Existence of solution for a fractional advection dispersion equation in RN

This paper is concerned with the existence of solution to the following fractional advection dispersion equation-∫|θ|=1DθDθβuM(dθ)+b(x)u=f(x,u),x∈RN,u∈Hα(RN),where N>1,infRNb(x)>0, f:RN×R→R is continuous, the constant β∈(0,1),α=β+12,M(dθ) is a Borel probability measure on the unit sphere in RN, Dθβ denotes directional fractional derivative of order β in the direction of the unit vector θ. We focus our investigation on the existence of solution to the problem when M is symmetric and nonsymmetric by the Mountain Pass theorem and iterative technique. The main results of this paper emphasize the central role played by the general Borel probability measure.