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Compactness of conformal metrics with positive Gaussian curvature in \(\mathbb{R}^2\). (English) Zbl 0933.35056
The authors consider the entire solution \(u\) of the equation \(\Delta u+ K(x)\exp (2u)=0\) in \({\mathbb R}^2\), where \(K(x)\) is positive everywhere, and decays like \(|x|^{-b}\) as \(|x|\rightarrow\infty\) for some \(b>0\). Let \(\alpha (u)=-\frac 1{2\pi} \int_{{\mathbb R}^2} K(x)\exp(2u) dx\). They prove the following three results: (1) Assume \(K(x)=|x|^{-b}\) for \(|x|\geq 1\) and \(K(x)\equiv 1\) for \(|x|\leq 1\). Then any solution \(u\), provided that \(\alpha (u)\) is finite, satisfies \(-2<\alpha(u)<\min\{0,b-2\}\). Conversely for any \(\alpha\) such that \(-2<\alpha<\min\{0,b-2\}\), there exists a unique radially symmetric solution \(u\) satisfying \(\alpha (u)=\alpha\). (2) Suppose \(A|x|^{-b}\leq K(x)\leq B|x|^{-b}\) \((0<b<2)\) for \(|x|\geq 1\). Then \(\sup\{\alpha (u)\); there exists a solution \(u\) s.t. \(\alpha(u)\) is finite\(\}<{{b-2}\over 2}\). (3) Assume \({\lim_{|x|\rightarrow\infty} K(x)|x|^{b}}=1\) \((0<b<2)\) . If a sequence \(\{u_n\}\) of solutions satisfies the condition \(-2< \lim_{n\rightarrow\infty} \alpha (u_{n})\neq b-2\), then \(\{u_n\}\) is bounded in \(W^{2,p}({\mathbb R}^2)\) for any \(p>1\).

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B45 A priori estimates in context of PDEs
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