Dual ground state solutions for the critical nonlinear Helmholtz equation

On a fourth‐order nonlinear Helmholtz equation

“…The associated function u was shown to be a strong solution of (1.4) lying in W 2,q (R N )∩C 1,α (R N ) for all q ∈ [p, ∞) and α ∈ (0, 1). Notice that existence results have been obtained also in the Sobolev-critical case p = 2N N −2 in [18]. The lower bound for p is related to the mapping properties of the resolvent type operator R k 2 , which in turn is linked with the Stein-Tomas Theorem (see Theorem 3.1).…”

Dual variational methods for a nonlinear Helmholtz system

“…Evéquoz further generalizes these results in [3]; for instance, it is shown that the dual variational techniques apply for any p ∈ (2, 2 * ) if Q satisfies suitable integrability conditions. In [10], Evéquoz and Yeşil prove the existence of a dual ground state in the critical case p = 2 * for N ≥ 4 and the non-existence for N = 3 where, again, Q is assumed to be the sum of a decaying and a periodic term. For continuous, nonnegative Q and 2(N +1) N −1 < p < 2 * , Evéquoz proves existence, concentration and multiplicity of ground states of the dual problem in the high-frequency limit λ ր ∞ in [5] based on a comparison of energies with a suitable limit problem.…”

“…for 1 ≤ p ≤ 2(N + 1) N + 3 (10) where g : R → C is a Schwartz function and dσ denotes the surface measure on the sphere S N −1 ⊆ R N . It is a consequence of the Stein-Tomas Theorem, see p. 375 and p. 414 in [19] for N ≥ 3 and N = 2, respectively.…”