| Abstract: |
| In 2023, H. Brezis published 28 open problems which he described as `raised throughout my career and
that have resisted so far`. In this talk, we present complete solutions to the first and seventh problems:
Problem 1.1. Let $lambda in (0, frac{lambda_1}{4})$ and $B_1$ be the unit ball in $R^3$. Are there nontrivial solutions to
$$ -Delta u= u^5 +lambda u mbox{in} B_1; u=0 mbox{on} partial B_1 ? $$
Our answer: Yes. There are infinitely many non-radial sign-changing solutions. (Joint work with Liming Sun and Wen Yang.)
Problem 3.1. Let $B_1$ be the unit disk in $R^2$. Consider the harmonic map problem $u: B_1 o S^2$
$$ -Delta u =|
abla u|^2 u mbox{in} B_1; u= (Rx, Ry, sqrt{1-R^2}) mbox{on} partial B_1 $$
It is well-known that there are at least two explicit solutions. Are there more solutions?
Our answer: Yes. There are infinitely many distinct solutions. (Joint work with Fanghua Lin.) |
|