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

Abstract

In this paper, we consider the Brezis-Nirenberg problem $$ -Δu=λu+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, Ω,\quad u=0,\quad\mbox{on}\,\, \partialΩ, $$ where $λ\in\mathbb{R}$, $Ω\subset\mathbb R^N$ is a bounded domain with smooth boundary $\partialΩ$ and $N\geq3$. We prove that every eigenvalue of the Laplacian operator $-Δ$ 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 $λ$ 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 $λ$ 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.