We study qualitative positivity properties of quasilinear equations of the formwhere Ω is a domain inis a symmetric and locally uniformly positive definite matrix, V is a real potential in a certain local Morrey space (depending on p), andOur 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 Hölder 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 behavior 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 Ω.

We prove sharp homogeneous improvements to L 1 weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball we also obtain further improvements.Mathematics Subject Classification: 35P15 · 52A30 · 49Q20 · 51M16

Let Ω be an open connected cone in R n with vertex at the origin. Assume that the operatoris subcritical in Ω, where δΩ is the distance function to the boundary of Ω and µ ≤ 1/4. We show that under some smoothness assumption on Ω, the following improved Hardy-type inequalityholds true, and the Hardy-weight λ(µ)|x| −2 is optimal in a certain definite sense. The constant λ(µ) > 0 is given explicitly.

We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy's inequality in the case p = n ≥ 2. In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for p = 2

We strengthen the classical inequality of C.B. Morrey concerning the optimal Hölder continuity of functions in W 1,p when p > n, by replacing the L p -modulus of the gradient with the sharp Hardy difference involving distance to the boundary. When p = n we do the same strengthening in the integral form of a well known inequality due to F. John and L. Nirenberg.