Nontrivial boundary structure in a Neumann problem on balls with radii tending to infinity

TL;DR

This paper studies nonlinear Neumann problems on expanding balls, revealing concentration phenomena and detailed asymptotic behavior influenced by boundary curvature and nonlinear boundary conditions.

Contribution

It introduces a class of problems on expanding domains, providing refined asymptotics and uncovering nontrivial boundary structures due to nonlinear and curvature effects.

Findings

Established concentration phenomena for solutions.

Derived precise asymptotic expansions with first two order terms.

Identified effects of boundary curvature and nonlinear boundary conditions.

Abstract

This note introduces a class of nonlinear Neumann problems on balls expanding with the radii tending towards infinity. Performing singular perturbation arguments, we establish the corresponding concentration phenomenon and refined asymptotic expansions with the precise first two order terms. In doing so, we obtain the nontrivial boundary structure of solutions with effects coming from the nonlinear Neumann boundary condition and the boundary mean curvature varied with expanding domains.