On the periodic and asymptotically periodic nonlinear Helmholtz equation

In the first part of this paper, the existence of infinitely many Lp-standing wave solutions for the nonlinear Helmholtz equation −Δu−λu=Q(x)∣u∣p−2u  in  RN is proven for N≥2 and λ>0, under the assumption that Q be a nonnegative, periodic and bounded function and the exponent p lies in the Helmholtz subcritical range. In a second part, the existence of a nontrivial solution is shown in the case where the coefficient Q is only asymptotically periodic.