A Concentration Phenomenon for p-Laplacian Equation

Abstract

It is proved that if the bounded function of coefficient $Q_n$ in the following equation $-\text{div}\hspace{0.17em}\{{\left|\nabla u\right|}^{p-\mathrm{2}}\nabla u\}+V(x){\left|u\right|}^{p-\mathrm{2}}u=Q_n(x){\left|u\right|}^{q-\mathrm{2}}u,\hspace{0.17em}\hspace{0.17em}u(x)=\mathrm{0}\hspace{0.17em}\hspace{0.17em}\text{as}\hspace{0.17em}\hspace{0.17em}x\in \partial \mathrm{\Omega }.u(x)⟶\mathrm{0}\hspace{0.17em}\hspace{0.17em}\text{as}\hspace{0.17em}\hspace{0.17em}\left|x\right|⟶\infty$ is positive in a region contained in Ω and negative outside the region, the sets $\{Q_n>0\}$ shrink to a point $x_0\in \mathrm{\Omega }$ as $n\to \mathrm{\infty }$, and then the sequence $u_n$ generated by the nontrivial solution of the same equation, corresponding to $Q_n$, will concentrate at $x_0$ with respect to $W_0^{1,p}(\mathrm{\Omega })$ and certain $L^s(\mathrm{\Omega })$-norms. In addition, if the sets $\{Q_n>0\}$ shrink to finite points, the corresponding ground states $\{u_n\}$ only concentrate at one of these points. These conclusions extend the results proved in the work of Ackermann and Szulkin (2013) for case $p=2$.