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

TL;DR

The paper proves that every Dirichlet Laplacian eigenvalue $ ext{λ}_ ext{κ}$ is a bubbling concentration value for the Brezis–Nirenberg problem in bounded domains in dimensions $N=4,5$, by constructing precise multi-bubble solutions near $ ext{λ}_ ext{κ}$ via a Ljapunov–Schmidt reduction. The authors introduce a multi-bubble ansatz with bubbles $ ext{W}_{μ_j,ξ_j}$, eigenfunction modes $e_i$, and a small correction $oldsymbol{φ}_ε$, then split the problem into a nonlinear projected problem on $oldsymbol{𝕂}_ε^ot$ and a finite-dimensional problem on its kernel $oldsymbol{𝕂}_ε$. In $N=4$ the reduced finite-dimensional system is fully coupled, allowing arbitrarily many bubbles, with the bubble count governed by critical points of the eigenfunction combination; in $N=5$ the limit problem decouples and only finitely many bubbles occur, determined by nodal domain structure. The results extend prior radial and near-first-eigenvalue constructions to general bounded domains and provide new insights into the role of eigenfunctions in bubbling near eigenvalues, with detailed asymptotics and a variational framework for selecting bubble locations and heights.

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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 3.1
• proof
• Proposition 3.1
• Proposition 3.2
• proof
• Lemma 4.1
• proof