Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913229026885632 |
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| author | Ambrosio, Vincenzo |
| author_facet | Ambrosio, Vincenzo |
| contents | In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-Δ+m^{2})^{s}$ is the fractional relativistic Schrödinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_13632 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth Ambrosio, Vincenzo Analysis of PDEs In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-Δ+m^{2})^{s}$ is the fractional relativistic Schrödinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$. |
| title | Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2105.13632 |