Abstract
In this paper, the authors investigate the following fractional Kirchhoff boundary value problem: \t\t\t{(a+b∫0T(0Dtαu)2dt)tDTα(0Dtαu)+λV(t)u=f(t,u),t∈[0,T],u(0)=u(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} ( {a+b\\int_{0}^{T} {({ }_{0}D_{t}^{\\alpha }u)^{2}\\,dt} } ){ }_{t}D_{T}^{\\alpha }({}_{0}D_{t}^{\\alpha }u)+\\lambda V(t)u=f(t,u),\\quad t\\in [0,T], \\\\ u(0)=u(1)=0, \\end{cases} $$\\end{document} where the parameter lambda >0 and constants a, b>0. By applying the mountain pass theorem and the linking theorem, some existence results on the above fractional boundary value problem are obtained. It should be pointed out that the potential V may be sign-changing.
Highlights
In recent ten years, the fractional differential equations have been extensively studied by many researchers due to their various applications in science and engineering [1–5]
We show that u+v ∗≤ u ∗+ v ∗
(i) We show that ∃ρλ(> rλ) and bλ > 0 such that Iλ(u) < 0 as u ∈ Xλ,1 ⊕ Re0 with u λ = ρλ and b < bλ
Summary
The fractional differential equations have been extensively studied by many researchers due to their various applications in science and engineering [1–5]. Due to the appearance of both left and right fractional derivatives in equations, the fixed point theory is generally not suitable for the study of the existence of solution to such problems. The readers can refer to [19–30] and the references therein Motivated by these works mentioned above and combining the fractional equations with left-right derivatives and the Kirchhoff equations, the authors will investigate the following fractional Kirchhoff boundary value problem (BVP for short):. In order to study the Kirchhoff-type boundary value problem with sign-changing potential V , we need the following work frame. Lemma 2.8 (Mountain pass theorem [32]) Let E be a Banach space, I ∈ C1(E, R) satisfies that max{I(0), I(e)} ≤ μ < η ≤ inf u =ρ I(u) for some μ < η, ρ > 0, and e ∈ E with e > ρ.
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