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Qualitative properties of solutions to some nonlinear elliptic equations in \(\mathbb{R}^2\). (English) Zbl 0923.35055

We investigate properties of the solutions to the elliptic equations \(-\Delta u= R(x)e^{u(x)}\), \(x\in\mathbb{R}^2\) for functions \(R(x)\) which are positive near infinity. Equations of this kind arise from a variety of situations, such as from prescribing Gaussian curvature in geometry and from combustion theory in physics.
We consider general functions \(R(x)\). First, we obtain the asymptotic behavior of the solution near infinity. Consequently, we prove that all the solutions satisfy an identity, which is somewhat of a generalization of the well-known Kazdan-Warner condition. Finally, using the asymptotic behavior together with the further development of the method employed in our previous paper, we show that all the solutions are radially symmetric provided \(R\) is radially symmetric and nonincreasing.

MSC:

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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