Existence and Uniqueness for Double-Phase Poisson Equations with Variable Growth

TL;DR

This paper investigates the existence and uniqueness of solutions for a class of nonlinear elliptic equations involving double-phase operators with variable exponents, relevant for modeling heterogeneous materials with phase transitions.

Contribution

It introduces a variational approach within modular function spaces to establish existence and uniqueness results for double-phase Poisson equations with variable growth.

Findings

Proved existence of weak solutions under minimal assumptions.

Established uniqueness of solutions in the variable exponent setting.

Demonstrated the uniform convexity of the associated modular function.

Abstract

We study a class of nonlinear elliptic problems driven by a double-phase operator with variable exponents, arising in the modeling of heterogeneous materials undergoing phase transitions. The associated Poisson problem features a combination of two distinct growth conditions, modulated by a measurable weight function \( \mu \), leading to spatially varying ellipticity. Working within the framework of modular function spaces, we establish the uniform convexity of the modular associated with the gradient term. This structural property enables a purely variational treatment of the problem. As a consequence, we prove existence and uniqueness of weak solutions under natural and minimal assumptions on the variable exponents and the weight.