Fengliu Li | Searcharxiv

arXiv2025

On Brezis-Nirenberg problems: open questions and new results in dimension six

In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left\{\begin{aligned} &-\Delta u = \lambda u+|u|^{2^*-2}u, \quad &\mbox{in}\,\Omega,\\ &u=0,\quad &\mbox{on}\, \partial\Omega, \end{aligned}\right. \end{equation*} where $\Omega $ is a smoothly bounded domain of $\mathbb R^N$ with $N\geq 3$, $\lambda>0$ is a parameter and $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenberg problem.

math.AP↗

arXiv2025

Construction of bubbling solutions of the Brezis-Nirenberg problem in general bounded domains (I): the dimensions 4 and 5

In this paper, we consider the Brezis-Nirenberg problem $$ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, \Omega,\quad u=0,\quad\mbox{on}\,\, \partial\Omega, $$ where $\lambda\in\mathbb{R}$, $\Omega\subset\mathbb R^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $N\geq3$. We prove that every eigenvalue of the Laplacian operator $-\Delta$ with the Dirichlet boundary is a concentration value of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ by constructing bubbling solutions with precisely asymptotic profiles via the Ljapunov-Schmidt reduction arguments. Our results suggest that the bubbling phenomenon of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ as the parameter $\lambda$ is close to the eigenvalues are governed by crucial functions related to the eigenfunctions, which has not been observed yet in the literature to our best knowledge. Moreover, as the parameter $\lambda$ is close to the eigenvalues, there are arbitrary number of multi-bump bubbing solutions in dimension $N=4$ while, there are only finitely many number of multi-bump bubbing solutions in dimension $N=5$, which are also new findings to our best knowledge.

math.AP↗