| Abstract: |
| My talk is concerned with the following question: How do sequences of positive solutions $u_\varepsilon$ to the Brezis-Nirenberg-type equation $-\Delta u_\varepsilon + (a + \varepsilon V) u_\varepsilon = u_\varepsilon^\frac{N+2}{N-2}$ with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb R^N$ blow up? Brezis and Peletier (1989) conjectured that the blow-up location and speed of single-peak sequences should be universally characterized by the geometry of the domain, with explicit formulas. This was confirmed shortly afterwards by Rey (1989) and Han (1991) for the case of dimension $N \geq 4$, which is non-critical in the sense of the Brezis-Nirenberg.
In critical dimension $N=3$, an additional cancellation phenomenon makes the analysis substantially more challenging. In recent work with R. Frank (Munich) and H. Kovarik (Brescia) we overcome this difficulty and prove the Brezis-Peletier conjecture for single-peak sequences in dimension three. Subsequently with P. Laurain (Paris/Champs-sur-Marne) we develop a multi-peak analogue of the Brezis-Peletier conjecture, thus providing a complete picture of blow-up in the Brezis-Nirenberg problem at arbitrary levels of energy. In my talk I will present an overview of these results and the main proof ideas. |
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