Bull. Korean Math. Soc. 2016; 53(5): 1585-1596
Online first article September 21, 2016 Printed September 30, 2016
https://doi.org/10.4134/BKMS.b150857
Copyright © The Korean Mathematical Society.
Ziheng Zhang
Tianjin Polytechnic University
In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem $$ \left\{ \begin{array}{ll} _tD^{\alpha}_T(_0D^{\alpha}_t u(t))=\nabla W(t,u(t)),\quad t\in [0,T],\\[0.1cm] u(0)=u(T)=0, \end{array} \right. \leqno(\mbox{FBVP}) $$ where $\alpha\in (1/2,1)$, $u\in \R^n$, $W\in C^1([0,T]\times\R^n,\R)$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, when the nonlinearity $W(t,u)$ involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.
Keywords: fractional boundary value problems, critical point, variational methods, mountain pass theorem, minimizing method
MSC numbers: 34C37, 35A15, 35B38
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