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Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. (English) Zbl 1056.53026

In the paper under review, the authors consider the prescribed zero scalar curvature and mean curvature problem on the \(n\)-dimensional Euclidean ball for \(n \geq 3\). Let \((B^n,\delta_{ij})\) be the unit ball in the \(n\)-dimensional Euclidean space with \(n\geq 3\). Given a function \(h : \partial B^n \to {\mathbb R}\), one can ask whether there exists a scalar flat metric \(g\) conformal to the flat metric \(g_0\) such that \(h\) is the mean curvature of \(\partial B^n\) with respect to the metric \(g\). This is equivalent to the problem of finding a positive smooth function \(u : B^n \to {\mathbb R}\) satisfying \[ \left\{\begin{aligned} &\Delta u = 0 \quad \text{in} \quad B^n \\ &\frac{\partial u}{\partial \eta} + \frac{n-2}{2} u = \frac{n-2}{2} h u^{\frac{n}{n-2}} \quad \text{on} \quad \partial B^n \end{aligned} \right.\tag{1} \] Trying to find solutions, the authors consider a general exponent \(q\) instead of the critical exponent \(n/n-2\). Namely, multiplying the function \(u\) by a suitable constant, equation \((1)\) above can be rewritten as \[ \left\{\begin{aligned} &\Delta u = 0 \quad \text{in}\quad B^n \\ &\frac{\partial u}{\partial \eta} + \frac{n-2}{2} u = \frac{n-2}{2} \sigma_{n-1} \left(J_q(u)\right)^{-1} h u^q \quad \text{on} \quad \partial B^n, \end{aligned} \right.\tag{2} \] where \(\sigma_{n-1} =\text{vol}(S^{n-1}), q \in (0, \frac{n}{n-2}]\) and \[ J_q(u) = \int_{\partial B^n}\, h u^{q-1}\, d \sigma. \] Observe that every solution for equation \((2)\) satisfies \[ E(u) = \int_{B^n} | \nabla u| ^2\, dx + \frac{n-2}{2} \int_{\partial B^n}\, u^2\, d \sigma = E(1). \] The authors obtain a priori estimates concerning the blow-up of solutions of equation \((2)\) and from it, they prove any smooth nonnegative function on \(\partial B^n\) can be approximated in the \(C^{0, \beta}\)-norm, where \(0 < \beta < 1\), by a sequence of smooth positive functions which are the mean curvature functions of smooth conformal metrics of the form \(u^{\frac{4}{n-2}} \delta_{ij}\) with zero scalar curvature. Using these properties, the authors study the limits of solutions of the regularization obtained by decreasing the critical exponent and characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate. Finally, they prove that, for \(n = 3\), equation has at least one solution and, for \(n \geq 4\), they give conditions on the function \(h\) to guarantee there is only one simple blow-up point.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J32 Boundary value problems on manifolds
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