Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities
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| Format: | Preprint |
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2024
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| author | Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo |
| author_facet | Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo |
| contents | We study the existence of solutions $(\underline u,λ_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $ψ_0$ close to $\underline u$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05194 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo Analysis of PDEs 35Q55, 35Q40, 35J20 We study the existence of solutions $(\underline u,λ_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with $ψ_0$ close to $\underline u$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution. |
| title | Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities |
| topic | Analysis of PDEs 35Q55, 35Q40, 35J20 |
| url | https://arxiv.org/abs/2405.05194 |