Semiclassical states for fractional logarithmic Schr\"{o}dinger equations
In this paper, we consider the following fractional logarithmic Schr\"odinger equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penali...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
18.06.2020
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| Subjects | |
| Online Access | Get full text |
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| Summary: | In this paper, we consider the following fractional logarithmic Schr\"odinger
equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log |u|^2\ \
\text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in
C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we
show that the problem has a positive solution $u_{\varepsilon}$ concentrating
at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on
decay rates of $V$, especially it can be compactly supported. |
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| DOI: | 10.48550/arxiv.2006.10338 |