Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature

Abstract

We show that the prescribed Gaussian curvature equation in ${\mathbb{R}}^2$ $$-\Delta u=(1-|x{|}^p)e^{2u},$$ has solutions with prescribed total curvature equal to $\Lambda :={\int }_{{\mathbb{R}}^2}(1-|x{|}^p)e^{2u}dx\in \mathbb{R}$, if and only if $$p\in (0,2)\text{and}(2+p)\pi \le \Lambda <4\pi$$ and prove that such solutions remain compact as $\Lambda \to \bar{\Lambda }\in [(2+p)\pi ,4\pi )$, while they produce a spherical blow-up as $\Lambda \uparrow 4\pi$.

References in corpus (3)

• Concentration-compactness phenomena in the higher order Liouville's equation
• Quantization for the prescribed Q-curvature equation on open domains
• Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature