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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.
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참고문헌 (21)
참고문헌 신청
R. Agarwal / 2009 / Boundary value problems for fractional differential equations / Georgian Math. J 16(3):401~411
O. Agrawal / 2004 / Fractional Derivatives and Their Application: Nonlinear dynamics / Springer-Verlag
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Anal 14(4):349~381
Z. B. Bai / 2005 / Positive solutions for boundary value problem of nonlinear fractional differential equation / J. Math. Anal. Appl 311(2):495~505
D. G. de Figueiredo / 2003 / Local superlinearity and sublinearity for indefinite semilinear elliptic problems / J. Functional Anal 199(2):452~467
O. Agrawal / 2004 / Fractional Derivatives and Their Application: Nonlinear dynamics / Springer-Verlag
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Anal 14(4):349~381
Z. B. Bai / 2005 / Positive solutions for boundary value problem of nonlinear fractional differential equation / J. Math. Anal. Appl 311(2):495~505
D. G. de Figueiredo / 2003 / Local superlinearity and sublinearity for indefinite semilinear elliptic problems / J. Functional Anal 199(2):452~467
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