论文标题
非线性标量场方程的基态非平稳状态，涉及三个和四个维度的高频在高频上
Nondegeneracy of ground states for nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies in three and four dimensions
论文作者
论文摘要
我们考虑涉及高频$ω$的非线性标量字段方程。由于基态的限制曲线为$ω\至\ infty $是aubin-talenti函数，并且从某种意义上说，从某种意义上说，从扰动方法的角度来看，高频下基态的非修复性问题是微妙的。此外，由于限制配置文件（Aubin-Talenti函数）未能躺在$ l^{2}（\ Mathbb {r}^{d}）$ for $ d = 3,4 $中，因此$ d = 3,4 $的非排效问题比$ d \ ge 5 $ and a ge 5 $和不知道的方法更困难。在本文中，我们通过修改[2，3]中的论点来解决$ d = 3,4 $的非统一问题。我们还表明，基态周围的线性操作员完全具有一个负特征值。
We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies $ω$. Since the limiting profile of the ground state as $ω\to \infty$ is the Aubin-Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin-Talenti function) fails to lie in $L^{2}(\mathbb{R}^{d})$ for $d=3,4$, the nondegeneracy problem for $d=3,4$ is more difficult than that for $d\ge 5$ and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for $d=3,4$ by modifying the arguments in [2, 3]. We also show that the linearized operator around the ground state has exactly one negative eigenvalue.
