论文标题
归一化解决方案，以降低临界choquard方程，并具有局部扰动
Normalized solutions to lower critical Choquard equation with a local perturbation
论文作者
论文摘要
在本文中，我们研究了使用局部扰动\ begin {equation*} \ begin {case}的归一化解决方案的存在和不存在的解决方案的存在和不存在-Δu+λu=γ（I_α\ AST | U |^{\ frac {n+α} {n}}} {n}}）| U |^{\ frac {n+α} {n} {n} {n} -2} -2} u++++++++| \int_{\mathbb{R}^N}|u|^2dx=c^2, \end{cases} \end{equation*} where $γ, μ, c>0$, $2<q\leq 2+\frac{4}{N}$, and $λ\in\mathbb{R}$ is an unknown parameter that appears as a拉格朗日乘数。有关该方程式的本文的结果回答了Yao，Chen，RǎDulescu和Sun提出的一些问题[Siam J. Math。肛门，54（3）（2022），3696-3723]。此外，根据获得的结果，我们研究了对非自主choquard方程\ begin {equation*} \ begin {case}}-ΔU+λu=（I_α\ ast）的多样化。 [h（εx）| u |^{\ frac {n+α} {n}}]）h（εx）| u |^{\ frac {n+α} {n} {n} {n} { \ int _ {\ mathbb {r}^n} | u | u |^2dx = c^2，\ end {cases} \ end {equation*}其中$ε> 0 $，$ 2 <q <q <q <q <q <2+ \ frac {4} {n} {n} {n} {n} $，$ h $是一个积极且连续的函数。事实证明，当$ε$足够小时，归一化解决方案的数量至少是$ h $的全球最大点的数量。
In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -Δu+λu=γ(I_α\ast|u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u+μ|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=c^2, \end{cases} \end{equation*} where $γ, μ, c>0$, $2<q\leq 2+\frac{4}{N}$, and $λ\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. The results of this paper about this equation answer some questions proposed by Yao, Chen, Rǎdulescu and Sun [Siam J. Math. Anal., 54(3) (2022), 3696-3723]. Moreover, based on the results obtained, we study the multiplicity of normalized solutions to the non-autonomous Choquard equation \begin{equation*} \begin{cases} -Δu+λu=(I_α\ast [h(εx)|u|^{\frac{N+α}{N}}])h(εx)|u|^{\frac{N+α}{N}-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=c^2, \end{cases} \end{equation*} where $ε>0$, $2<q<2+\frac{4}{N}$, and $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enough.
