Let Ω be a bounded domain in Rn with n ≥ 3. In this paper we are concerned with the problem of finding u ∈ H01 (Ω) satisfying the nonlinear elliptic problemsΔu + |u|(n+2/n-2) + f(x) = 0  in Ω and u(x) = 0 on ∂Ω, andΔu + u + |u|(n+2/n-2) + f(x) = 0  in Ω and u(x) = 0 on ∂Ω, when of f ∈ L∞(Ω).

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.