Electron. J. Differential Equations, Vol. 2018 (2018), No. 05, pp. 1-12.

Fractional Schrodinger equations with new conditions

Abderrazek Benhassine

In this article, we study the nonlinear fractional Schrodinger equation
 (-\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.

Submitted July 28, 2017. Published January 4, 2018.
Math Subject Classifications: 35B38, 35G99.
Key Words: Fractional Schrodinger equations; critical point theory; symmetric mountain pass theorem.

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Abderrazek Benhassine
Department of Mathematics
Higher Institut for Informatics and Mathematics
5000, Monastir, Tunisia
email: ab.hassine@yahoo.com

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