### A multiplicity result for a semilinear Maxwell type equation

DOI: http://dx.doi.org/10.12775/TMNA.2008.005

#### Abstract

In this paper we look for solutions of the equation

$$

\delta d\text{\bf A}=f'(\langle\text{\bf 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.

$$

\delta d\text{\bf A}=f'(\langle\text{\bf 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.

#### Keywords

Semilinear Maxwell equations; strongly indefinite functional; Strong convexity

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