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On the structure of the conformal Gaussian curvature equation on \({\mathbb{R}}^ 2\). (English) Zbl 0733.35037
The object of investigation of this paper is the semilinear elliptic equation \(\Delta u+Ke^{2u}=0\) in \({\mathbb{R}}^ 2\), where K is a nonpositive function. Such an equation arises in Riemannian geometry. There where known non-existence results given some assumptions on the behavior of K at \(\infty\). This work deals with existence problems. Under fairly general assumptions on K the authors prove the existence of a maximal solution - provided the equation has a (non-maximal) solution. This rather abstract result is then used to study the existence for two quite general types of asymptotic behavior of K at \(\infty\). The consequences are twofold: first an exhaustive classification of the solutions becomes possible for a general class of K’s and on the other hand the geometry of the solutions can be understood when K is assumed radially symmetric. Namely in this case - given that K has a certain asymptotic behavior - it can be shown that all solutions are radially symmetric. The proofs are based on comparison arguments using maximum principle.

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53B20 Local Riemannian geometry
35B50 Maximum principles in context of PDEs
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