Asymptotics for a high-energy solution of a supercritical problem

In this paper we deal with the equation $$-{\mathrm{\Delta }}_{p}u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$$ for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of ${\mathbb{R}}^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to \mathrm{\infty }$ 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.