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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
##### Keywords:
finite mass conditions; Rellich-Pokhozaev identity
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##### References:
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