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

TL;DR

This paper introduces a new class of double phase variable exponent elliptic problems, establishing foundational properties of the associated function spaces and proving existence and uniqueness of solutions under broad conditions.

Contribution

It develops the theory of Musielak-Orlicz Sobolev spaces for double phase operators with variable exponents and proves existence and uniqueness of solutions with gradient-dependent data.

Findings

Properties of Musielak-Orlicz Sobolev spaces are established.

The double phase operator is shown to be continuous and strictly monotone.

Existence and uniqueness of solutions with convection terms are proven.

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.