On some Elliptic and Parabolic Problems Involving the Anisotropic $p(u)$-Laplacian

TL;DR

The paper analyzes elliptic and parabolic PDEs involving the anisotropic p(u)-Laplacian, where the diffusion exponent depends on the unknown and direction. It develops the anisotropic variable-exponent Sobolev framework and employs a p^+-regularized perturbed problem together with pseudomonotone operator theory to prove elliptic existence, then applies time discretization, a Schauder fixed-point argument, and careful limit passages to obtain parabolic existence. The results extend nonstandard growth PDE theory to anisotropic, solution-dependent diffusion with nonlocal temporal coupling, under natural Carathéodory and growth conditions on the nonlinearity. The approach yields nontrivial weak solutions and provides a robust variational framework adaptable to related nonhomogeneous operators and nonlocal-in-time problems.

Abstract

We investigate a class of elliptic and parabolic partial differential equations driven by p(u) laplacian. This dependence necessitates the use of variable exponent Sobolev spaces specifically tailored to the anisotropic framework. For the elliptic case, we establish the existence of a weak solution by employing the theory of pseudomonotone operators in conjunction with suitable approximation techniques. In the parabolic setting, the existence of a weak solution is obtained via a time discretization scheme and Schauder fixed-point theorem, supported by a priori estimates and compactness arguments.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.1
• Definition 2.1
• Theorem 2.4