A Concentration Phenomenon for p-Laplacian Equation

[摘要] It is proved that if the bounded function of coefficientQnin 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 pointx0∈Ωasn→∞, and then the sequenceungenerated by the nontrivial solution of the same equation, corresponding toQn, will concentrate atx0with respect toW01,p(Ω)and certainLs(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 inthe work of Ackermann and Szulkin (2013) for casep=2.

[学科分类] 应用数学