Principal eigenvalues and asymptotic behavior for the weighted $p$-Laplacian with Robin boundary conditions on exterior domains

TL;DR

The paper studies a weighted $p$-Laplacian eigenproblem with Robin boundary on the exterior of the unit ball in $\mathbb{R}^N$, $N>p$, under a Lorentz-class weight. It develops a robust variational framework on the constrained manifold to establish existence, uniqueness, simplicity, and isolation of the principal eigenvalue, together with $C^{1,\alpha}_{\text{loc}}$ regularity. It then analyzes how the principal eigenvalue and eigenfunction vary with the Robin parameter $\beta$, demonstrating concavity and monotonicity, and establishes Neumann and Dirichlet limits. The authors derive a universal far-field decay and a $\beta$-dependent near-boundary scaling, identify a unique interior gradient maximum, and introduce unified gradient estimates using a characteristic length scale $L\sim\beta^{-1/(N-1)}$ that connects boundary and exterior-region behavior. The results provide a comprehensive description of how Robin boundary effects propagate into exterior domains for quasilinear operators with critical Lorentz weights.

Abstract

The spectral theory of the p-Laplacian is well developed for classical Dirichlet and Neumann boundary conditions, but the transitional Robin regime on exterior domains remains largely unexplored. This paper studies a weighted p-Laplacian eigenvalue problem with Robin boundary conditions on the exterior of the unit ball in Euclidean space of dimension N, with N greater than p. The weight function belongs to a critical Lorentz class and decays at infinity. Under natural assumptions on the weight, we prove the existence, uniqueness, simplicity, and isolation of a positive principal eigenvalue and establish local first-order regularity of the associated eigenfunction. We analyze the dependence of the principal eigenvalue on the Robin parameter and recover the Neumann and Dirichlet limits as the parameter approaches zero and infinity, respectively. The far-field behavior of the eigenfunction exhibits a universal algebraic decay rate that is independent of the Robin parameter, while the near-boundary structure displays an explicit scaling with respect to the parameter. We further investigate the gradient behavior of the eigenfunction, showing the existence of a unique critical radius and providing quantitative bounds on both the critical radius and the boundary value in terms of the Robin parameter. The main contribution of this work is the derivation of unified gradient estimates that connect the near-boundary and far-field regions through a characteristic length scale determined by the Robin parameter, yielding a global description of how boundary effects penetrate into the exterior domain.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2: Asymptotic Behavior
• Theorem 1.3: Gradient Estimates
• Theorem 1.4: Boundary Value and Critical Point Characterization
• Theorem 1.5: Unified Gradient Estimates
• Definition 2.1
• Lemma 2.2: Generalized Hardy-Sobolev inequality
• proof
• Lemma 2.3
• proof