发表机构
School of Mathematics East China University of Science and Technology; Department of Mathematics Faculty of Science and Technology Keio University（华东理工大学数学学院; 庆应义塾大学理工学部数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带周期势与局域非线性的L²超临界非线性薛定谔方程的归一化解，结合单调性技巧等方法证明小μ>0时解存在，f在原点附近具质量超临界增长时所有μ>0时解存在。
AI 中文摘要
本文研究如下带周期势的L²超临界非线性薛定谔方程的归一化解的存在性：在ℝⁿ中，-Δu + V(x)u + λu = χ_Ω(x)f(u)；在ℝⁿ中，u>0；在ℝⁿ中，∫|u|²dx = μ。其中N≥1，μ>0为给定值，λ∈ℝ为拉格朗日乘子，V∈C(ℝⁿ)对x₁,…,x_N是1周期的，f∈C¹(ℝ)在无穷远处具一般的质量超临界增长，Ω⊂ℝⁿ是具光滑边界∂Ω的非空有界开集，χ_Ω是Ω的特征函数。我们证明对所有足够小的μ>0，归一化解存在；若f在原点附近还具质量超临界增长，则存在结果可扩展至所有μ>0。该结果通过结合单调性技巧、带约束泛函莫尔斯指数信息的极小极大原理及爆破分析得到。
英文摘要
In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential \begin{equation*} \begin{dcases} -Δu +V (x)u + λu=χ_{Ω}(x)f(u)\quad \text{in } \mathbb{R}^N, u>0 \quad \text{in} \ \mathbb{R}^N, \int_{\mathbb{R}^N}\abs{u}^2\, dx =μ. \end{dcases} \end{equation*} Here $N \geq 1$, $μ>0$ is prescribed, $λ\in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $Ω\subset \mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\partial Ω$ or the whole space \(\mathbb{R}^N\), and $χ_{Ω}$ is the characteristic function of $Ω$. In both cases, we prove the existence of normalized solutions of mountain pass type when $μ>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N<p<2^*$ at infinity. Moreover, if $Ω$ is bounded, $1 \leq N \leq 4$ and $f$ has $L^2$-supercritical growth near the origin, then the existence of normalized solutions is obtained for all $μ>0$. On the other hand, when $Ω=\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $μ>0$ is small. When $Ω$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $Ω= \mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.
Comments53 pages
