On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems
We study the structure of positive solutions to the equation ɛmΔmu-um-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ɛ→0+ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ɛ for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m⩽2 holds and ɛ is sufficiently large, any positive solution must be a constant.
& Zhao, C.
(2005). On the Structure of Solutions to a Class of Quasilinear Elliptic Neumann Problems. Journal of Differential Equations, 212 (1), 208-233.