Critical p-biharmonic problems and applications to Hamiltonian systems

Abstract: We study fourth-order quasilinear elliptic problems that involve the p-biharmonic operator and Navier boundary conditions. The nonlinear term grows at the critical Sobolev rate. Starting from a Hamiltonian system of two second-order equations, we use an inversion step to reduce it to a single p-biharmonic equation with a lower-order perturbation. We handle both non-resonant and resonant cases and show that the problem admits non-trivial solutions when the forcing term and the superscaled perturbation are small enough. The proof combines concentration-compactness with an abstract critical point method based on the cohomological index. Our theorems cover both homogeneous and nonhomogeneous settings and extend Tarantello's classical results for the Laplacian, improving earlier work on p-biharmonic equations (including the case p = 2) and on critical Hamiltonian systems.