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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.02049 |
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| _version_ | 1866912218872807424 |
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| author | Liu, Jianlun Sun, Hong-Rui Zhang, Ziheng |
| author_facet | Liu, Jianlun Sun, Hong-Rui Zhang, Ziheng |
| contents | This paper is concerned with the following focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases}
Δ^2u-Δu-λu-μ|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm]
\int_{\mathbb{R}^N} u^2 dx = c, \end{cases}
$$ where $N \geq 5$, $μ,c>0$, $2<p<4^*:=\frac{2N}{N-4}$ and $λ\in \mathbb{R}$ is a Lagrange multiplier. For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concentration-compactness principle and the genus theory presented by C.O. Alves et al. (Arxiv, (2021), doi: 2103.07940v2). Compared to the results of C.O. Alves et al. we obtain a more general result after removing the further assumptions given in (3.2) of their paper. In the case of $L^2$-supercritical perturbation ($2+\frac{8}{N}<p<4^*$), we explore the existence results of normalized solutions by applying the constrained variational methods and the mountain pass theorem. Moreover, we propose a novel method to address the effects of the dispersion term $Δu$. This approach allows us to extend the recent results obtained by X. Chang et al. (Arxiv, (2023), doi: 2305.00327v1) to the mixed dispersion situation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to focusing Sobolev critical biharmonic Schrödinger equation with mixed dispersion Liu, Jianlun Sun, Hong-Rui Zhang, Ziheng Analysis of PDEs This paper is concerned with the following focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases} Δ^2u-Δu-λu-μ|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $N \geq 5$, $μ,c>0$, $2<p<4^*:=\frac{2N}{N-4}$ and $λ\in \mathbb{R}$ is a Lagrange multiplier. For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concentration-compactness principle and the genus theory presented by C.O. Alves et al. (Arxiv, (2021), doi: 2103.07940v2). Compared to the results of C.O. Alves et al. we obtain a more general result after removing the further assumptions given in (3.2) of their paper. In the case of $L^2$-supercritical perturbation ($2+\frac{8}{N}<p<4^*$), we explore the existence results of normalized solutions by applying the constrained variational methods and the mountain pass theorem. Moreover, we propose a novel method to address the effects of the dispersion term $Δu$. This approach allows us to extend the recent results obtained by X. Chang et al. (Arxiv, (2023), doi: 2305.00327v1) to the mixed dispersion situation. |
| title | Normalized solutions to focusing Sobolev critical biharmonic Schrödinger equation with mixed dispersion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.02049 |