题目：Global Branch Approach -- A unified framework for semilinear and quasilinear elliptic equations with mass constraints
摘要：
This talk presents a novel global branch approach as a unified framework for studying normalized solutions, distinct from traditional constrained variational methods. The methodology, initially developed in collaborations with T. Bartsch & W. Zou [Math. Ann., 2021] and later with L. Jeanjean & J. Zhang [J. Math. Pures Appl., 2024], involves constructing global solution branches, analyzing their evolution, and precisely characterizing solutions along branches (particularly w.r.t. the evolution limit of the mass)
I will report on recent advances with T. Bartsch and J. Zhang that refine and extend this framework to quasilinear problems, overcoming Liouville-type barriers. This yields a unified treatment for both semilinear and quasilinear problems across all mass regimes. Specifically, we consider
$$-\Delta v - \Delta\big[\ell(v^2)\big]\ell'(v^2)v + \lambda v = f(|v|^2)v \quad \text{in } \mathbb{R}^N,$$
with $\int_{\mathbb{R}^N} |v|^2 \mathrm{d}x = c$. The framework applies to broad classes of $\ell \in C^1([0,+\infty))$, including physically significant cases like $\ell(s) \equiv \text{constant}$, $\ell(s) = s$, and $\ell(s) = \sqrt{1+s}$.
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2025-06-24 08:34:15
2025-10-10
