Speaker
Description
(Jacopo Schino - University of Warsaw, Polland)
Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties: one of them is the conservation of mass, which gives rise to the search for normalised solutions. In this talk, I will explain a possible approach to solve
$$\begin{cases}
(-\Delta)^s u + \mu u = g(u)\\
\int_{\mathbb{R}^N} u^2~dx = m\\
(μ,u) \in \mathbb{R} \times H^s(\mathbb{R}^N),
\end{cases}$$
where $N \geqslant 2$, $0<s<1$, and $m>0$ is is prescribed, in cases that include the so-called strongly sublinear regime:
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\lim_{t \to 0} \dfrac{g(t)}{t}=\infty~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$
This makes a direct variational approach impossible because the energy functional is not well-defined in $H^s(\mathbb{R}^N)$. In the proposed approach, when $m$ is sufficiently large, a family of approximating problems is considered so that the energy functional is of class $\mathcal{C}^1$ and a corresponding family of solutions is obtained, which eventually converge to a solution to the original problem. When $(1)$ holds, the previous result for a suitably translated problem is exploited to obtain a solution for any $m>0$.
This is joint work with Marco Gallo (Catholic University of the Sacred Heart, Brescia, Italy).
