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arXiv 2106.07567math.AP
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上半空间中与Yamabe边值问题有关的非线性Laplace方程

On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space

Gael Diebou Yomgne

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This document is an update of the previous version with a use of a different framework, including changes in the main results
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摘要：

本文考虑具有Neumann边界条件的非线性椭圆型方程\begin{align*} \begin{cases} Δu=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{η-1}u+f\,\,\mbox{ on }\,\,\partial\rnp. \end{cases} \end{align*}.对于$a,b\neq 0$，$m>\frac{n+1}{n-1}$，$(n>1)$，$η=\frac{m+1}{2}$和小数据$f\in L^{\frac{nq}{n+1},\infty}(\partial\rnp)$，$q=\frac{(n+1)(m-1)}{m+1}$，我们证明了问题是可解的.更准确地说，我们建立的存在性，唯一性和连续依赖的边界数据$f$在函数空间$\X^{q}_{\infty}$，其中\[\|u\|_{\X^{q}_{\infty}}=\sup_{t>0}t^{\frac{n+1}{q}-1}\|u(\cdot,t)\|_{L^{\infty}(\partial\rnp)}+\|u\|_{L^{\frac{q(m+1)}{2},\infty}(\rnp)}+\|\nabla u\|_{L^{q,\infty}(\rnp)}. \]作为一个直接的后果，我们得到的局部正则性$C^{1,ν}_{loc}$，$ν\in (0,1)$这些解决方案以及能量估计$m$的某些值。边界值衰减速度比$|x|^{-(m+1)/(m-1)}$，$x\in \rn\setminus\{0\}$产量的可解性和这种衰减性能被证明是尖锐的正非线性。此外，我们能够表明，解决方案继承定性特征的边界数据，如积极性，旋转对称性相对于$(n+1)$ -轴，径向单调的切向变量和均匀性。当$a,b>0$时，确定了正解存在的临界指数$m_c$，即$m_c=(n+1)/(n-1)$ .

英文摘要：

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} Δu=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{η-1}u+f\,\,\mbox{ on }\,\,\partial\rnp. \end{cases} \end{align*} For $a,b\neq 0$, $m>\frac{n+1}{n-1}$, $(n>1)$, $η=\frac{m+1}{2}$ and small data $f\in L^{\frac{nq}{n+1},\infty}(\partial\rnp)$, $q=\frac{(n+1)(m-1)}{m+1}$ we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data $f$ in the function space $\X^{q}_{\infty}$ where \[\|u\|_{\X^{q}_{\infty}}=\sup_{t>0}t^{\frac{n+1}{q}-1}\|u(\cdot,t)\|_{L^{\infty}(\partial\rnp)}+\|u\|_{L^{\frac{q(m+1)}{2},\infty}(\rnp)}+\|\nabla u\|_{L^{q,\infty}(\rnp)}. \] As a direct consequence, we obtain the local regularity property $C^{1,ν}_{loc}$, $ν\in (0,1)$ of these solutions as well as energy estimates for certain values of $m$. Boundary values decaying faster than $|x|^{-(m+1)/(m-1)}$, $x\in \rn\setminus\{0\}$ yield solvability and this decay property is shown to be sharp for positive nonlinearities. Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the $(n+1)$-axis, radial monotonicity in the tangential variable and homogeneity. When $a,b>0$, the critical exponent $m_c$ for the existence of positive solutions is identified, $m_c=(n+1)/(n-1)$.

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