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arXiv · 2103.08928

A new class of double phase variable exponent problems: Existence and uniqueness

Ángel Crespo-Blanco·Leszek Gasiński·Petteri Harjulehto·Patrick Winkert
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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.

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Ángel Crespo-Blanco, Leszek Gasiński, Petteri Harjulehto, Patrick Winkert. 2021-03-16. A new class of double phase variable exponent problems: Existence and uniqueness. https://arxiv.org/abs/2103.08928

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