Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on R^{N+1}
Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on R^{N+1}
复制标题
R^{N 1} 上齐次椭圆方程部分定域准周期解的存在性
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Darío A. Valdebenito
中科院分区：
文献类型：
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作者：
P. Polácik;Darío A. Valdebenito
我们考虑方程 Δu+ uyy + f(u) = 0, (x, y) ∈ R × R, (1) 其中 N ≥ 2 并且 f 是满足 f(0) = 0 且 f ′(0) < 0 的光滑函数。我们证明，对于这种形式的方程 (1) 的合适非线性 f 拥有无数正解，这些正解在 y 中是准周期的，在 x 中是径向对称的，并且衰减为 |x| → y 上均匀一致。我们的方法基于中心流形和 KAM 型结果，并涉及对 y 独立解 u*(x)（R 上方程 Δu+ f(u) = 0 的基态）附近的 (1) 解进行分析。
We consider the equation ∆u+ uyy + f(u) = 0, (x, y) ∈ R × R, (1) where N ≥ 2 and f is a smooth function satisfying f(0) = 0 and f ′(0) < 0. We show that for suitable nonlinearities f of this form equation (1) possesses uncountably many positive solutions which are quasiperiodic in y, radially symmetric in x, and decaying as |x| → ∞ uniformly in y. Our method is based on center manifold and KAM-type results and involves analysis of solutions of (1) in a vicinity of a y-independent solution u∗(x)—a ground state of the equation ∆u+ f(u) = 0 on R .