论文标题
choquard方程阳性解决方案的先验估计，独特性和非分类性
A priori estimates, uniqueness and non-degeneracy of positive solutions of the Choquard equation
论文作者
论文摘要
我们考虑非本地Choquard方程$ - ΔU + u - （| \ cdot |^{ - α} * | U |^p）| U |^{p-2} u = 0 $ in $ \ mathbb {r}^d $。与接地状态相比，阳性解决方案构成了较大的解决方案，并且缺乏各种信息。在Ma-Zhao结果的参数范围内[Ma-Zhao，2010]在对称性上，我们证明了阳性溶液的先验估计值，从而推广了figueiredo-Lions-russbaum的经典方法[de figueiredo-lions-nussbaum，1982]对未结合的域和我们的非结合域和我们模型中的非结合域中。作为一个应用程序，当$ d \ in \ {3、4、5 \} $，$ p \ ge 2 $和$（α，p）$接近$（d-2，2）$时，我们显示了Choquard方程的正面解决方案的唯一性和非分类结果。
We consider the positive solutions for the nonlocal Choquard equation $- Δu + u - (|\cdot|^{-α} * |u|^p) |u|^{p-2} u = 0$ in $\mathbb{R}^d$. Compared with ground states, positive solutions form a larger class of solutions and lack variational information. Within the range of parameters of Ma-Zhao's result [Ma-Zhao, 2010] on symmetry, we prove a priori estimates for positive solutions, generalizing the classical method of De Figueiredo-Lions-Russbaum [De Figueiredo-Lions-Nussbaum, 1982] to the unbounded domain and the nonlocal nonlinearity in our model. As an application, we show uniqueness and non-degeneracy results for the positive solution of the Choquard equation when $d \in \{ 3, 4, 5\}$, $p \ge 2$ and $(α, p)$ close to $(d-2, 2)$.
