Local behavior of solutions of quasi-linear equations

Classification of positive D1,p(RN)-solutions to the critical p

“…Let us recall that any solution u ∈ D 1,p (R N ) of P * belongs to L ∞ (R N ) as it follows by [15,17,22]. Consequently we have that u is locally of class C 1,α by C 1,α estimates (see Partially supported by ERC-2011-grant: Elliptic PDE's and symmetry of interfaces and layers for odd nonlinearities.…”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…By local regularity for quasilinear elliptic equations (see [19]) and standard argument, {u β,R } converges, as R → ∞, in C 1 loc (R N ) to a function u β which is a weak solution of (1.1) in R N and satisfies U β ≤ u β ≤ s 0 . Since u β (0) ≥ U β (0) = δ > 0, by Harnack inequality (see [29], [31]), we obtain u β > 0 in R N . Since u β ≤ s 0 , it follows that u β ∈ S p .…”

“…Take an arbitrary subset G Ω then there exists n G > 0 such that G Ω n for every n ≥ n G . By Harnack's inequality (see, e.g., [29,Theorem 5], [31, Theorem 1]), one can find a constant C G > 0 independent of n such that…”

A singular perturbation problem for the p-Laplace operator

“…p−1 for some constants c 0 and c 1 , the conclusion of the Lemma 2.2 follows directly from Serrin's Harnack inequality for nonhomogeneous quasilinear operators, see [16]. Our proof, however, uses only Harnack inequality for homogeneous operators and is based on compactness rather than energy methods, which allows to generalize it to a broad range of operators.…”