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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
##### Keywords:
elliptic equation; entire solution; compactness
Full Text:
##### References:
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