[Paper Review] Symmetry and monotonicity results for positive solutions of p-Laplace systems

Céline Azizieh|ArXiv.org|Aug 7, 2001

Nonlinear Partial Differential Equations9 references3 citations

TL;DR

This paper extends symmetry and monotonicity results for positive solutions of scalar p-Laplace equations to systems of p-Laplace equations using the moving hyperplanes method. It establishes that solutions are symmetric and strictly decreasing in directionally symmetric domains when the exponents $ p_1, p_2 \in (1,2) $ or one is 2, under increasing, locally Lipschitz nonlinearities.

ABSTRACT

We extend to the case of a system involving p-Laplacians, the monotonicity and symmetry results of Damascelli and Pacella obtained in the case of a scalar p-Laplace equation with $1

Motivation & Objective

• To extend the monotonicity and symmetry results of Damascelli and Pacella from scalar p-Laplace equations to coupled p-Laplace systems.
• To analyze the behavior of positive weak solutions to a system involving two p-Laplace operators with different exponents $ p_1, p_2 \in (1,2) $ or one equal to 2.
• To establish directional monotonicity and symmetry of solutions under minimal regularity assumptions on the nonlinearities $ f $ and $ g $, which are nondecreasing and locally Lipschitz continuous.
• To prove that solutions are symmetric and strictly decreasing in the direction of symmetry when the domain is symmetric and the nonlinearities are strictly increasing.
• To resolve the technical challenge of applying the moving hyperplane method in systems where both $ p_1, p_2 \neq 2 $, where standard comparison principles are less effective.

Proposed method

• Employs the moving hyperplanes method to analyze symmetry and monotonicity of solutions in a bounded domain $ \Omega \subset \mathbb{R}^N $ with $ C^1 $ boundary.
• Defines reflection $ x_{\lambda}^{\nu} $ with respect to hyperplanes $ T_{\lambda}^{\nu} $, and uses the sets $ \Omega_{\lambda}^{\nu} $ and $ (\Omega_{\lambda}^{\nu})' $ to compare solutions across the hyperplane.
• Applies the strong maximum principle and Hopf lemma for the p-Laplacian to deduce strict positivity of normal derivatives in the moving plane setup.
• Uses comparison principles adapted to systems, particularly in the case $ p_1 \in (1,\infty), p_2 = 2 $, and in the case $ p_1, p_2 \in (1,2) $, where special care is taken due to weaker comparison tools.
• Establishes the existence of a critical hyperplane $ \lambda_1(\nu) $ beyond which symmetry and monotonicity fail, using compactness and continuity arguments.
• Proves that the set of points where $ u $ and $ v $ achieve their maximum on the boundary of a reflection region must be a single point or a set of measure zero, leading to contradiction if symmetry is broken.

Experimental results

Research questions

• RQ1 Can the moving hyperplane method be extended from scalar p-Laplace equations to systems of p-Laplace equations with different exponents?
• RQ2 Under what conditions on $ p_1, p_2 $ and the nonlinearities $ f, g $ do positive solutions of the system exhibit monotonicity and symmetry?
• RQ3 How does the behavior of the system change when $ p_1, p_2 \in (1,2) $ compared to when one exponent is 2?
• RQ4 What role do the local Lipschitz continuity and monotonicity of $ f $ and $ g $ play in ensuring the symmetry and monotonicity of solutions?
• RQ5 Can radial symmetry be deduced for solutions in radially symmetric domains like balls, under suitable conditions on $ f $ and $ g $?

Key findings

• For $ p_1 \in (1,\infty), p_2 = 2 $, and nondecreasing, locally Lipschitz $ f, g $, solutions $ u, v $ satisfy $ u(x) \leq u(x_{\lambda}^{\nu}) $ and $ v(x) \leq v(x_{\lambda}^{\nu}) $ for all $ x \in \Omega_{\lambda}^{\nu} $, and $ \partial v / \partial \nu > 0 $ in $ \Omega_{\lambda}^{\nu} $ for all $ \lambda < \lambda_1(\nu) $.
• When $ p_1, p_2 \in (1,2) $, and $ f, g $ are strictly increasing and locally Lipschitz, the same monotonicity and symmetry results hold for all $ \lambda \in (a(\nu), \lambda_1(\nu)] $.
• If the domain $ \Omega $ is symmetric with respect to a hyperplane $ T_0^{\nu} $ and $ \lambda_1(\nu) = \lambda_1(-\nu) = 0 $, then $ u $ and $ v $ are symmetric and strictly decreasing in the $ \nu $-direction in $ \Omega_0^{\nu} $.
• In the case of a ball $ B_R(0) $, the solutions $ u $ and $ v $ are radially symmetric and strictly decreasing in $ r = |x| $, provided $ f(x) > 0 $ and $ g(x) > 0 $ for all $ x > 0 $.
• The proof relies on contradiction: if $ u $ is constant on a set of reflection points, then $ \nabla u = 0 $ on that set, which contradicts the strong maximum principle unless symmetry is fully achieved.
• The image of the reflection map over a compact set of directions contains a full ball in $ \mathbb{R}^{N-1} $, which ensures that the set of symmetric points has nontrivial interior, leading to contradiction if symmetry fails.