Generalized superharmonic functions with strongly nonlinear operator

Abstract

We study properties of A-harmonic and A-superharmonic functions involving an operator having generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. In particular, Harnack's Principle and Minimum Principle are provided for A-superharmonic functions and boundary Harnack inequality is proven for A-harmonic functions.

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

“…It follows that u ∈ W 1,G 0 (Ω) (see e.g. [24]), and therefore µ ∈ (W 1,G 0 (Ω)) ′ . As a matter of fact, then the distributional solutions are weak solutions and the classical regularity theory provide more information by completely other strong tools.…”

“…is well defined on a reflexive and separable Banach space W 1,G (Ω) and A G (W 1,G (Ω)) ⊂ (W 1,G (Ω)) ′ . Indeed, when u ∈ W 1,G (Ω) and φ ∈ C ∞ 0 (Ω), growth conditions (6), Hölder's inequality (24), and Lemma 3.1 justify that…”

“…Our main inspirations are [48,69] and [50]. The tools of potential analysis used extensively in the proof have been elaborated lately in [24] for A-superharmonic problems with more general growth of Musielak-Orlicz type.…”

“…Similar scheme was used also in [38,52]. This approach requires a basic toolkit of potential analysis provided in [24].…”

“…[5,7,12,13,19,21,70]. Favorable versions of Theorems 1 and 2 in the generality of [24] would be also very interesting.…”