Double phase problems with variable growth

TL;DR

This paper investigates double phase variational integrals with variable growth, analyzing eigenvalues and their existence, extending known frameworks for operators like p(x)-Laplace and capillarity operators, and contributing to the theory of unbalanced growth variational problems.

Contribution

It introduces two Rayleigh quotients related to eigenvalues, extending the framework to variable growth operators and unbalanced growth variational integrals.

Findings

Existence of an infinite interval of eigenvalues.

Nonexistence of eigenvalues under certain conditions.

Extension of the abstract framework to variable exponent operators.

Abstract

We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fu\v{c}ik, Ne\v{c}as, Sou\v{c}ek, and Sou\v{c}ek. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the $p(x)$-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.