Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916078382219264 |
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| author | Shen, Liejun Squassina, Marco |
| author_facet | Shen, Liejun Squassina, Marco |
| contents | We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth
$$ -Δu +λu+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad
\partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad
\partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad
\int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $λ\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case Shen, Liejun Squassina, Marco Analysis of PDEs 35J20, 35J61, 35B06 We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth $$ -Δu +λu+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $λ\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution. |
| title | Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case |
| topic | Analysis of PDEs 35J20, 35J61, 35B06 |
| url | https://arxiv.org/abs/2401.00623 |