A Lagrangian approach for prescribed mass solutions of cubic-quintic Schrödinger equations and $L^2$-supercritical problems
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arXiv
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| Natura: | Preprint |
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2026
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| author | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| author_facet | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| contents | We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$):
$$ (*)_m \quad - Δu + μu = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb R}^N} u^2\, dx = m,$$
where $g(s) \in C({\mathbb R},{\mathbb R})$, $m > 0$ and $μ\in {\mathbb R}$ is an unknown Lagrangian multiplier. We take an approach using a Lagrangian formulation of $(*)_m$:
$$J_m(μ,u)={1\over 2}\int_{{\mathbb R}^N} |\nabla u|^2\,dx -\int_{{\mathbb R}^N} G(u)\,dx +μ\left({1\over 2}\int_{{\mathbb R}^N} u^2\, dx-m\right) \in C^1((0,\infty)\times H_r^1({\mathbb R}^N), {\mathbb R})$$
and we give new general existence results through the function:
$$ b_m:\, (0,\infty) \to {\mathbb R};\ μ\mapsto \text{Mountain Pass minimax value for}\ (u\mapsto J_m(μ,u)).$$
We will show the existence of solutions of $(*)_m$ related to local minima and local maxima of $b_m(μ)$. As applications, we study cubic-quintic type equations and $L^2$-supercritical problems. In particular, when $N=2,3$, we show new existence results of normalized solutions without assuming global Ambrosetti-Rabinowitz type conditions, which partially improve the preceding results due to Jeanjean [24] and Jeanjean-Lu [26, 28]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19751 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Lagrangian approach for prescribed mass solutions of cubic-quintic Schrödinger equations and $L^2$-supercritical problems Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga Analysis of PDEs 35J20, 35A01, 35B38, 58E05, 35J91, 35Q40, 35Q55, 47J30, 49J35 We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$): $$ (*)_m \quad - Δu + μu = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb R}^N} u^2\, dx = m,$$ where $g(s) \in C({\mathbb R},{\mathbb R})$, $m > 0$ and $μ\in {\mathbb R}$ is an unknown Lagrangian multiplier. We take an approach using a Lagrangian formulation of $(*)_m$: $$J_m(μ,u)={1\over 2}\int_{{\mathbb R}^N} |\nabla u|^2\,dx -\int_{{\mathbb R}^N} G(u)\,dx +μ\left({1\over 2}\int_{{\mathbb R}^N} u^2\, dx-m\right) \in C^1((0,\infty)\times H_r^1({\mathbb R}^N), {\mathbb R})$$ and we give new general existence results through the function: $$ b_m:\, (0,\infty) \to {\mathbb R};\ μ\mapsto \text{Mountain Pass minimax value for}\ (u\mapsto J_m(μ,u)).$$ We will show the existence of solutions of $(*)_m$ related to local minima and local maxima of $b_m(μ)$. As applications, we study cubic-quintic type equations and $L^2$-supercritical problems. In particular, when $N=2,3$, we show new existence results of normalized solutions without assuming global Ambrosetti-Rabinowitz type conditions, which partially improve the preceding results due to Jeanjean [24] and Jeanjean-Lu [26, 28]. |
| title | A Lagrangian approach for prescribed mass solutions of cubic-quintic Schrödinger equations and $L^2$-supercritical problems |
| topic | Analysis of PDEs 35J20, 35A01, 35B38, 58E05, 35J91, 35Q40, 35Q55, 47J30, 49J35 |
| url | https://arxiv.org/abs/2602.19751 |