Asymptotics for a high-energy solution of a supercritical problem

In this paper we deal with the equation -Delta 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-decreasing solutions, whose existence for q large is proved in Colasuonno et al. (2022). We detect the limit profile as q -> infinity 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.