A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type
Marco Gallo, Jacopo Schino
TL;DR
This work extends the theory of normalized solutions for fractional Schrödinger equations with general, possibly sublinear nonlinearities of Berestycki–Lions type under a mass constraint. By combining an $\varepsilon$-perturbation of the nonlinearity with a Lagrangian formulation on a product space and a Pohožaev-based minimization, the authors obtain existence of L^2-minima and Pohožaev minima, linking energy, action, and frequency through a Legendre-type relation. They show existence of radially symmetric minimizers for large mass and, under stronger origin behavior, for all masses, with Nehari and Pohožaev identities, and they establish symmetry and positivity properties under various assumptions. The results include new insights even in the local case $s=1$ and accommodate logarithmic and sublinear nonlinearities, broadening the scope of normalized solutions in both fractional and classical settings. The methods offer potential for multiplicity, low-mass regimes, and extensions to other subcritical or critical frameworks.
Abstract
In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schrödinger equation $$(-Δ)^s u + μu = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint $\int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = m>0$; here, $N\geq 2$, $s \in (0,1)$, and $μ$ 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 Pohožaev 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.