DOI: 10.1515/gmj-2026-3004 ISSN: 1072-947X

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

Guofeng Feng, Meiyan Wang, Bin Ge
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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.

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