A new class of double phase variable exponent problems: Existence and uniqueness
Abstract
In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.
Cite
@article{arxiv.2103.08928,
title = {A new class of double phase variable exponent problems: Existence and uniqueness},
author = {Ángel Crespo-Blanco and Leszek Gasiński and Petteri Harjulehto and Patrick Winkert},
journal= {arXiv preprint arXiv:2103.08928},
year = {2022}
}