Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities
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2025
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| author | He, Qihan Wang, Hao |
| author_facet | He, Qihan Wang, Hao |
| contents | In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -Δu-Δ(|u|^2)u+λu=|u|^{p-2}u+τ|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $λ\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,τ$ are all positive constants. We are concerned about the mass-mixed case $2<q<2+\frac{4}{N}$ and $4+\frac{4}{N}<p<2\cdot2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq3$, while $2^*:=\infty$ for $N=1,2$. We show the existence of normalized ground state solution and normalized solution of mountain pass type. Our results can be regarded as a supplement to Lu et al. ( Proc. Edinb. Math. Soc., 2024) and Jeanjean et al. ( arXiv:2501.03845). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities He, Qihan Wang, Hao Analysis of PDEs In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -Δu-Δ(|u|^2)u+λu=|u|^{p-2}u+τ|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $λ\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,τ$ are all positive constants. We are concerned about the mass-mixed case $2<q<2+\frac{4}{N}$ and $4+\frac{4}{N}<p<2\cdot2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq3$, while $2^*:=\infty$ for $N=1,2$. We show the existence of normalized ground state solution and normalized solution of mountain pass type. Our results can be regarded as a supplement to Lu et al. ( Proc. Edinb. Math. Soc., 2024) and Jeanjean et al. ( arXiv:2501.03845). |
| title | Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.00375 |