Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929217387626496 |
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| author | Zhang, Xue Squassina, Marco Zhang, Jianjun |
| author_facet | Zhang, Xue Squassina, Marco Zhang, Jianjun |
| contents | We get multiplicity of normalized solutions for the fractional Schrödinger equation
$$
(-Δ)^su+V(\varepsilon x)u=λu+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$},
\qquad\int_{\mathbb{R}^N}|u|^2dx=a,
$$ where $(-Δ)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $λ\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00621 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials Zhang, Xue Squassina, Marco Zhang, Jianjun Analysis of PDEs 35A15 35B33 35Q55 We get multiplicity of normalized solutions for the fractional Schrödinger equation $$ (-Δ)^su+V(\varepsilon x)u=λu+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-Δ)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $λ\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough. |
| title | Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials |
| topic | Analysis of PDEs 35A15 35B33 35Q55 |
| url | https://arxiv.org/abs/2401.00621 |