A Poho\v{z}aev minimization for normalized solutions: fractional sublinear equations of logarithmic type

Abstract

In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schr\"odinger equation $$(-\Delta)^s u + \mu u = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint $\int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = m>0$; here, $N\geq 2$, $s \in (0,1)$, and $\mu$ is a Lagrange multiplier. We study the case of $L^2$-subcritical nonlinearities $g$ of Berestycki--Lions type, without assuming that $g$ is superlinear at the origin, which allows us to include examples like a logarithmic term $g(u)= u\log(u^2)$ or sublinear powers $g(u)=u^q-u^r$, $0<r<1<q$. Due to the generality of $g$ and the fact that the energy functional might be not well-defined, we implement an approximation process in combination with a Lagrangian approach and a new Poho\v{z}aev minimization in the product space, finding a solution for large values of $m$. In the sublinear case, we are able to find a solution for each $m$. Several insights on the concepts of minimality are studied as well. We highlight that some of the results are new even in the local setting $s=1$ or for $g$ superlinear.