Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909127652933632 |
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| author | Deng, Ting Squassina, Marco Zhang, Jianjun Zhong, Xuexiu |
| author_facet | Deng, Ting Squassina, Marco Zhang, Jianjun Zhong, Xuexiu |
| contents | We are concerned with solutions of the following quasilinear Schrödinger equations \begin{eqnarray*} -{\mathrm{div}}\left(φ^{2}(u) \nabla u\right)+φ(u) φ^{\prime}(u)|\nabla u|^{2}+λu=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $λ\in \mathbb{R}$ appears as the Lagrange multiplier and $φ\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} \right )$ is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity Deng, Ting Squassina, Marco Zhang, Jianjun Zhong, Xuexiu Analysis of PDEs 35J62, 35B40, 35B09 We are concerned with solutions of the following quasilinear Schrödinger equations \begin{eqnarray*} -{\mathrm{div}}\left(φ^{2}(u) \nabla u\right)+φ(u) φ^{\prime}(u)|\nabla u|^{2}+λu=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $λ\in \mathbb{R}$ appears as the Lagrange multiplier and $φ\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} \right )$ is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case. |
| title | Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity |
| topic | Analysis of PDEs 35J62, 35B40, 35B09 |
| url | https://arxiv.org/abs/2403.01338 |