### A minimax inequality for a class of functionals and applications to the existence of solutions for two-point boundary-value problems.

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Using Ricceri's variational principle, we establish the existence of infinitely many solutions for a class of two-point boundary value Kirchhoff-type systems.

Using a recent critical point theorem due to Bonanno, the existence of a non-trivial solution for a class of systems of n fourth-order partial differential equations with Navier boundary conditions is established.

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧${\Delta \left(\right|\Delta u|}^{p-2}{\Delta u\left)-div\right(\left|\nabla u\right|}^{p-2}\nabla u)=\lambda f(x,u)+\mu g(x,u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $\Omega \subset {\mathbb{R}}^{N}$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧$\Delta \left(\right|\Delta {u}_{i}{|}^{p-2}\Delta {u}_{i})=\lambda {F}_{{u}_{i}}(x,u\u2081,\dots ,u\u2099)$ in Ω, ⎨ ⎩${u}_{i}=\Delta {u}_{i}=0$ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.

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