归一化解的 Nehari-Pankov类型对质量超临界不定变分问题
Normalized solutions of Nehari-Pankov type to mass-supercritical indefinite variational problems
AI总结：
本文研究了形如A u = λu + I'(u)的非线性方程，提出了一种不依赖质量亚临界假设的新方法，用于检测具有指定范数的解，并在各类非线性Schrödinger方程和生物谐振方程中应用，证明了存在多个具有指定小质量的解。
摘要：
我们考虑形如A u = λu + I'(u)的抽象非线性方程，其中A是具有紧谱的自共轭算子，λ∈R是参数，I'(u)是变分性质的超线性项。在这一抽象设定中，我们开发了一种新方法来检测H中的指定范数解，而无需任何质量亚临界假设。我们随后考虑了该方法的各种应用。首先，在一般假设下，包括完整的质量超临界参数区域，我们得到了在紧致图G上的一类非线性Schrödinger方程的解，具有任意大的质量，从而改进了之前仅覆盖小质量的结果。此外，我们还为在2- тор上的一类生物谐振方程推导出类似结果。对于更广泛的二阶及更高阶方程在有界域中带有Dirichlet边界条件的更大类中，我们还证明了存在多个具有指定小质量的解。我们得到的解被检测为对应λ依赖的行动功能的Nehari-Pankov类型基态，其中λ在足够大的A的特征值之间变化。该抽象框架中的关键新观察是，这些λ依赖的解族的H范数形成连通集，尽管解族本身可能不连通。为了在特定情况下估计这些连通集的大小，我们使用Weyl类型的谱间隙长度估计、特征值的变分刻画、相关特征函数的界以及解析数论中的界。
英文摘要：
We consider abstract nonlinear equations of the form $A u = λu + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $λ\in \mathbb{R}$ is a parameter, and $u \mapsto I'(u)$ is a superlinear term of variational nature. In this abstract setting, we develop a new approach to detect prescribed norm solutions in $H$ which does not rely on any mass-subcriticality assumptions. We then consider various applications of this approach. First, we obtain, under general assumptions including the full mass-supercritical parameter regime, the existence of (infinitely many) solutions to a class of nonlinear Schrödinger equations on a compact graph $\mathcal{G}$ with prescribed arbitrarily large mass, thereby improving previous results which only cover small masses. Moreover, we derive a similar result for a biharmonic Schrödinger equation in the $2$-torus. For a larger class of second order and higher order equations in a bounded domain with Dirichlet boundary conditions, we also show the existence of multiple solutions with prescribed small mass. The solutions we obtain are detected as ground states of Nehari-Pankov type for the associated $λ$-dependent action functional, where $λ$ varies in a spectral gap between sufficiently large eigenvalues of $A$. The key new observation in this abstract framework is the fact that the $H$-norms of these $λ$-dependent solution families form connected sets even though the solution families themselves may be disconnected. To estimate the size of these connected sets in specific settings, we use Weyl type estimates for the length of spectral gaps, variational characterizations of eigenvalues, bounds for associated eigenfunctions and a bound from analytic number theory.
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