Liouville type theorem for quasi-linear elliptic inequality Δpu + uσ ≤ 0 on Riemannian manifolds

We studied the uniqueness of a nonnegative solution of the quasi-linear elliptic inequality Delta(p)u + u(sigma) <= 0 (*) on a connected geodesically complete Riemannian manifold X, where sigma is a parameter and Delta(p)u = div(vertical bar del u vertical bar(p-2)del u). We proved that if p> 1, sigma > p - 1, and for some x(0) is an element of X, lim(t -> 0+) inf t sigma/sigma-p+1 integral(infinity)(1) mu(B(x(0),r))/r((p+1)sigma-p+1/sigma-p+1 + pt/2(p-1)) dr < infinity, then inequality (*) has no nontrivial nonnegative weak solutions. We also studied the weighed case and the sharpness of the main result. (C) 2015 Elsevier Inc. All rights reserved.