Spike-layered solutions with compact support to some singularly perturbed quasilinear elliptic problems in general smooth domains

Abstract

In this paper we study the structure of nonnegative nontrivial solutions of the following problem: −εΔ pu=f(u) in Ω u⩾0 in Ω, u=0 on ∂ Ω as ε→0 +, where Δ p u=div(| Du| p−2 Du) with p>1. ε>0 is a small parameter and Ω is a bounded smooth domain in R N ( N⩾1). f is a class of logistic-type nonlinearities satisfying f(0)= f( z 1)= f( z 2)=0 with 0< z 1< z 2, f<0 in (0, z 1), f>0 in ( z 1, z 2) and lim ̄ u→0 + f(u)/u p−1=−∞ . By virtue of the sub- and supersolution method, we prove that there are many nonnegative nontrivial solutions and they are spike-layered solutions. Moreover, the measure of each spike layer is estimated as ε→0 +.