Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with $N\ge 3$ is a bounded domain, $q\in (2,2^{\ast })$ and $2^{\ast }=\frac{2N}{N-2}$ 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 $\epsilon >0$ 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 $\epsilon \to 0$ 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 $\mathrm{\Omega }$. 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 $\mathrm{\Omega }$, but also depends on the exponent $q\in (\max \{2,\frac{4}{N-2}\},2^{\ast })$ in a non-trivial way.