#### Title

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

#### Document Type

Article

#### Publication Date

5-1-2005

#### 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 Δmu − u m−1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of (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 C1, 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 C1, α sense as ε → ∞.

#### Repository Citation

Zhao, C.,
& Li, Y.
(2005). On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems. Part II. *Calculus of Variations and Partial Differential Equations Calculus of Variations and Partial Differential Equations, 212* (1), 208-233.

https://corescholar.libraries.wright.edu/math/70

#### DOI

10.1016/j.jde.2004.07.021