Normalized solutions of mass supercritical Schrodinger-Poisson equation with potential
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916521114075136 |
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| author | Peng, Xueqin Rizzi, Matteo |
| author_facet | Peng, Xueqin Rizzi, Matteo |
| contents | In this paper we prove the existence of normalized solutions $(λ,u)\subset (0,\infty)\times H^1(\mathbb{R}^3)$ to the following Schrödinger-Poisson equation $$ \begin{cases} -Δu+V(x)u+λu+(|x|^{-1}\ast u^2)u=|u|^{p-2}u&\text{in}\,\mathbb{R}^{3},\\ u>0,\quad \int_{\mathbb{R}^{3}}u^2dx=a^2, \end{cases} $$ where $a>0$ is fixed, $p\in(\frac{10}{3},6)$ is a given exponent and the potential $V$ satisfies some suitable conditions. Since the $L^2(\mathbb{R}^3)$-norm of $u$ is fixed, $λ$ appears as a Lagrange multiplier. For $V(x)\geq0$, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al. For $V(x)\leq0$, we directly construct an entire mountain-pass solution with positive energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of mass supercritical Schrodinger-Poisson equation with potential Peng, Xueqin Rizzi, Matteo Analysis of PDEs In this paper we prove the existence of normalized solutions $(λ,u)\subset (0,\infty)\times H^1(\mathbb{R}^3)$ to the following Schrödinger-Poisson equation $$ \begin{cases} -Δu+V(x)u+λu+(|x|^{-1}\ast u^2)u=|u|^{p-2}u&\text{in}\,\mathbb{R}^{3},\\ u>0,\quad \int_{\mathbb{R}^{3}}u^2dx=a^2, \end{cases} $$ where $a>0$ is fixed, $p\in(\frac{10}{3},6)$ is a given exponent and the potential $V$ satisfies some suitable conditions. Since the $L^2(\mathbb{R}^3)$-norm of $u$ is fixed, $λ$ appears as a Lagrange multiplier. For $V(x)\geq0$, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al. For $V(x)\leq0$, we directly construct an entire mountain-pass solution with positive energy. |
| title | Normalized solutions of mass supercritical Schrodinger-Poisson equation with potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.07277 |