Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$

We are interested in the nonlinear, time-harmonic Maxwell equation $$\nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3$$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$h(x, \alpha w) = f(x, \alpha) w - g(x, \alpha) w$$ for $w \in \mathbb{R}^3$, $|w|=1$ and $\alpha \in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schr\"odinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schr\"odinger equation.