Eigenvalue problem for a p-Laplacian equation with trapping potentials

Consider the following eigenvalue problem of p-Laplacian equation - Delta(P)u + V(x)vertical bar u vertical bar(p-2)u = mu vertical bar u vertical bar(p-2)u + a vertical bar u vertical bar(s-2)u, x epsilon R-n, (P) where a >= 0, p is an element of (1,n) and is an element of R. V(x) is a trapping type potential, e.g., inf(x)is an element of R-n. V(x) < lim(vertical bar x vertical bar ->+infinity) 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 vertical bar u vertical bar L-p = 1 for some mu is an element of R and all a is an element of [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 NE arrow a*. The optimal rate of blowup is obtained for V(x) being a polynomial type potential. (C) 2016 Elsevier Ltd. All rights reserved.