We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails when the dimension of the manifold is not less than $62$.

中文翻译：

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常标量曲率和常边界平均曲率方程的爆破现象

我们首先提出了一个带边界的翘曲积流形，以显示正常数标量曲率和正常数边界平均曲率方程的非唯一性。接下来，我们构造一个平滑的反例来证明当流形的维数不小于 $62$ 时，上述方程组的“低能量”解的紧致性失败。