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

Source abstract

Abstract This paper deals with the following fractional advection dispersion equation in ℝ N {\mathbb{R}^{N}} : - ∫ | θ | = 1 D θ ⁢ D θ β ⁢ u ⁢ M ⁢ ( d ⁢ θ ) + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , x ∈ ℝ N . see text {-\int_{|\theta|=1}D_{\theta}D_{\theta}^{\beta}uM(d\theta)+V(x)u=f(x,u),\quad x% \in\mathbb{R}^{N}.} Here, N > 1 {N>1} , β ∈ ( 0 , 1 ) {\beta\in(0,1)} , α = β + 1 2 {\alpha=\frac{\beta+1}{2}} , M ⁢ ( d ⁢ θ ) {M(d\theta)} is a Borel probability measure on the unit sphere in ℝ N {\mathbb{R}^{N}} , and D θ β {D_{\theta}^{\beta}} denotes the directional fractional derivative of order β in the direction of the unit vector θ. We establish a Bartsch–Wang-type compact embedding theorem for fractional Sobolev spaces when V vanishes at infinity. Then the existence of multiple solutions is established by using the mountain pass theorem and the fountain theorem.