Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
In this paper, the authors study the existence and asymptotics of solutions of the equation \[ (-\Delta)^\frac{3}{2}u = 2 e^{3u}\quad\text{on}\quad \mathbb R^3\tag{1} \] with \[ V:=\int_{\mathbb R^3} e^{3u}\, dx <\infty.\tag{2} \] Note that (1) can be considered as a prescribed \(Q\)-curvature equation, in the sense that if a smooth function \(u\) solves the equation \((-\Delta)^\frac{3}{2}u = K e^{3u}\) for some function \(K\), then the metric \( \tilde g:= e^{2u}g_0\) has \(Q\)-curvature \(K\) and the quantity \(V\) appearing in (2) is simply the volume of the conformal metric \(\tilde g\). Here, \(g_0\) is the flat Euclidean metric on \(\mathbb R^3\) and \((-\Delta)^\frac{3}{2}\) is interpreted as \((-\Delta)^\frac{1}{2} \circ (-\Delta)\). To define \((-\Delta)^{\frac{1}{2}}v\) for a function \(v\), it is required that \[ v \in L_{1/2}(\mathbb R^3):= \left\{v \in L^1_{\operatorname{loc}}: \int_{\mathbb R^3} \frac{| v(x)|}{1+| x|^4}dx < \infty\right\}, \] which makes \((-\Delta)^{\frac{1}{2}}v\) be a tempered distribution. Given a tempered distribution \(f\) on \(\mathbb R^3\), we say that \(u\) is a solution of \((-\Delta)^\frac{3}{2}u = f\) if \(u \in W^{2,1}_{\operatorname {loc}}(\mathbb R^3), \Delta u \in L_{1/2}(\mathbb R^3)\) and \(\int_{\mathbb R^3} (-\Delta u)(-\Delta)^{\frac{1}{2}} \varphi\), \(dx = \langle f\), \(\varphi\rangle\) for every \(\varphi \in \mathcal{S}(\mathbb R^3)\), where \(\mathcal{S}(\mathbb R^3)\) is the Schwarz space of rapidly decreasing smooth functions on \(\mathbb R^3\). One of the main results in this paper is the following. For every \(V \in (0, 2\pi^2)\), the problem (1)-(2) has at least one solution \(u \in C^\infty(\mathbb R^3)\) and \(u\) is of the form \(u = v+p\), where \(p\) is a polynomial of degree 0 or 2 and \[ v(x) = \frac{1}{2\pi^2}\int_{\mathbb R^3} \log\left(\frac{| y|}{| x-y|}\right)e^{3u(y)}\, dy. \] Using this decomposition property for the solutions, the authors investigate the behavior of the solution \(u\) at infinity. For example, the authors prove that \(\deg p = 0\) is equivalent to the fact that \(u\) is a spherical solution of the form \[ u(x) = \log\left(\frac{2\lambda}{1+\lambda^2| x-x_0|^2}\right) \] for some \(x_0 \in\mathbb R^3\) and \(\lambda >0\), or equivalently \(u(x) = {\text o}(| x|^2)\) as \(| x| \to \infty\). Finally, the authors show that if \(u\) is a non-spherical solution of (1)-(2), then \(V < 2\pi^2\).
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