Boundary value problem with fractional \(p\)-Laplacian operator. (English) Zbl 1337.26019
Summary: The aim of this paper is to obtain the existence of solution for the fractional \(p\)-Laplacian Dirichlet problem with mixed derivatives
\[
\begin{aligned} _tD_T^\alpha(|_0D_t^\alpha u(t)|^{p-2}_0D_t^\alpha u(t)) = f(t,u(t)), \quad t\in [0,T],\\ u(0) = u(T) = 0,\end{aligned}
\]
where \(\frac{1}{p} <\alpha < 1\), \(1 < p < \infty\) and \(f : [0,T] \times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function which satisfies some growth conditions. We obtain the existence of nontrivial solutions by using the direct method in variational methods and mountain pass theorem.
MSC:
26A33 | Fractional derivatives and integrals |
58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
65L10 | Numerical solution of boundary value problems involving ordinary differential equations |