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In this work, we investigate the following fractional boundary value problems \begin{eqnarray*} \left\{\begin{array}{ll} _{t}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}(u(t))|^{p-2} {_0 D}_{t}^{\alpha}u(t)\right)\\ =\nabla W(t,u(t))+\lambda g(t) |u(t)|^{q-2}u(t),\;t\in (0,T),\\[0.2em] u(0)=u(T)=0, \end{array} \right. \end{eqnarray*} where $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$ and $W\in C([0,T]\times \mathbb{R}^{n},\mathbb{R})$ is homogeneous of degree $r$, $\lambda$ is a positive parameter, $g\in C([0,T])$, $1<r<p<q$ and $\frac{1}{p}<\alpha<1$. Using the Fibering map and Nehari manifold, for some positive constant $\lambda_0$ such that $0<\lambda<\lambda_0$, we prove the existence of at least two non-trivial solutions.
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참고문헌 (33)
참고문헌 신청
R. P. Agarwal / 2009 / Boundary value problems for fractional dierential equations / Georgian Math. J 16(3):401~411
O. P. Agrawal / 2004 / Fractional Derivatives and Their Application: Nonlinear Dynamics / Springer-Verlag
Ricardo Almeida / 2018 / Analysis of a fractional SEIR model with treatment / Applied Mathematics Letters 84:56~62
Zhanbing Bai / 2005 / Positive solutions for boundary value problem of nonlinear fractional differential equation / Journal of Mathematical Analysis and Applications 311(2):495~505
Loïc Bourdin / 2013 / Existence of a weak solution for fractional Euler–Lagrange equations / Journal of Mathematical Analysis and Applications 399(1):239~251
O. P. Agrawal / 2004 / Fractional Derivatives and Their Application: Nonlinear Dynamics / Springer-Verlag
Ricardo Almeida / 2018 / Analysis of a fractional SEIR model with treatment / Applied Mathematics Letters 84:56~62
Zhanbing Bai / 2005 / Positive solutions for boundary value problem of nonlinear fractional differential equation / Journal of Mathematical Analysis and Applications 311(2):495~505
Loïc Bourdin / 2013 / Existence of a weak solution for fractional Euler–Lagrange equations / Journal of Mathematical Analysis and Applications 399(1):239~251
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