A Concentration Phenomenon For Quasilinear P-Laplacian Type Problems

In this paper, we study a concentration phenomenon for the following p-Laplacian type problems where0∈Ω(?) RN,Δpu=div(|▽u|p-2▽u),n,N∈N,2≤p<N,p<q<p:=Np/N-p.Here Ω is an open set, possibly unbounded, with smooth or empty boundary, V≥0. For every n∈N, Qn is uniformly bounded respect to n, Qn is positive in a neighbourhood of0and negative outside. The sets{x∈Ω:Qn>0} with positive measure shrink to0∈Ω as n→∞. For n large enough, we prove that if un is a ground state solution corresponding to Qn, then the sequence{un} concentrates at0with respect to H1and Lq norms. We also show that if the sets{x∈Ω:Qn>0} shrink to two different points in Ω, then the sequence{un} concentrates at only one of these two points.Our results generalize some similar results in [18] for semilineax elliptic equations to the nonlinear p-Laplacian type problems.