Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem. (English) Zbl 07896319
This article is concerned with the study of positive solutions to \(-\Delta u+\varepsilon V u=N(N-2)u^{\frac{N+2}{N-2}}\) in a bounded open set \(\Omega\subset\mathbb{R}^N\), \(N\geq 4\). The solutions are assumed to satisfy an homogeneous Dirichlet boundary condition while the potential \(V\in C^1(\overline{\Omega})\) is negative. The main result of the article establishes the existence of solutions exhibiting multiple concentration points. It is shown that such concentration points are isolated and the authors provide a characterization of the vector of these points as a critical point of a suitable function derived from the Green function of \(-\Delta\) on \(\Omega\).
Reviewer: Marius Ghergu (Dublin)
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
semilinear equation with Laplacian; Dirichlet problem; existence of solutions with multiple concentration pointsReferences:
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