Eigenvalue problem for a p-Laplacian equation with trapping potentials

Abstract

Consider the following eigenvalue problem of p-Laplacian equation (P)−Δpu+V(x)|u|p−2u=μ|u|p−2u+a|u|s−2u,x∈Rn, where 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 |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.