Fractional Schrodinger equations with new conditions
In this article, we study the nonlinear fractional Schrodinger equation $$\displaylines{ (-\Delta)^{\alpha}u+ V(x)u= f(x,u)\cr u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), }$$ where $(-\Delta)^{\alpha}(\alpha \in (0, 1))$ stands for the fractional Laplacian of order $\alpha$, $x\in \mathbb{R...
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Texas State University
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doaj-d1c5a0658acc4620a421931c521904fb2020-11-25T02:35:52ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912018-01-01201805,112Fractional Schrodinger equations with new conditionsAbderrazek Benhassine0 Higher Inst. for Informatics, Monastir, Tunisia In this article, we study the nonlinear fractional Schrodinger equation $$\displaylines{ (-\Delta)^{\alpha}u+ V(x)u= f(x,u)\cr u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), }$$ where $(-\Delta)^{\alpha}(\alpha \in (0, 1))$ stands for the fractional Laplacian of order $\alpha$, $x\in \mathbb{R}^{n}$, $V\in C(\mathbb{R}^{n},\mathbb{R})$ may change sign and f is only locally defined near the origin with respect to u. Under some new assumptions on V and f, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result.http://ejde.math.txstate.edu/Volumes/2018/05/abstr.htmlFractional Schrodinger equationscritical point theorysymmetric mountain pass theorem |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Abderrazek Benhassine |
spellingShingle |
Abderrazek Benhassine Fractional Schrodinger equations with new conditions Electronic Journal of Differential Equations Fractional Schrodinger equations critical point theory symmetric mountain pass theorem |
author_facet |
Abderrazek Benhassine |
author_sort |
Abderrazek Benhassine |
title |
Fractional Schrodinger equations with new conditions |
title_short |
Fractional Schrodinger equations with new conditions |
title_full |
Fractional Schrodinger equations with new conditions |
title_fullStr |
Fractional Schrodinger equations with new conditions |
title_full_unstemmed |
Fractional Schrodinger equations with new conditions |
title_sort |
fractional schrodinger equations with new conditions |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2018-01-01 |
description |
In this article, we study the nonlinear fractional Schrodinger equation
$$\displaylines{
(-\Delta)^{\alpha}u+ V(x)u= f(x,u)\cr
u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}),
}$$
where $(-\Delta)^{\alpha}(\alpha \in (0, 1))$ stands for the
fractional Laplacian of order $\alpha$, $x\in \mathbb{R}^{n}$,
$V\in C(\mathbb{R}^{n},\mathbb{R})$ may change sign and f is
only locally defined near the origin with respect to u.
Under some new assumptions on V and f, we show that the above system
has infinitely many solutions near the origin. Some examples are also
given to illustrate our main theoretical result. |
topic |
Fractional Schrodinger equations critical point theory symmetric mountain pass theorem |
url |
http://ejde.math.txstate.edu/Volumes/2018/05/abstr.html |
work_keys_str_mv |
AT abderrazekbenhassine fractionalschrodingerequationswithnewconditions |
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1724802949416222720 |