# On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems, Part II

## Document Type

Article

## Publication Date

1-2010

## Publication Title

Calculus of Variations and Partial Differential Equations

## DOI

10.1007/s00526-009-0261-2

## ISSN

1432-0835

## Abstract

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208–233, 2005) to study the structure of positive solutions to the equation *ε ^{m}* Δ

_{m}*u − u*

^{m}^{−1}+

*f*(

*u*) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of R

*N*(

*N*≥ 2). First, we study subcritical case for 2

*< m < N*and show that after passing by a sequence positive solutions go to a constant in

*C*

^{1, α}sense as

*ε*→ ∞. Second, we study the critical case for 1

*< m < N*and prove that there is a uniform upper bound independent of

*ε*∈ [1, ∞) for the least-energy solutions. Third, we show that in the critical case for 1 <

*m*≤ 2 the least energy solutions must be a constant if

*ε*is sufficiently large and for 2

*< m < N*the least energy solutions go to a constant in

*C*sense as

^{1, α}*ε*→ ∞.

## Recommended Citation

Zhao, Chunshan, Yi Li.
2010.
"On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems, Part II."
*Calculus of Variations and Partial Differential Equations*, 37 (1-2): 237-258.
doi: 10.1007/s00526-009-0261-2

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/247