Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents
Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents
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DOI：
10.4171/jems/473
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发表时间：
2011-07
期刊：
影响因子：
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通讯作者：
M. Pino;F. Mahmoudi;M. Musso
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文献类型：
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作者：
M. Pino;F. Mahmoudi;M. Musso
我们考虑方程$d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $，在零诺伊曼边界条件下，其中$\Omega$是开的，光滑的，有界的，$d$是一个小的正参数。我们假设$\partial\Omega$存在一个$k$维闭嵌最小子流形$K$，它是非简并的，并且$\partial\Omega$的截面曲率沿$K$有一定的正加权平均值。然后我们证明了一个序列$d=d_j\to 0$和一个正解$u_d$的存在性，使得$$ d^2 |\nabla u_{d} |^2 \rightharpoonup S, \delta_K \ass d \to 0 $$在测度的意义上，其中$\delta_K$表示在$K$上支持的狄拉克测度，$S$是一个正常数。
We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial\Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial\Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $$ d^2 |\nabla u_{d} |^2 \rightharpoonup S, \delta_K \ass d \to 0 $$ in the sense of measures, where $\delta_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.