Auxiliary principle for generalized nonlinear variational-like inequalities.
We prove existence of small amplitude, 2p/v-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.
We deal with a bifurcation result for the Dirichlet problem ⎧ a.e. in Ω, ⎨ ⎩. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number such that for every the above problem admits a nonzero weak solution in satisfying .