Title

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 εm Δmu − um−1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of RN (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 → ∞.

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