Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 106, pp. 1-15.
Title: Existence of solutions to boundary value problems arising
from the fractional advection dispersion equation
Author: Lingju Kong (the Univ. of Tennessee at Chattanooga, TN, USA)
Abstract:
We study the existence of multiple solutions to the
boundary value problem
$$\displaylines{
\frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))+\frac12{}_tD_T^{-\beta}(u'(t))
\Big)+\lambda \nabla F(t,u(t))=0,\quad t\in [0,T],\cr
u(0)=u(T)=0,
}$$
where $T>0$, $\lambda>0$ is a parameter, $0\leq\beta<1$, ${}_0D_t^{-\beta}$
and ${}_tD_T^{-\beta}$ are, respectively, the left and right Riemann-Liouville
fractional integrals of order $\beta$,
$F: [0,T]\times\mathbb{R}^N\to\mathbb{R}$ is a given function.
Our interest in the above system arises from studying the steady fractional
advection dispersion equation.
By applying variational methods, we obtain sufficient conditions
under which the above equation has at least three solutions.
Our results are new even for the special case when $\beta=0$.
Examples are provided to illustrate the applicability of our results.
Submitted December 17, 2012. Published April 24, 2013.
Math Subject Classifications: 34B15, 34A08.
Key Words: Three solutions; fractional boundary value problem;
variational methods.