Fractional advection dispersion equation in ℝ N with a potential vanishing at infinity

This paper deals with the following fractional advection dispersion equation in ℝ N {RN} : - ∫ | θ | = 1 D θ ⁢ D θ β ⁢ u ⁢ M ⁢ ( d ⁢ θ ) + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , x ∈ ℝ N . see text {-∫|θ|=1DθDθβuM(dθ)+V(x)u=f(x,u), x% N.} Here, N > 1 {N>1} , β ∈ ( 0 , 1 ) {β∈(0,1)} , α = β + 1 2 {α=β+1/2} , M ⁢ ( d ⁢ θ ) {M(dθ)}