ON POSITIVE SOLUTIONS OF THE (p,A)-LAPLACIAN WITH POTENTIAL IN MORREY SPACE

We study qualitative positivity properties of quasilinear equations of the form Q' (A,p,V) [v]:= -div(vertical bar del nu vertical bar(p-2)(A) A(x)del nu)+ V(x)vertical bar nu vertical bar(p-2) nu=0; x is an element of Omega, where Omega is a domain in R-n, 1 < p < infinity, A = (a(ij)) is an element of L-loc(infinity)(Omega; R (n x n)) is a symmetric and locally uniformly positive definite matrix, V is a real potential in a certain local Morrey space (depending on p), and vertical bar xi vertical bar(2)(A) :=A(x)xi center dot xi = [GRAPHICS] a(ij)(x)xi(i)xi(j), x is an element of Omega, xi=(xi(1,) ... , xi(n)) is an element of R-n. Our assumptions on the coefficients of the operator for p >= 2 are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Holder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when p < 2. We prove an Allegretto-Piepenbrink-type theorem for the operator Q' (A, p, V), and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case 1 < p <= n, we examine the behaviour of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator Q' (A,p,V) [u] in Omega