  Consider the following eigenvalue problem of p-Laplacian equation
\begin{equation}\label{P}
  -\Delta_{p}u+V(x)|u|^{p-2}u=\mu|u|^{p-2}u+a| u|^{s-2}u, x\in \mathbb{R}^{n},
\tag{P} \end{equation} where $a\geq0$, $p\in (1,n)$ and $\mu\in\mathbb{R}$.
$V(x)$ is a trapping type potential, e.g., $\inf\limits_{x \in
\mathbb{R}^n}V(x)< \lim\limits_{|x|\rightarrow+\infty}V(x)$. By using
constrained variational methods, we proved that there is $a^*>0$, which can be
given explicitly, such that problem (\ref{P}) has a ground state $u$ with
$\|u\|_{L^p}=1$ for some $\mu \in \mathbb{R}$ and all $a\in [0,a^*)$, but
(\ref{P}) has no this kind of ground state if $a\geq a^*$. Furthermore, by
establishing some delicate energy estimates we show that the global maximum
point of the ground states of problem (\ref{P}) approach to one of the global
minima of $V(x)$ and blow up if $a\nearrow a^*$. The optimal rate of blowup is
obtained for $V(x)$ being a polynomial type potential.
