The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2020-07-26 , DOI: 10.1007/s12220-020-00483-2 Qiongfen Zhang , Canlin Gan , Ting Xiao , Zhen Jia
The existence of nontrivial solutions for the following kind of Klein–Gordon–Maxwell system
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u-(2\omega +\phi )\phi u=f(x,u),&{}\quad x\in {{\mathbb {R}}}^{3},\\ \Delta \phi =(\omega +\phi )u^{2},&{}\quad x\in {{\mathbb {R}}}^{3}, \end{array}\right. \end{aligned}$$is investigated, where \(\omega >0\) is a constant, \(V\in C({{\mathbb {R}}}^{3},{{\mathbb {R}}})\) is either periodic or coercive and is allowed to be sign-changing, \(f\in C({{\mathbb {R}}}^{3}\times {{\mathbb {R}}},{{\mathbb {R}}})\) and f is subcritical and local super-linear. Using local super-quadratic conditions and other suitable assumptions on the nonlinearity f(x, u) and the potential V(x), the existence of nontrivial solutions for the above system is established. The obtained results in this paper improve the related ones in the literature.
中文翻译：
具有局部超二次条件的Klein-Gordon-Maxwell系统非平凡解的一些结果
下列Klein-Gordon-Maxwell系统的非平凡解的存在
$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta u + V（x）u-（2 \ omega + \ phi）\ phi u = f（x，u），＆ {} \ quad x \ in {{\ mathbb {R}}} ^ {3}，\\ \ Delta \ phi =（\ omega + \ phi）u ^ {2}，＆{} \ quad x \ in { {\ mathbb {R}}} ^ {3}，\ end {array} \ right。\ end {aligned} $$进行了研究，其中\（\欧米加> 0 \）是一个常数，\（V \在C（{{\ mathbb {R}}} ^ {3}，{{\ mathbb {R}}}）\）是周期性或强制性的，并且可以进行符号转换，\（f \ in C（{{\ mathbb {R}}} ^ {3} \ times {{\ mathbb {R}}}，{{\ mathbb { R}}}）\）和f是亚临界且局部超线性的。使用局部超二次条件和非线性f（x， u）和势V（x）的其他适当假设，可以建立上述系统的非平凡解的存在。本文获得的结果改进了文献中的相关结果。
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