On Liouville problems for $(p,q)$-Laplacian inequalities on Finsler measure spaces

Abstract

We study the Liouville property for nonnegative weak solutions of the $(p,q)$-Laplacian elliptic differential inequality $${\Delta }_p^{m}u(x)+{\Delta }_q^{m}u(x)+V(x)u^s(x)\le 0$$ and the associated parabolic differential inequality $${\partial }_{t}u(x,t)\ge {\Delta }_p^{m}u(x,t)+{\Delta }_q^{m}u(x,t)+V(x,t)u^s(x,t)$$ on a forward geodesically complete noncompact Finsler measure space $(M,F,m)$ with finite reversibility. Under several sets of integral growth conditions on the positive potential function over certain annular domains, we prove that any nonnegative weak solution vanishes almost everywhere. The proofs of our results are essentially based on the nonlinear capacity method.

32 pages; all comments are welcome!