Consider the following eigenvalue problem of p-Laplacian equation
where
d63b7a83231f7d78b8736cb0b" title="Click to view the MathML source">a≥0,
p∈(1,n) and
μ∈R.
V(x) is a trapping type potential, e.g.,
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
d63af5e7f9665f497235f24a584" 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
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
V(x) and blows up if
a↗a∗. The optimal rate of blowup is obtained for
V(x) being a polynomial type potential.