Critical points of the Moser-Trudinger functional on a disk

By A. Malchiodi, L. Martinazzi. 2012

67 citations recorded by the source corpus.

On the unit disk B-1 subset of R-2 we study the Moser-Trudinger functional E(u) = integral(B1) (e(u2) - 1)dx, is an element of H-0(1) (B-1) and its restrictions E vertical bar M-Lambda, where M-Lambda := {u is an element of H-0(1) (B-1): parallel to u parallel to(2)(H01) = Lambda} for Lambda > 0. We prove that if a sequence u(k) of positive critical points of E vertical bar(M Lambda k) ( for some Lambda(k) > 0) blows up as k -> infinity, then Lambda(k) -> 4 pi, and u(k) -> 0 weakly in H-0(1) (B-1) and strongly in C-loc(1) ((B) over bar (1) \ {0}). Using this fact we also prove that when Lambda is large enough, then E vertical bar M-Lambda has no positive critical point, complementing previous existence results by Carleson-Chang, Struwe and Lamm-Robert-Struwe.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6d0bcbde43df5d22ecbc4c629eddbb117a888422

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Gabriele Mancini
2. L. Martinazzi
3. G. Csató
4. Prosenjit Roy
5. Xiaobao Zhu
6. Y. Yang
7. Matteo Cozzi

In Geometry and Topology

1. Gabriele Mancini
2. L. Martinazzi

Citations in the corpus

• Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions (2014, background): “See also Malchiodi-Martinazzi [9] and the references therein for some recent developments on the subject.”
• The Moser–Trudinger inequality and its extremals on a disk via energy estimates (2016, methodology): “Inspired from [13] (where the Taylor expansion was made only up to order μ−4 k ), we will prove uniform bounds on the error term φk up to sufficiently large scales.”