论文标题
存在电磁作用的系统的解决方案
Existence of solutions for systems arising in electromagnetism
论文作者
论文摘要
在本文中，我们研究以下$ p（x）$ - 卷曲系统：\ begin {eqnarray*} \ begin {cases} \ nabla \ times（| \ nabla \ nabla \ times \ times \ mathbf {u} \ MathBf {U}）+A（X）| \ MathBf {U} |^{P（X）-2} \ MathBf {U} =λf（X，X，\ MathBf {U}）+μg（X，X，X，X，\ Mathbf {u}），\ Quad \ quad \ quad \ nabla \ albaf \ \ \ \ \ \ \ cd \ cdot \ cdot \ cd \ cdot \ cd \ cdot \ cd \ cdot \ cdot \ cdot; \ mbox {in}ω，\\ | \ nabla \ times \ MathBf { } \partialΩ，\ end {case} \ end {eqnarray*}其中$ω\ subset \ subset \ mathbb {r}^{3} $是一个仅通过$ c^{1,1} $的连接域，由$ c^{1,1} $ - 边界，由$ \ poartial op p：\ $ p：\ yline of（$ pline）表示，\ yline of（\ yline）\ inline \ inline \ inline \ inline cy（函数，$ a \ in l^\ infty（ω）$，$ f，g：ω\ times \ mathbb {r}^{3}^{3} \ to \ mathbb {r}^{3} $是carathéodory函数，$λ，μ$是两个参数。使用基于喷泉定理和双喷泉定理的变分参数，我们为解决方案建立了一些存在和不存在的结果。我们的主要结果推广了Xiang等人的结果。 （J.Math。Anal。Appl。，2017年），Bahrouni和Depovš（复杂的Var. Elliptic Equ。，2018）和GE and Lu（Mediterr。J.Math。，2019）。
In this paper, we study the following $p(x)$-curl systems: \begin{eqnarray*} \begin{cases} \nabla\times(|\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u})+a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u}=λf(x,\mathbf{u})+μg(x,\mathbf{u}),\quad\nabla\cdot \mathbf{u}=0,\; \mbox{ in } Ω, \\ |\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u}\times \mathbf{n}=0,\quad \mathbf{u}\cdot \mathbf{n}=0, \mbox{ on } \partialΩ, \end{cases} \end{eqnarray*} where $Ω\subset \mathbb{R}^{3}$ is a bounded simply connected domain with a $C^{1,1}$-boundary, denoted by $\partial Ω$, $p:\overlineΩ\to (1, +\infty)$ is a continuous function, $a \in L^\infty(Ω)$, $f,g : Ω\times \mathbb{R}^{3}\to \mathbb{R}^{3}$ are Carathéodory functions, and $λ,μ$ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang et al. (J. Math. Anal. Appl., 2017), Bahrouni and Repovš (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).
