Asymptotic behavior of multi-peak solutions to the Brezis-Nirenberg problem. The sub-critical perturbation case
Jinkai Gao, Shiwang Ma
TL;DR
This work advances the understanding of multi-peak blow-up for the Brezis-Nirenberg problem with a subcritical perturbation by providing a complete asymptotic description as $\varepsilon\to0$. It combines finite-dimensional Lyapunov-Schmidt reduction, Green's function techniques, and precise Pohozaev identities to derive the exact blow-up rate, concentration speeds, and locations, and to classify blow-up configurations via the reduced energy $\Phi_n$. The authors prove both the existence and finiteness of blow-up configurations, characterize when these configurations are unique and nondegenerate, and establish a sharp link between the PDE blow-up and the critical points of the interaction energy. The results highlight how the subcritical exponent $q$ qualitatively alters the asymptotics, affecting uniqueness, nondegeneracy, and the multiplicity of blow-up patterns, with implications for the Brezis-Peletier conjecture in the subcritical regime.
Abstract
In this paper, we consider the following well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{2^*-1}+\varepsilon u^{q-1}, \quad u>0, &{\text{in}~Ω},\\ \quad \ \ u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\geq 3$, $Ω$ 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 $Ω$ and space dimension $N$ which is already known in the literature.