论文标题
非线性培养基中的旋转波和临界退化Sobolev不等式
Rotating waves in nonlinear media and critical degenerate Sobolev inequalities
论文作者
论文摘要
我们调查了非线性波方程的旋转波解$ \ partial_t^2 v -ΔV +m v = | v | v |^{p -2} v $ in $ \ mathbb {r} \ times \ times \ times \ mathbf {b} $在$ \ mathbb {r} \ times \ partial \ mathbf {b} $上。根据旋转的规定角速度$α$的规定，这导致了半椭圆形或退化椭圆方程的差异问题。我们表明，这个问题受半空间中相关的关键归化sobolev不平等的影响。在证明了这种不平等和相关的极端功能的存在之后，我们为存在基态解决方案的存在推导了必要和充分的条件。此外，我们分析了这些基础状态的$α$，$ m $和$ p $的条件是非自由的，因此产生了真正的旋转波。我们的方法延续到了环和具有边界（包括半球）的更通用的Riemannian模型中的相应的Dirichlet问题。我们简要讨论这些问题，并表明它们与较大的关联批判性堕胎家族有关。
We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_t^2 v - Δv +m v = |v|^{p-2} v$ in $\mathbb{R} \times \mathbf{B}$, where $\mathbf{B} \subset \mathbb{R}^N$ is the unit ball, complemented with Dirichlet boundary conditions on $\mathbb{R} \times \partial\mathbf{B}$. Depending on the prescribed angular velocity $α$ of the rotation, this leads to a Dirichlet problem for a semilinear elliptic or degenerate elliptic equation. We show that this problem is governed by an associated critical degenerate Sobolev inequality in the half space. After proving this inequality and the existence of associated extremal functions, we then deduce necessary and sufficient conditions for the existence of ground state solutions. Moreover, we analyze under which conditions on $α$, $m$ and $p$ these ground states are nonradial and therefore give rise to truly rotating waves. Our approach carries over to the corresponding Dirichlet problems in an annulus and in more general Riemannian models with boundary, including the hemisphere. We briefly discuss these problems and show that they are related to a larger family of associated critical degenerate Sobolev inequalities.
