论文标题
部分可观测时空混沌系统的无模型预测
Bounded solutions for quasilinear modified Schrödinger equations
论文作者
论文摘要
在本文中，我们为Quasilinear椭圆问题\ [ - {\ rm div}（a（x，x，u）| \ nabla u |^{p-2} \ nabla u） + \ frac1p a_t（x，x，u）| g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$, $p>1$ and $V:\mathbb{R}^N\to\mathbb{R}$ suitable measurable positive function, which generalizes the modified Schrödinger equation.在这里，我们假设$ a：\ mathbb {r}^n \ times \ mathbb {r} \ rightarrow \ mathbb {r} $是$ \ mathcal {c}^{1} $ - carathéodory-carathéodory函数给定的carathéodory函数$ g：\ mathbb {r}^n \ times \ times \ mathbb {r} \ rightarrow \ rightarrow \ mathbb {r} $具有亚临界增长，并满足Ambrosetti-rabinowitz条件。 由于主要部分的系数也取决于解决方案本身，因此我们研究了合适的Banach空间中两个不同规范的相互作用，以获得“良好”的变异方法。因此，通过对有限集的近似参数，我们可以说明存在非平凡的弱界解决方案的存在。
In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$, $p>1$ and $V:\mathbb{R}^N\to\mathbb{R}$ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that $A:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a $\mathcal{C}^{1}$-Carathéodory function such that $A_t(x,t) = \frac{\partial A}{\partial t} (x,t)$ and a given Carathéodory function $g:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ has a subcritical growth and satisfies the Ambrosetti-Rabinowitz condition. Since the coefficient of the principal part depends also on the solution itself, we study the interaction of two different norms in a suitable Banach space so to obtain a "good" variational approach. Thus, by means of approximation arguments on bounded sets we can state the existence of a nontrivial weak bounded solution.
