Solutions to a nonlinear Maxwell equation with two competing nonlinearities in
Tom 69 / 2021
Streszczenie
We are interested in the nonlinear, time-harmonic Maxwell equation \nabla (\nabla \mathbf {E} ) + V(x) \mathbf {E} = h(x, \mathbf {E})\quad \ \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 \lt q \lt p \lt 6. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of a special form are in one-to-one correspondence with weak solutions to a Schrödinger equation with a singular potential. Using this equivalence result we show the existence of a least energy solution among cylindrically equivariant solutions to the Maxwell equation of a particular form, as well as to the Schrödinger equation.