论文标题
对于具有Neumann边界条件的纯车道填充系统的关键双曲线上存在溶液
Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions
论文作者
论文摘要
我们研究以下车道填充系统\ [-ΔU= | v |^{q-1} v \ quad \ text {in}ω，\ qquad-ΔV= | u | u |^{p-1} u \ quad \ quad \ quad \ text {in}Ω $ \ Mathbb {r}^n $，$ n \ ge 4 $的$ω$和属于所谓关键的trigical blitla $ 1/（p+1）+1/（q+1）=（n-2）=（n-2）/n $的有限的常规域。我们表明，在$ P，Q $，最小能量（签名）解决方案的适当条件下，它们是经典的。在证明中，我们利用了双重变异表述，该配方允许处理问题的强烈不确定特征。我们建立了一个基于新的Cherrier类型不平等的紧凑条件。然后，我们通过使用AS测试功能证明了这种条件，使整个空间中系统的解决方案并进行精致的渐近估计值。如果$ n \ ge 5 $，$ p = 1 $，则以上的系统还原为双旋转方程，我们也证明存在最小能量的解决方案。最后，我们证明了$ω$的一些部分对称性和对称性的结果是一个球或一个环。
We study the following Lane-Emden system \[ -Δu=|v|^{q-1}v \quad \text{ in } Ω, \qquad -Δv=|u|^{p-1}u \quad \text{ in } Ω, \qquad u_ν=v_ν=0 \quad \text{ on } \partial Ω, \] with $Ω$ a bounded regular domain of $\mathbb{R}^N$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $Ω$ is a ball or an annulus.
