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Abstract  In this paper, we study the mixed dispersion fourth‐order nonlinear Helmholtz equationΔ  2   u  −  β  Δ  u  +  α  u  =  Γ    |  u  |    p  −  2    u    inR  N   ,    for positive, bounded and    Z  N   ‐periodic functions   Γ   in the following three cases:(  a  )α  <  0  ,  β  ∈  R    or(  b  )α  >  0  ,  β  <  −  2   αor(  c  )α  =  0  ,  β  <  0  .    Using the dual method of Evéquoz and Weth, we find solutions to this equation and establish some of their qualitative properties.

On a fourth‐order nonlinear Helmholtz equation