# Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

## TL;DR

This paper investigates the behavior and boundary localization of least-energy solutions to a singularly perturbed quasilinear elliptic problem with Neumann boundary conditions, revealing their boundary approach rate and decay properties.

## Contribution

It introduces an intrinsic variation method to analyze the boundary localization and decay of solutions, providing new insights into their asymptotic shape and boundary behavior.

## Key findings

- Maximum points approach the boundary faster than linear rate.
- Boundary points tend to where mean curvature is maximized.
- Solutions exhibit exponential decay.

## Abstract

In this paper we study the shape of least-energy solutions to a singularly perturbed quasilinear problem with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that at limit, the global maximum point of least-energy solutions goes to a point on the boundary faster than the linear rate and this point on the boundary approaches to a point where the mean curvature of the boundary achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.