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We provide extensions of the dual variational method for the nonlinear Helmholtz equation from Evéquoz and Weth. In particular we prove the existence of dual ground state solutions in the Sobolev critical case, extend the dual method beyond the standard Stein Tomas and Kenig Ruiz Sogge range and generalize the method for sign changing nonlinearities.
We establish weighted Lp-Fourier extension estimates for O(N−k)×O(k)-invariant functions defined on the unit sphere SN−1, allowing for exponents p below the Stein–Tomas critical exponent 2(N+1)/N−1. Moreover, in the more general setting of an arbitrary closed subgroup G⊂O(N) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation −Δu−u = Q(x)|u|p−2u,u∈W2,p(RN), where Q is a nonnegative bounded and G-invariant weight function.
We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form − delta u−k2u=Q(x)/u/p−2u, uEW2, p(RN) – delta u − k2u=Q(x)/u/p−2u, uEW2, p(RN) with k>0, k>0, N≥3N≥3, pE[2(N+1)N − 1, 2NN − 2)pE[2(N+1)N − 1, 2NN−2) and QEL ∞ (RN)QEL ∞ (RN). Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.