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On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in $$\mathbb R^ 2$$. (English) Zbl 0871.35014
We study the asymptotic behavior at infinity of a solution $$u$$ of $$\Delta u+ K(x) e^{2u} =0$$ in $$\mathbb R^2$$. With some mild assumptions on $$K$$ and $$u$$, we conclude that $$u(x)= \alpha\log |x|+O(1)$$ at infinity for some real $$\alpha$$. Applying this result, we prove that solutions $$u$$ of the equation must be radially symmetric provided that $$K=K(|x|)$$ is radial and nonincreasing in $$|x|$$. We also prove that the above equation has no solution with a finite total curvature provided that $$K$$ is not identically a constant, bounded between two positive constants and monotonic along some direction.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35J60 Nonlinear elliptic equations 45G10 Other nonlinear integral equations