Infinitely many cylindrically symmetric solutions of nonlinear Maxwell equations with concave and convex nonlinearities

In this paper, we will study the nonlinear Maxwell equations del x (mu(x)(-1)del x E) - omega(2)epsilon(x)E - sigma(x)E = xi(1)Q(x)vertical bar E vertical bar Eq-2 + xi P-2(x)vertical bar E vertical bar Ep-2 in R-3, where epsilon(x), mu(x) are the permittivity and magnetic permeability of the material, respectively, and sigma(x) is the conductivity of the media. 1 < q < p/p-1 < 2 < p <= 6, Q(x),P(x) is an element of C(R-3 ,R+), xi(1), xi(2) are two real numbers. When mu(x) mu is a constant, and (2 vertical bar mu vertical bar(P-q)vertical bar xi(1)vertical bar(p-2)vertical bar xi(2)vertical bar(2-q) <= eta(p,q) for some eta(p,q) > 0, using variational methods, we prove the existence of infinitely many large and small energy cylindrical symmetry solutions which have free divergence.

Automated summary

This paper studies the solutions of the nonlinear Maxwell equations under certain conditions and proves the existence of infinitely many cylindrical symmetry solutions that satisfy the conditions.