论文标题
至少质量关键问题的归一化解决方案：奇异的多谐和方程和相关的卷曲 - 卷曲问题
Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems
论文作者
论文摘要
我们对问题的存在感兴趣\ begin {qore*} \ begin {cases}（-Δ）^m u + + \fracμ{| y |^{2m}} u + +λu= g（u），\ quad x =（y quad x =（y，z） \ Mathbb {r}^{n-k}，\\ \\ int _ {\ Mathbb {r}^n} | u |^2 \，dx =ρ> 0，\ End {cases} \ end {cases} \ end {equination {equination*}至少在所谓的质量关键的至少质量关键的范围内。我们利用最近引入的变异技术，涉及$ l^2 $ -Ball的最小化。此外，我们还找到了相关卷曲问题的解决方案\ begin {equation*} \ begin {case} \ nabla \ times \ nabla \ times \ times \ mathbf {u}+λ\ mathbf {u} = f（\ mathbf {u}），\ quad x \ in \ in \ mathbb {r}^n，\\ \ int _ {\ mathbb {r}^n} | \ mathbf {u} |^2 \，dx =ρ， \ end {cases} \ end {equation*}，它来自麦克斯韦方程系统，在非线性光学中非常重要。
We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*} \begin{cases} \nabla\times\nabla\times\mathbf{U}+λ\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=ρ, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.
