On the structure of solutions to a class of quasilinear elliptic Neumann problems. Part II.
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 epsilon(m) Delta(m)u - u(m-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 C(1, alpha) sense as epsilon --> infinity. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of epsilon in [1, infinity) 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 epsilon is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C(1, alpha) sense as epsilon --> infinity.
Zhao
Chunshan
C
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA.
Li
Yi
Y