论文标题
对二阶关键椭圆方程的签名解决方案的稳定性和不稳定性结果
Stability and instability results for sign-changing solutions to second-order critical elliptic equations
论文作者
论文摘要
在平滑，封闭的Riemannian歧管$ \ left（m，g \右）$的$ n \ ge3 $的$中，我们考虑固定的schrödinger方程$Δ_GU+H_0U = \ weft | u \ u \ u \ right | u \ right |^{2^* - 2^*-2} - 2} - 2} - 2} u $，wher c^1 \ left（m \ right）$和$ 2^*：= \ frac {2n} {n-2} $。我们证明，在$ c^1 \ left（m \右）$中的潜在函数$ h_0 $的扰动之后，以$ h^1 \ left（m \右）为$ c^2 $ topology topaction中的签名更改解决方案集。我们在假设$ \ left（m，g \右）$的假设下获得了此结果，$ n \ ge7 $和$ h_0 \ ne \ ne \ ne \ frac {n-2} {4 \ left（n-1 \ right）} \ fript（n-1 \ right）} \ text} {scal} _g $在$ m $中的所有点，$ \ cl scale as clate scale as cl scale}然后，我们在每个维度$ n \ ge3 $中提供反例，以显示这些假设的最佳性。
On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schrödinger equation $Δ_gu+h_0u=\left|u\right|^{2^*-2}u$, where $Δ_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^* :=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge7$ and $h_0\ne\frac{n-2}{4\left(n-1\right)}\text{Scal}_g$ at all points in $M$, where $\text{Scal}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge3$ showing the optimality of these assumptions.
