Normalized solutions to mass supercritical Schrödinger equations with radial potentials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910106886602752 |
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| author | Carrillo, P. Jeanjean, L. |
| author_facet | Carrillo, P. Jeanjean, L. |
| contents | We study the stationary nonlinear Schrödinger equation \begin{equation}-Δu+V(x)u+λu=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the $L^2$-supercritical regime, we show the existence of an explicit $μ_0 >0$ such that, for any $μ\in (0, μ_0)$, the equation admits two solutions having $L^2$ norm $μ$. The potential $V$ is not assumed to have a sign, nor a specific behavior at infinity and only a low regularity is required. Our proof relies on the use of Morse type information, on some spectral arguments, and on a blow-up analysis developed in a radial setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06390 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to mass supercritical Schrödinger equations with radial potentials Carrillo, P. Jeanjean, L. Analysis of PDEs We study the stationary nonlinear Schrödinger equation \begin{equation}-Δu+V(x)u+λu=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the $L^2$-supercritical regime, we show the existence of an explicit $μ_0 >0$ such that, for any $μ\in (0, μ_0)$, the equation admits two solutions having $L^2$ norm $μ$. The potential $V$ is not assumed to have a sign, nor a specific behavior at infinity and only a low regularity is required. Our proof relies on the use of Morse type information, on some spectral arguments, and on a blow-up analysis developed in a radial setting. |
| title | Normalized solutions to mass supercritical Schrödinger equations with radial potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.06390 |