Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations

• Published in: Nonlinear analysis Vol. 256; p. 113796
• Main Author: Garain, Prashanta
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.07.2025
• Subjects: De Giorgi theory; Elliptic and doubly nonlinear parabolic mixed anisotropic and nonlocal quasilinear equation; Energy estimates; Existence; Pointwise behavior; Regularity; Semicontinuity; System of mixed singular equations; Weak Harnack inequality; 35J62; 35B45; 35A01; 35K59; 35P30; 35M10; Weak Harnack inequality; Elliptic and doubly nonlinear parabolic mixed anisotropic and nonlocal quasilinear equation; Pointwise behavior; Energy estimates; 35B65; 35R11; 35B51; System of mixed singular equations; 35B50; Regularity; Semicontinuity; Existence; De Giorgi theory
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2025.113796

This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal p-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal p-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal p-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest.