Asymptotics for a high-energy solution of a supercritical problem

Abstract

In this paper we deal with the equation −∆pu + |u| p−2 u = |u| q−2 u for 1 < p < 2 and q > p, under Neumann boundary conditions in the unit ball of R N . We focus on the three positive, radial and radially non-deacreasing solutions, whose existence for q large is proved in [13]. We detect the limit profile as q → ∞ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant 1. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest.