论文标题
一半球的Nirenberg问题的非简单爆炸
Non simple blow ups for the Nirenberg problem on half spheres
论文作者
论文摘要
在本文中，我们研究了标准半领域的Nirenberg类型问题$（\ Mathbb {s}^n _+，g_0）$，包括在边界$ \ poartial $ \ partial \ saptial \ mathbb {s}^n _ _+$上找到规定的标量曲率和零边界曲率的共形指标。此问题等于解决涉及关键Sobolev指数的以下边界值问题：\ begin {equination*}（\ Mathcal {p}）\ quad \ quad \ begin {case} - \ d_ {g_0} u \，+\，\ frac {n（n-2）} {4} u \，= k \，u^，u^{\ frac {n+2} {n-2} {n-2}} {n-2}}，\，\，\，\，u> 0 \ frac {\ partial u} {\partialν} \，= \，0＆\ mbox {on} \ partial \ mathbb {s}^n_+。 \ end {cases} \ end {equation*}其中$ k \ in c^2（\ mathbb {s}^n _+）$是一个正函数。 我们在函数$ k $的一般条件下构建了$（\ MATHCAL {p}）$的有限能量解决方案，在尺寸$ n \ geq 5 $的一半领域上，在同一边界点上表现出\ emph {cluster-type}的多重爆炸。这些解决方案可能具有零或非弱极限，并且可能在不同边界点发展簇。这种爆炸的现象在半球上的现象与球体上的Nirenberg问题的情况形成了鲜明的对比，在这些情况下，不可能进行有限的能量亚物种的不简单爆炸，并且在流体动力学和平均值方程中在Euler方程中引起的涡流方程中出现的涡流类型问题的意外联系。 我们还构建了$ \ partial \ mathbb {s}^n _+$的$ k $限制的适当条件下
In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} where $K \in C^2(\mathbb{S}^n_+)$ is a positive function. We construct, under generic conditions on the function $K$, finite energy solutions of a subcritical approximation of $(\mathcal{P})$ on half spheres of dimension $n \geq 5$, which exhibit multiple blow up of \emph{cluster-type} at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy subsolutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of $K$ on $\partial \mathbb{S}^n_+$, approximate solutions of arbitrarily large energy and Morse index
