Normalized solutions to Schrödinger equations in the strongly sublinear regime
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910468190240768 |
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| author | Mederski, Jarosław Schino, Jacopo |
| author_facet | Mederski, Jarosław Schino, Jacopo |
| contents | We look for solutions to the Schrödinger equation \[ -Δu + λu = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = ρ^2$, with $N\ge2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim_{s\to0}g(s)/s = -\infty$, which includes the case \[ g(s) = αs \ln s^2 + μ|s|^{p-2} s \] with $α> 0$ and $μ\in \mathbb{R}$, $2 < p \le 2^*$ properly chosen. We consider a family of approximating problems that can be set in $H^1(\mathbb{R}^N)$ and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1(\mathbb{R}^N)$, we prove the existence of infinitely many solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06015 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to Schrödinger equations in the strongly sublinear regime Mederski, Jarosław Schino, Jacopo Analysis of PDEs 34J10, 35J20, 35J61, 58E05 We look for solutions to the Schrödinger equation \[ -Δu + λu = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = ρ^2$, with $N\ge2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim_{s\to0}g(s)/s = -\infty$, which includes the case \[ g(s) = αs \ln s^2 + μ|s|^{p-2} s \] with $α> 0$ and $μ\in \mathbb{R}$, $2 < p \le 2^*$ properly chosen. We consider a family of approximating problems that can be set in $H^1(\mathbb{R}^N)$ and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1(\mathbb{R}^N)$, we prove the existence of infinitely many solutions. |
| title | Normalized solutions to Schrödinger equations in the strongly sublinear regime |
| topic | Analysis of PDEs 34J10, 35J20, 35J61, 58E05 |
| url | https://arxiv.org/abs/2306.06015 |