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- W2896964562 abstract "The purpose of this paper is to study the solutions of $$ Delta u +K(x) e^{2u}=0 quad{rm in};; mathbb{R}^2 $$ with $Kle 0$. We introduce the following quantity: $$alpha_p(K)=supleft{alpha in mathbb{R}:, int_{mathbb{R}^2} |K(x)|^p(1+|x|)^{2alpha p+2(p-1)} dx -infty$ for some $p>1$ and $alpha_1(K) > 0$, we show that for any $0 1$ and $alpha_1(K_0) > 0$, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution $u_*$ such that $u_* -alpha_*ln|x| = O(1)$ at infinity for some $alpha_* > 0$, but who does not converge to a constant at infinity. This example exhibits a new phenomenon of solutions with logarithmic growth and non-uniform behavior at infinity." @default.
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- W2896964562 date "2018-10-17" @default.
- W2896964562 modified "2023-09-27" @default.
- W2896964562 title "On Gaussian curvature equations in $mathbb{R}^2$ with prescribed non-positive curvature." @default.
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