On Gaussian curvature equation in R² with prescribed nonpositive curvature

The purpose of this paper is to study the solutions of ∆u + K(x)e^2u = 0 in R² with K ≤ 0. We introduce the following quantities: αp(K) = sup { α ∈ R: ^Z _R2 |K(x)|^p(1 + |x|)²αp+2(p−¹⁾dx < +∞ } , ∀ p ≥ 1. Under the assumption (H₁): αp(K) > −∞ for some p > 1 and α1(K) > 0, we show that for any 0 < α < α1(K), there is a unique solution uα with uα(x) = 2β αln |x|+cα+o(|x|^{− 1+2}β ) at infinity and β ∈ (0, α1(K)−α). Furthermore, we show an example K₀ ≤ 0 such that αp(K₀) = −∞ for any p > 1 and α1(K₀) > 0, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution uα_∗ such that uα_∗ − α∗ ln |x| = O(1) at infinity for some α∗ > 0, which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.