Spike-layered Solutions of Singularly Perturbed Quasilinear Dirichlet Problems on Ball

Abstract

We consider the singularly perturbed quasilinear Dirichlet problems of the form  {-∈Δ_pu = f(u) in Ω  u ≥ 0 in , u = 0 on ∂ Ω  where Δ_pu = div(|Du|^{p-2}Du), p > 1, f is subcritical. ∈ > 0 is a small parameter and  is a bounded smooth domain in R^N (N ≥ 2). When Ω = B_1 = {x; |x| < 1} is the unit ball, we show that the least energy solution is radially symmetric, the solution is also unique and has a unique peak point at origin as ∈ → 0.