Conformally invariant systems of nonlinear PDE of Liouville type
type: Item
instance of: scholarly article
title: Conformally invariant systems of nonlinear PDE of Liouville type
zbMATH Open document ID: 0858.35035
DOI: 10.1007/BF01902215
author: Sagun Chanillo
author: Michael K.-H. Kiessling
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-03-23T00:00:00Z
full work available at URL: https://eudml.org/doc/58213
review text: The question is under which conditions on \(\gamma\) all solutions \(u_i\) \((i=1,\dots,N)\) of \[ -\Delta u_i(x)=\prod_{j\in{\mathcal I}}\exp(\gamma^{i,j}u_j), \qquad x\in\mathbb{R}^2, \] with \(N\) finite mass conditions \(\int_{\mathbb{R}^2} \exp(u_i)dx<\infty\), and \(\sum_{j\in{\mathcal I}}\gamma^{i,j}=1\), \(i\in{\mathcal I}\), are radially symmetric and decreasing about some point. For the scalar case \(N=1\) it is shown that the theorem in question can be proved by use of an isoperimetric inequality in its strict form and a Rellich-Pohozaev identity. Main theme of the paper is that this technique applies under certain conditions on the matrix \(\gamma=(\gamma^{i,j})\) to the above system as well to get an analog theorem.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 881240
zbMATH Keywords: finite mass conditions
zbMATH Keywords: Rellich-Pokhozaev identity
reviewed by: Joachim Jaenicke
MaRDI profile type: Publication
cites work: Q3876353
cites work: A new multicomponent Poincaré-Beckner inequality
cites work: Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)euin two dimensions
cites work: A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description
cites work: Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on \(S^ n\)
cites work: Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry
cites work: Continuity of solutions of uniformly elliptic equations in \(\mathbb{R}^ 2\)
cites work: Classification of solutions of some nonlinear elliptic equations
cites work: Asymptotic radial symmetry for solutions of \(\Delta u+e^ u=0\) in a punctured disc
cites work: Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence
cites work: Symmetry and related properties via the maximum principle
cites work: Q5187597
cites work: Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities
cites work: Statistical mechanics of classical particles with logarithmic interactions
cites work: Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains
cites work: The conjectures on conformal transformations of Riemannian manifolds
cites work: On the positivity of the effective action in a theory of random surfaces
cites work: Extremals of determinants of Laplacians
cites work: Q5516356
cites work: Darstellung der Eigenwerte von \(\Delta u+\lambda u=0\) durch ein Randintegral
OpenAlex ID: W2066359405
description: scientific article; zbMATH DE number 881240