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.