论文标题
$ \ mathbb {r}^{2} $中指数增长的choquard方程的归一化解决方案
Normalized solutions for a Choquard equation with exponential growth in $\mathbb{R}^{2}$
论文作者
论文摘要
在本文中，我们研究了以下非线性choquard方程的归一化解决方案，并具有指数增长\ begin {align*} \ left \ {\ begin {aligned}＆-ΔU+λu=（i_α\ ast f（u））f（u），\ quad \ quad \ quad \ hbox {in} \ mathbb {r} ＆\ int _ {\ mathbb {r}^{2}} | u |^{2} dx = a^{2}，\ end {aligned} \ right。 \ end {align*}其中规定了$ a> 0 $，$λ\ in \ mathbb {r} $，$α\ in（0,2）$，$i_α$表示riesz的潜力，$ \ ast $指示卷积运算符，函数$ f（t）$ f（t）$ f（t）$ f（t）$ \ mathbb} $ f（t）= \ int^{t} _ {0} f（τ）dτ$。使用Pohozaev歧管和变异方法，我们确定了上述问题的归一化解决方案。
In this paper, we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} &-Δu+λu=(I_α\ast F(u))f(u), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$ is prescribed, $λ\in \mathbb{R}$, $α\in(0,2)$, $I_α$ denotes the Riesz potential, $\ast$ indicates the convolution operator, the function $f(t)$ has exponential growth in $\mathbb{R}^{2}$ and $F(t)=\int^{t}_{0}f(τ)dτ$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.
