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 $\Omega \subset {ℝ}^N$ (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.

In this paper, we consider the following boundary value problem $$\left\{\begin{array}{c}{D}_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }\left(D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)\right)=f(t,u\left(t\right)),t\in [0,T],\hfill \\ u\left(0\right)=u\left(T\right)=0\hfill \\ D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(0\right)\right)=D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(T\right)\right)=0,\hfill \end{array}\right.$$ where $0<\alpha \le 1$ and $f:[0,T]\times ℝ\to ℝ$ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

This paper is devoted to the existence of conformal metrics on $S^3$ with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron.

We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative kown up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on technique developed by W. Veech for Rauzy induction.We study...

In the paper, we deal with the equation of a rectangular thin plate with a simply supported boundary. The restoring force being an odd superlinear function of the vertical displacement, the existence of infinitely many nonzero time-periodic solutions is proved.

We consider the eigenvalue problem$$\begin{array}{c}-\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega \hfill \\ u=0\text{on}\partial \Omega ,\hfill \end{array}$$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda \&gt;0$ are eigenvalues.

We consider the eigenvalue problem $$\begin{array}{c}{\displaystyle -\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega u=0\text{on}\partial \Omega ,}\hfill \end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.