Normalized solutions of nonlinear elliptic problems

Abstract: The talk will be concerned with the existence of $L^2$ normalized solutions to nonlinear elliptic equations and systems. A model problem is the system of nonlinear Schrödinger equations

$$-\Delta u+\lambda_1 u = \mu_1 u^3 + \beta u v^2 \qquad \in \mathbb{R}^3$$ $$-\Delta v+\lambda_2 v = \mu_2 v^3 + \beta u^2 v \qquad \in \mathbb{R}^3$$

with normalization constraints

$$\int_{\mathbb{R}^3} u^2 = a^2 \quad \text{and}\quad \int_{\mathbb{R}^3} v^2 = b^2 \, .$$

Whereas nonlinear elliptic equations and systems have been investigated intensively since the 1960s, in comparison surprisingly little is known about solutions with prescribed $L^2$ norms. We discuss this problem and survey recent results. The talk is based on joint work with Louis Jeanjean, Yanyan Liu, Zhaoli Liu, Nicola Soave, Xuexiu Zhong, Wenming Zou.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysis

Audience: researchers in the topic

"Partial Differential Equations and Applications" Webinar