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Eigenvalue problem for a p-Laplacian equation with trapping potentials

Consider the following eigenvalue problem of p-Laplacian equation \begin{equation}\label{P} -Δ_{p}u+V(x)|u|^{p-2}u=μ|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 $μ\in\mathbb{R}$. $V(x)$ 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 $μ\in \mathbb{R}$ 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.