论文标题
部分可观测时空混沌系统的无模型预测
Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type
论文作者
论文摘要
本文的目的是调查梯度类型的耦合的准椭圆形系统的存在\ [（P）\ qquad \ left \ left \ {\ stray {array} {ll} {ll} {ll} - {\ rm div}（\ rm div}（a（x，x，x，x，U） + \ frac {1} {p_1} a_u（x，u）\ vert \ nabla u \ vert^{p_1} = g_u（x，x，u，v）＆\ hbox {in $ω$，} $，} \ \ \ \ \ \ \ \ \ [5pt] - {5pt] - -2} \ nabla v） +\ frac {1} {p_2} b_v（x，v）\ vert \ nabla v \ vert^{p_2} = g_v \ left（x，x，x，u，u，v \ right） $ \partialΩ$，} \ end {array} \ right。 \]其中$ω\ subset \ mathbb {r}^n $是一个开放式界面，$ p_1 $，$ p_2> 1 $和$ a（x，u）$，$ b（x，v）$是$ \ mathcal {c}^1 $-Carathéodory函数，$-Carathéodory函数$-Carathéodorywitter $ω\ times \ times \ times \ times \ times \ times \ bb bb bb^r} $ a_u（x，u）$，分别$ b_v（x，v）$，而$ g_u（x，u，v）$，$ g_v（x，x，u，u，v）$均在$ \ times \ times \ times \ times \ mathbb {r} \ times \ times \ times \ mathbb {r} $ parte parte $ nforivives a parctecarathéodory映射上carathéodory映射。我们证明，即使系数使变异方法变得更加困难，在适当的假设功能$ \ cal {j} $下，与问题$ $（p）$有关，也承认了“右” banach Space $ x $中的至少一个关键点。此外，如果$ \ cal {j} $均匀，则$（p）$具有无限的许多弱界解决方案。利用两种不同规范之间相互作用的证明是基于Cerami-Palais-Smale条件的弱版本，Banach Space $ X $的“良好”分解以及Ambrosetti-Rabinowitz Mountain的合适概括。
The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x, u)\vert\nabla u\vert^{p_1} = G_u(x, u, v) &\hbox{ in $Ω$,}\\[5pt] - {\rm div} (B(x, v)\vert\nabla v\vert^{p_2 -2} \nabla v) +\frac{1}{p_2}B_v(x, v)\vert\nabla v\vert^{p_2} = G_v\left(x, u, v\right) &\hbox{ in $Ω$,}\\[5pt] u = v = 0 &\hbox{ on $\partialΩ$,} \end{array} \right. \] where $Ω\subset \mathbb{R}^N$ is an open bounded domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are $\mathcal{C}^1$-Carathéodory functions on $Ω\times \mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$, $G_v(x,u,v)$ are given Carathéodory maps defined on $Ω\times \mathbb{R}\times \mathbb{R}$ which are partial derivatives of a function $G(x,u,v)$. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional $\cal{J}$, related to problem $(P)$, admits at least one critical point in the ''right'' Banach space $X$. Moreover, if $\cal{J}$ is even, then $(P)$ has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ''good'' decomposition of the Banach space $X$ and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.
