一个统一的零质量方程

中文摘要 AI 辅助

我们引入非局部零质量方程 \\[ - \Delta u + \Phi_u(|x|)\\, |u|^{p-2}\\, u = f(u) \quad \text{in } {\mathbb R}^N, \qquad \Phi_u(r) = a \int_r^\infty \rho^{-b}\\, h_u^{q-1}(\rho)\\, d\rho, \\] 其中 $h_u(\rho)$ 是 $|u|^p$ 在半径为 $\rho$ 的球内的质量。对于径向函数，该方程将散焦逆幂薛定谔方程、陈-西蒙斯-薛定谔方程和薛定谔-泊松-斯莱特方程作为特例包含在内。我们发展了相关的球质量勒贝格和索伯列夫空间，这些空间是一致凸的，并证明了在新的临界指数之上的尖锐紧致径向嵌入、Brézis-Lieb 型分裂以及 Pohožaev 恒等式。利用算子的尺度不变性，我们使用 Fadell-Rabinowitz 上同调指标构造了尺度化特征值问题的一个无界特征值序列，并在亚尺度、超尺度和临界区域获得了存在性和多重性结果。我们的主要结果是一个 Brézis-Nirenberg 型定理，通过一个新的尺度化链接定理证明，该定理对每个不是特征值的 $\lambda > 0$ 都给出一个非平凡的径向解。专门针对这三个模型，以统一的方式恢复了一些已知结果，并产生了新的结果，包括在维数 $N \ge 4$ 下薛定谔-泊松-斯莱特方程的 Brézis-Nirenberg 型结果。

英文摘要

We introduce the nonlocal zero-mass equation \[ - Δu + Φ_u(|x|)\, |u|^{p-2}\, u = f(u) \quad \text{in } {\mathbb R}^N, \qquad Φ_u(r) = a \int_r^\infty ρ^{-b}\, h_u^{q-1}(ρ)\, dρ, \] where $h_u(ρ)$ is the mass of $|u|^p$ in the ball of radius $ρ$. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every $λ> 0$ that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions $N \ge 4$.

发表机构

• Florida Institute of Technology（佛罗里达理工学院）

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