Asymptotic behavior of multi-peak solutions to the Brezis-Nirenberg problem. The sub-critical perturbation case

TL;DR

This paper investigates the detailed asymptotic behavior, blow-up rates, and solution uniqueness of multi-peak solutions to the Brezis-Nirenberg problem with sub-critical perturbations, extending understanding of solution concentration phenomena.

Contribution

It provides a complete characterization of multi-peak blow-up solutions, including their asymptotic profiles, blow-up rates, and solution count, for the sub-critical perturbation case.

Findings

Describes asymptotic profile of solutions as epsilon approaches zero.

Derives exact blow-up rates and concentration points.

Proves uniqueness, nondegeneracy, and counts solutions.

Abstract

In this paper, we consider the following well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u= u^{2^*-1}+\varepsilon u^{q-1}, \quad u>0, &{\text{in}~\Omega},\\ \quad \ \ u=0, &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $N\geq 3$, $\Omega$ is a smooth and bounded domain in $\R^{N}$, $\varepsilon>0$ is a small parameter, $q\in (2,2^*)$ and $2^*:=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. The existence of solutions to the above problem has been obtained by many authors in the literature. However, as far as the authors know, the asymptotic behavior of solutions to the above problem is still open. Here we first describe the asymptotic profile of solutions to the above problem as $\varepsilon\to 0$. Then, we derive the exact blow-up rate and characterize the concentration speed and the location of concentration points in the general case of multi-peak solutions. Finally, we prove the uniqueness, nondegeneracy and count the exact number of blow-up solutions. The main results in this paper give a complete picture of multi-peak blow-up phenomena in the framework of Brezis-Peletier conjecture in the case of sub-critical perturbation. On the other hand, compared with the special case $q=2$ previously studied in the literature, we observe that the exponent $q$ has a significant impact on the asymptotic behavior, uniqueness and nondegeneracy of solutions in addition to the geometry of domain $\Omega$ and space dimension $N$ which is already known in the literature.