Consider the following eigenvalue problem of p-Laplacian equation
where
a≥0,
e621d619720d84aeb3a9" title="Click to view the MathML source">p∈(1,n) and
μ∈R.
e6cbef02a0e60" title="Click to view the MathML source">V(x) is a trapping type potential, e.g.,
e680d6ac873083aca90ad54851570" title="Click to view the MathML source">infx∈RnV(x)<lim|x|→+∞V(x). By using constrained variational methods, we proved that there is
a∗>0, which can be given explicitly, such that problem
(P) has a ground state
u with
97235f24a584" title="Click to view the MathML source">|u|Lp=1 for some
μ∈R and all
a∈[0,a∗), but
(P) has no this kind of ground state if
e65906d387cf535bd113" title="Click to view the MathML source">a≥a∗. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground state of problem
(P) approaches one of the global minima of
e6cbef02a0e60" title="Click to view the MathML source">V(x) and blows up if
a↗a∗. The optimal rate of blowup is obtained for
e6cbef02a0e60" title="Click to view the MathML source">V(x) being a polynomial type potential.