Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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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.

Solutions for a class of fractional boundary value problem with mixed nonlinearities

Ziheng Zhang

Tianjin Polytechnic University

Abstract

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