Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation

We study layered solutions in a one-dimensional version of the scalar Ginzburg-Landau equation that involves a mixture of a second spatial derivative and a fractional half-derivative, together with a periodically modulated nonlinearity. This equation appears as the Euler-Lagrange equation of a suitably renormalized fractional Ginzburg-Landau energy with a double-well potential that is multiplied by a 1-periodically varying nonnegative factor $g(x)$ with ${\int }_0^1\frac{1}{g(x)}dx<\mathrm{\infty }.$ A priori this energy is not bounded below due to the presence of a nonlocal term in the energy. Nevertheless, through a careful analysis of a minimizing sequence we prove existence of global energy minimizers that connect the two wells at infinity. These minimizers are shown to be the classical solutions of the associated nonlocal Ginzburg-Landau type equation.