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Volume 35, Issue 3
Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type

Qing Guo & Lixiu Duan

DOI: 10.4208/jpde.v35.n3.5

J. Part. Diff. Eq., 35 (2022), pp. 259-280.

Published online: 2022-06

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  • Abstract

This paper, we study the multiplicity of solutions for the fractional Schrödinger equation

\begin{equation*}(-\Delta)^su+V(x)u=u^p,\ \ u>0,\ \ x\in\mathbb{R}^N,\ \ u\in H^s(\mathbb{R}^N),\end{equation*}

with $s\in(0,1),\  N\geq3,\  p\in(1,\frac{2N}{N-2s}-1)$ and $\lim_{|y|\rightarrow+\infty}V(y)>0$. By assuming suitable decay property of the radial potential $V(y)=V(|y|)$, we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium,  simultaneously. Another difficulty lies in the non-local property of the operator $(-\Delta)^s$ and the algebraic decay involving the approximation solutions make the estimates become more subtle.

  • Keywords

Fractional Schrödinger equations, infinitely many solutions, reduction method.

  • AMS Subject Headings

35J20, 35J60, 35B25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

guoqing0117@163.com (Qing Guo)

dlx18801183265@163.com (Lixiu Duan)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-35-259, author = {Guo , Qing and Duan , Lixiu}, title = {Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {35}, number = {3}, pages = {259--280}, abstract = {

This paper, we study the multiplicity of solutions for the fractional Schrödinger equation

\begin{equation*}(-\Delta)^su+V(x)u=u^p,\ \ u>0,\ \ x\in\mathbb{R}^N,\ \ u\in H^s(\mathbb{R}^N),\end{equation*}

with $s\in(0,1),\  N\geq3,\  p\in(1,\frac{2N}{N-2s}-1)$ and $\lim_{|y|\rightarrow+\infty}V(y)>0$. By assuming suitable decay property of the radial potential $V(y)=V(|y|)$, we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium,  simultaneously. Another difficulty lies in the non-local property of the operator $(-\Delta)^s$ and the algebraic decay involving the approximation solutions make the estimates become more subtle.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n3.5}, url = {http://global-sci.org/intro/article_detail/jpde/20775.html} }
TY - JOUR T1 - Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type AU - Guo , Qing AU - Duan , Lixiu JO - Journal of Partial Differential Equations VL - 3 SP - 259 EP - 280 PY - 2022 DA - 2022/06 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n3.5 UR - https://global-sci.org/intro/article_detail/jpde/20775.html KW - Fractional Schrödinger equations, infinitely many solutions, reduction method. AB -

This paper, we study the multiplicity of solutions for the fractional Schrödinger equation

\begin{equation*}(-\Delta)^su+V(x)u=u^p,\ \ u>0,\ \ x\in\mathbb{R}^N,\ \ u\in H^s(\mathbb{R}^N),\end{equation*}

with $s\in(0,1),\  N\geq3,\  p\in(1,\frac{2N}{N-2s}-1)$ and $\lim_{|y|\rightarrow+\infty}V(y)>0$. By assuming suitable decay property of the radial potential $V(y)=V(|y|)$, we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium,  simultaneously. Another difficulty lies in the non-local property of the operator $(-\Delta)^s$ and the algebraic decay involving the approximation solutions make the estimates become more subtle.

Qing Guo & Lixiu Duan. (2022). Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type. Journal of Partial Differential Equations. 35 (3). 259-280. doi:10.4208/jpde.v35.n3.5
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