A multiplicity result for a semilinear Maxwell type equation

Abstract

In this paper we look for solutions of the equation $$ \delta d\text{\bf A}=f'(\langle\text{\mathbf{A}},\text{\bf A}\rangle) \text{\bf A}\quad \text{in }\mathbb R^{2k}, $$ where $\text{\bf A}$ is a $1$-differential form and $k\geq 2$. These solutions are critical points of a functional which is strongly indefinite because of the presence of the differential operator $\delta d$. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.