This article is concerned with the existence and multiplicity of nontrival solutions for a fourth-order elliptic equation Δ 2 u - M ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) , in Ω , u = Δ u = 0 , on ∂ Ω by using the mountain pass theorem.

Combining the properties of the Green function with some point theorems, we consider the existence of nontrivial solutions for fractional equations with p-Laplacian operator D0+βϕp[D0+α(ptu′(t))]+f(t,u(t))=0, 0<t<1, au(0)-bp0u′(0)=0, and cu(1)+dp1u′(1)=0, D0+αptu′tt=0=0, where a,b,c,d are constants and p(·):[0,1]→(0,+∞) is continuous.

In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems $$\textstyle\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\Lambda(t)f(u(t))=0,\quad1 < q\leq2, t\in (0,1), \\ \alpha u(0)-\beta p(0)u'(0)=0,\qquad\gamma u(1)+\delta p(1)u'(1)=0, \end{cases} $$ where α, β, γ, δ are constants satisfying \(0\neq \vert\beta\gamma+\alpha\gamma\int_{0}^{1}\frac{1}{p(\tau)}\,d\tau +\alpha \delta\vert<+\infty\), \(p(\cdot)\) is positive and continuous on \([0,1]\). In addition, some existence results are given for the problem $$\textstyle\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\Lambda(t)f(u(t),\lambda)=0,\quad1 < q\leq2, t\in (0,1), \\ \alpha u(0)-\beta p(0)u'(0)=0,\qquad\gamma u(1)+\delta p(1)u'(1)=0, \end{cases} $$ where \(\lambda\geq0\) is a parameter. The proof is based on the fixed point theorems and the Leray-Schauder nonlinear alternative for single-valued maps.