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Conformally invariant systems of nonlinear PDE of Liouville type. (English) Zbl 0858.35035
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.

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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