Existence of solutions to Hamilton systems for the curl–curl operator with subcritical nonlinearities

This paper is concerned with the following Hamilton system for the curl–curl operator \(\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times (\nabla \times U_1)=f_2(x,U_2) \quad & \hbox {in}\;\Omega , \\ \nabla \times (\nabla \times U_2)=f_1(x,U_1) \quad & \hbox {in}\;\Omega , \\ \nu \times U_1=\nu \times U_2=0 \ \ & \hbox {on}\;\partial \Omega \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\nabla }\times (\mathrm{\nabla }\times U_1)=f_2(x,U_2)\hfill & \text{in}\mathrm{\Omega },\hfill \\ \mathrm{\nabla }\times (\mathrm{\nabla }\times U_2)=f_1(x,U_1)\hfill & \text{in}\mathrm{\Omega },\hfill \\ \nu \times U_1=\nu \times U_2=0\hfill & \text{on}\partial \mathrm{\Omega }\hfill \end{array}\right)\end{array}$$ in a simply connected bounded Lipschitz domain \(\Omega \subset {\mathbb {R}}^3\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with connected boundary, where \(\nabla \times \) $\mathrm{\nabla }\times$ denotes the curl operator in \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ and \(\nu :\partial \Omega \rightarrow {\mathbb {R}}^3\) $\nu :\partial \mathrm{\Omega }\to {\mathbb{R}}^3$ is the exterior normal. By using some variational approaches inspired by Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009) and Bartsch and Mederski (Arch Ration Mech Anal 215(1):283–306, 2015), we show that there exists a ground state solution for the above system if \(f_1\) $f_1$ and \(f_2\) $f_2$ are both subcritical and satisfy some other growth conditions and convexity conditions. Furthermore, if the nonlinearities are both even, we establish the existence of infinitely many solutions. Finally, we prove the existence of two types of cylindrically symmetric solutions under some symmetry conditions on the domain and the nonlinearities.