| Abstract: |
| Schr\{o}dinger-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 a fractional equation paired with a constraint on the $L^2$ norm. The context includes the so-called strongly sublinear regime, i.e., when the RHS has a negative sublinear growth at the origin. 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 the mass 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. In the strongly sublinear regime, the previous result for a suitably translated problem is exploited to obtain a solution for any mass. |
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