A Concentration Phenomenon for p-Laplacian Equation

It is proved that if the bounded function of coefficient qn in the following equation - div {|∇u|p-2∇u} + v (x)|u|p-2u = qn(x)|u|q-2u,u(x) = 0 as x ∈ ∂ω. u(x) → 0 as |x| → ∞ is positive in a region contained in ω and negative outside the region,the sets {qn > 0} shrink to a point x0 ∈ ω as n → ∞,and then the sequence un generated by the nontrivial solution of the same equation,corresponding to qn,will concentrate at x0 with respect to w01,p (ω) and certain ls (ω) -norms. in addition,if the sets {qn > 0} shrink to finite points,the corresponding ground states {un} only concentrate at one of these points. these conclusions extend the results proved in the work of ackermann and szulkin (2013) for case p = 2. © 2014 yansheng zhong.