A new class of anisotropic double phase problems: exponents depending on solutions and their gradients

TL;DR

This paper introduces two new classes of double phase elliptic equations with exponents depending on solutions and gradients, establishing their properties and existence of solutions using advanced variational methods.

Contribution

It presents the first analysis of double phase problems with exponents depending on solutions and gradients, expanding the mathematical framework and solution techniques.

Findings

Established properties of Musielak-Orlicz Sobolev spaces for these classes

Proved existence of solutions using variational and perturbation methods

Highlighted applications to physical processes like fluid flow and phase changes

Abstract

In this work, we introduce two novel classes of quasilinear elliptic equations, each driven by the double phase operator with variable exponents. The first class features a new double phase equation where exponents depend on the gradient of the solution. We delve into proving various properties of the corresponding Musielak-Orlicz Sobolev spaces, including the $\Delta_2$ property, uniform convexity, density and compact embedding. Additionally, we explore the characteristics of the new double phase operator, such as continuity, strict monotonicity, and the (S$_+$)-property. Employing both variational and nonvariational methods, we establish the existence of solutions for this inaugural class of double phase equations. In the second category, the treatment of exponents is dependent on the solution itself. This class differs from the first one due to the unavailability of suitable Musielak-Orlicz Sobolev spaces. For this reason, we employ a perturbation argument that leads to the classical double phase class. These two new classes highlight how different physical processes like the movement of special fluids through porous materials, phase changes, and fluid dynamics interact with each other. Our results are novel in this context and includes a self-contained techniques.