Variational and nonvariational solutions for double phase variable exponent problems
Mustafa Avci
TL;DR
The paper addresses the existence of nontrivial solutions for double-phase, variable-exponent PDEs in both nonvariational and variational settings. It combines Browder–Minty monotone operator theory to handle gradient-dependent nonlinearities with a Bonanno–Chinnì critical-point framework to treat a variational formulation, proving at least one nontrivial solution in each case. A detailed Musielak–Orlicz–Sobolev space foundation is developed, including embeddings and energy functionals, accompanied by concrete examples such as a Ginzburg–Landau–type model with convection. The dual treatment advances understanding of anisotropic, heterogeneous diffusion and provides a methodological template for similar double-phase problems in applications ranging from materials science to image processing.
Abstract
In this article, we examine two double-phase variable exponent problems, each formulated within a distinct framework. The first problem is non-variational, as the nonlinear term may depend on the gradient of the solution. The first main result establishes an existence property from the nonlinear monotone operator theory given by Browder and Minty. The second problem is set up within a variational framework, where we employ a well-known critical point result by Bonanno and Chinnì. In both cases, we demonstrate the existence of at least one nontrivial solution. To illustrate the practical application of the main results, we provide examples for each problem.