Three Solutions for Impulsive Fractional Boundary Value Problems with p-Laplacian

The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a $p$-Laplacian and Dirichlet conditions \begin{equation} \begin{cases} D^{\alpha}_{T^-}\Phi_p(^cD^{\alpha}_{0^+}u(t))+|u(t)|^{p-2}u(t)= \lambda f(t,u(t)), t\neq t_j,\ \ t\in(0,T),\\ \Delta(D^{\alpha-1}_{T^-}\Phi_p(^cD^{\alpha}_{0^+}u))(t_j)=I_j(u(t_j)),\\ u(0)=u(T)=0, \end{cases} \end{equation} where $\alpha\in(\frac{1}{p}, 1]$ and $p gt; 1$. Their approach is based on variational methods. The main result is illustrated with an example.