutions
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: \(Q\)-curvature
zbMATH Keywords: spherical solutions
zbMATH Keywords: tempered distribution
Wikidata QID: Q115387397
MaRDI profile type: Publication
OpenAlex ID: W3098747809
arXiv ID: 1309.4299
cites work: Blow-up Analysis in Dimension 2 and a Sharp Form of Trudinger–Moser Inequality
cites work: Global compactness properties of semilinear elliptic equations with critical exponential growth
cites work: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
cites work: Group representations arising from Lorentz conformal geometry
cites work: Sharp Inequalities, the Functional Determinant, and the Complementary Series
cites work: Nonlinear elliptic equations in conformal geometry
cites work: A note on a class of higher order conformally covariant equations
cites work: The zeta functional determinants on manifolds with boundary. I: The formula
cites work: On uniqueness of solutions of \(n\)-th order differential equations in conformal geometry
cites work: Classification of solutions of some nonlinear elliptic equations
cites work: Multibumps analysis in dimension 2: quantification of blow-up levels
cites work: Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth
cites work: Scattering matrix in conformal geometry
cites work: Conformal metrics on \(\mathbb{R}^{2m}\) with constant \(Q\)-curvature, prescribed volume and asymptotic behavior
cites work: The heat flow with a critical exponential nonlinearity
cites work: Compactness of solutions to some geometric fourth-order equations
cites work: A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\)
cites work: \(Q\)-curvature flow on \(S^4\)
cites work: Classification of solutions to the higher order Liouville's equation on \({\mathbb{R}^{2m}}\)
cites work: Concentration-compactness phenomena in the higher order Liouville's equation
cites work: A threshold phenomenon for embeddings of \({H^m_0}\) into Orlicz spaces
cites work: QUANTIZATION FOR THE PRESCRIBED Q-CURVATURE EQUATION ON OPEN DOMAINS
cites work: Conformal metrics on \(\mathbb R^{2m}\) with constant \(Q\)-curvature and large volume
cites work: Quantization for an elliptic equation of order \(2m\) with critical exponential non-linearity
cites work: Asymptotics and Quantization for a Mean-Field Equation of Higher Order
cites work: The principle of symmetric criticality
cites work: Concentration phenomena for a fourth-order equation with exponential growth: the radial case
cites work: Quantization effects for a fourth-order equation of exponential growth in dimension $4$
cites work: Asymptotic Profile for a Fourth Order PDE with Critical Exponential Growth in Dimension Four
cites work: Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition
cites work: Regularity of the obstacle problem for a fractional power of the laplace operator
cites work: Q5607484
cites work: A flow approach to Nirenberg's problem
cites work: Quantization for a fourth order equation with critical exponential growth
cites work: Classification of solutions of higher order conformally invariant equations
cites work: Nonradial solutions for a conformally invariant fourth order equation in \(\mathbb {R}^4\)
cites work: Uniqueness and non-existence theorems for conformally invariant equations
DOI: 10.1007/S00526-014-0718-9
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