Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities
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2025
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| _version_ | 1866909660275015680 |
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| author | Yu, Shubin Yang, Chen Tang, Chun-Lei |
| author_facet | Yu, Shubin Yang, Chen Tang, Chun-Lei |
| contents | We look for normalized solutions to the nonlinear Schrödinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-Δ)^{s_{1}} u+(-Δ)^{s_{2}} u=λu+μ|u|^{q-2}u+|u|^{p-2}u \ \text{in}\;{\mathbb{R}^{N}}, \\[0.1cm] \int_{\mathbb{R}^{N}}|u|^2\mathrm dx=a^2, \end{array} \right. $$ where $N\geq 2,\;0<s_2<s_1<1, μ>0$ and $λ\in\mathbb R$ appears as an unknown Lagrange multiplier. We mainly focus on some special cases, including fractional Sobolev subcritical or critical exponent. More precisely, for $2<q<2+\frac{4s_2}{N}<2+\frac{4s_1}{N}<p<2_{s_1}^{\ast}:=\frac{2N}{N-2s_1}$, we prove that the above problem has at least two solutions: a ground state with negative energy and a solution of mountain pass type with positive energy. For $2<q<2+\frac{4s_2}{N}$ and $p=2_{s_1}^{\ast}$, we also obtain the existence of ground states. Our results extend some previous ones of Chergui et al. (Calc. Var. Partial Differ. Equ., 2023) and Luo et al. (Adv. Nonlinear Stud., 2022). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities Yu, Shubin Yang, Chen Tang, Chun-Lei Analysis of PDEs 35R11, 35A15, 35B33, 35J60 We look for normalized solutions to the nonlinear Schrödinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-Δ)^{s_{1}} u+(-Δ)^{s_{2}} u=λu+μ|u|^{q-2}u+|u|^{p-2}u \ \text{in}\;{\mathbb{R}^{N}}, \\[0.1cm] \int_{\mathbb{R}^{N}}|u|^2\mathrm dx=a^2, \end{array} \right. $$ where $N\geq 2,\;0<s_2<s_1<1, μ>0$ and $λ\in\mathbb R$ appears as an unknown Lagrange multiplier. We mainly focus on some special cases, including fractional Sobolev subcritical or critical exponent. More precisely, for $2<q<2+\frac{4s_2}{N}<2+\frac{4s_1}{N}<p<2_{s_1}^{\ast}:=\frac{2N}{N-2s_1}$, we prove that the above problem has at least two solutions: a ground state with negative energy and a solution of mountain pass type with positive energy. For $2<q<2+\frac{4s_2}{N}$ and $p=2_{s_1}^{\ast}$, we also obtain the existence of ground states. Our results extend some previous ones of Chergui et al. (Calc. Var. Partial Differ. Equ., 2023) and Luo et al. (Adv. Nonlinear Stud., 2022). |
| title | Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities |
| topic | Analysis of PDEs 35R11, 35A15, 35B33, 35J60 |
| url | https://arxiv.org/abs/2506.20943 |