In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<2$ 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 \cite{BFGM}. We detect the limit profile as $q\to\infty$ 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.
Asymptotics for a high-energy solution of a supercritical problem
Francesca Colasuonno;
2023
Abstract
In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<2$ 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 \cite{BFGM}. We detect the limit profile as $q\to\infty$ 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.File | Dimensione | Formato | |
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2203.06940.pdf
embargo fino al 01/11/2023
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