Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case
Jinkai Gao, Shiwang Ma
TL;DR
The paper analyzes the Brezis–Nirenberg problem with a subcritical perturbation $-\,\Delta u = u^{2^*-1} + \varepsilon u^q$ on a smooth bounded domain, identifying sharp blow-up profiles and rates as $\varepsilon\to0$ for $q\in(\max\{2,4/(N-2)\},2^*)$. Through a detailed blow-up analysis, it shows the rescaled least energy solutions converge to the Aubin–Talenti bubble and that blow-up occurs at interior points which are critical points of the Robin function, with an explicit blow-up rate depending on $N$ and $q$. The work proves asymptotic uniqueness and nondegeneracy of least energy solutions under two sets of domain assumptions, using Lyapunov–Schmidt type reductions and local Pohožaev identities, and provides a precise energy expansion involving the Robin function at the concentration point. Overall, the results extend known $q=2$ theories to the subcritical regime, highlighting the nuanced roles of dimension, domain geometry, and exponents, and include a comprehensive appendix of technical tools.
Abstract
In this paper, we are concerned with the 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 $Ω\subset \mathbb R^N$ with $N\ge 3$ is a bounded domain, $q\in(2,2^*)$ and $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. It is well-known (H. Brézis and L. Nirenberg, \newblock {\em Comm. Pure Appl. Math.}, 36(4):437--477, 1983) that the above problem admits a positive least energy solution for all $\varepsilon >0$ and $q>\max\{2,\frac{4}{N-2}\}$. In the present paper, we first analyze the asymptotic behavior of the positive least energy solution as $\varepsilon\to 0$ and establish a sharp asymptotic characterisation of the profile and blow-up rate of the least energy solution. Then, we prove the uniqueness and nondegeneracy of the least energy solution under some mild assumptions on domain $Ω$. The main results in this paper can be viewed as a generalization of the results for $q=2$ previously established in the literature. But the situation is quite different from the case $q=2$, and the blow-up rate not only heavily depends on the space dimension $N$ and the geometry of the domain $Ω$, but also depends on the exponent $q\in(\max\{2,\frac{4}{N-2}\}, 2^*)$ in a non-trivial way.