Research on Sturm–Liouville boundary value problems of fractional p-Laplacian equation

• Main Authors: Tingting Xue, Fanliang Kong, Long Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Advances in Difference Equations
• Subjects: Fractional p-Laplacian equation; Sturm–Liouville boundary value conditions; Multiplicity of solutions; Variational methods; Critical point theory
• Online Access: https://doi.org/10.1186/s13662-021-03339-3

Abstract In this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e.  t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ where D t α 0 C ${}_{0}^{C}D_{t}^{\alpha }$ , D T α t ${}_{t}D_{T}^{\alpha }$ are the left Caputo and right Riemann–Liouville fractional derivatives of order α ∈ ( 1 2 , 1 ] $\alpha \in ( {\frac{1}{2},1} ]$ , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.