Consider the following eigenvalue problem of p-Laplacian equation
where
a≥0,
619720d84aeb3a9" title="Click to view the MathML source">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
618db0b28" title="Click to view the MathML source">a∗>0, which can be given explicitly, such that problem
(P) has a ground state
350875fbb552a162" title="Click to view the MathML source">u with
35f24a584" title="Click to view the MathML source">|u|Lp=1 for some
μ∈R and all
3524f" title="Click to view the MathML source">a∈[0,a∗), but
(P) has no this kind of ground state if
35bd113" 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
V(x) and blows up if
61444" title="Click to view the MathML source">a↗a∗. The optimal rate of blowup is obtained for
V(x) being a polynomial type potential.