Further results on quasiperiodic partially localized solutions of homogeneous elliptic equations on RN+1
Further results on quasiperiodic partially localized solutions of homogeneous elliptic equations on RN+1
复制标题
RN 1 上齐次椭圆方程准周期部分局部解的进一步结果
DOI：
10.1016/j.jfa.2022.109457
复制
发表时间：
2022
影响因子：
1.7
通讯作者：
Valdebenito, Darío A.
中科院分区：
文献类型：
--
作者：
Poláčik, Peter;Valdebenito, Darío A.
研究椭圆方程(1)Δ x u+ u y y+ f (u)= 0,(x, y)∈R nx R，其中N≥2且f是c1函数满足f(0)= 0且f '(0)< 0的正局部解。我们所说的部分局部解是指u （x, y）在y中随| x|→∞一致衰减到零的解。我们主要关注的是y中准周期的正部分局部解的存在性。我们在前面的工作中证明了这样的解可以存在于上述形式的方程中：我们证明了非线性f可以设计成方程(1)具有2个频率的正部分局部拟周期解。我们在本文中的主要贡献有两个方面。首先，我们改进了前面的结果，证明了对于某些非线性f，存在频率为n≥2的正的部分局部拟周期解。其次，我们给出了方程(1)存在这样的准周期解的一个有形的充分条件，该充分条件可能在f稍加扰动后存在。例如，当n= 2时，这个条件适用于一些组合幂非线性函数f (u)= u p+ λ u q−u，它们具有合适的指数p> q> 1和系数λ> 0。
We study positive partially localized solutions of the elliptic equation (1) Δ x u+ u y y+ f (u)= 0,(x, y)∈ R N× R, where N≥ 2 and f is a C 1 function satisfying f (0)= 0 and f′(0)< 0. By partially localized solutions we mean solutions u (x, y) which decay to zero as| x|→∞ uniformly in y. Our main concern is the existence of positive partially localized solutions which are quasiperiodic in y. The fact that such solutions can exist in equations of the above form was demonstrated in our earlier work: we proved that the nonlinearity f can be designed in such a way that equation (1) possesses positive partially localized quasiperiodic solutions with 2 frequencies. Our main contributions in the present paper are twofold. First, we improve the previous result by showing that positive partially localized quasiperiodic solutions with any prescribed number n≥ 2 of frequencies exist for some nonlinearities f. Second, we give a tangible sufficient condition on f which guarantees that equation (1) has such quasiperiodic solutions, possibly after f is perturbed slightly. The condition, with n= 2, applies, for example, to some combined-powers nonlinearities f (u)= u p+ λ u q− u with suitable exponents p> q> 1 and coefficient λ> 0.