A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type
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2025
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| author | Gallo, Marco Schino, Jacopo |
| author_facet | Gallo, Marco Schino, Jacopo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type Gallo, Marco Schino, Jacopo Analysis of PDEs 35B06, 35B09, 35B38, 35D30, 35J20, 35Q40, 35Q55, 35R09, 35R11 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. |
| title | A Pohožaev minimization for normalized solutions: fractional sublinear equations of logarithmic type |
| topic | Analysis of PDEs 35B06, 35B09, 35B38, 35D30, 35J20, 35Q40, 35Q55, 35R09, 35R11 |
| url | https://arxiv.org/abs/2503.24080 |