Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups

Abstract

Abstract In this paper, we investigate the nonnegative solutions of the differential inequality u p ≤ ℒ s ⁢ u u^{p}\leq\mathcal{L}^{s}u on the Grushin space 𝔾 α n {\mathbb{G}_{\alpha}^{n}} for ( p , s , α ) ∈ ( 1 , ∞ ) × ( 0 , 1 ) × ( 0 , ∞ ) {(p,s,\alpha)\in(1,\infty)\times(0,1)\times(0,\infty)} , where the ℒ s {\mathcal{L}^{s}} are the fractional powers of the Grushin operator ℒ {\mathcal{L}} . We show that any nonnegative solution of the fractional order differential inequality displayed above is zero if and only if p ≤ Q Q - 2 ⁢ s {p\leq\frac{Q}{Q-2s}} , where Q is the homogeneous dimension of 𝔾 α n {\mathbb{G}_{\alpha}^{n}} . Moreover, we also consider the similar problems of nonnegative weak solutions of the fractional sub-Laplacian differential inequality on nilpotent Lie groups.