Electron. J. Differential Equations, Vol. 2019 (2019), No. 24, pp. 1-22.

Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth

Xudong Shang, Jihui Zhang

In this article we consider the multiplicity and concentration behavior of positive solutions for the fractional nonlinear Schrodinger equation
 \varepsilon^{2s}(-\Delta)^{s}u  + V(x)u= u^{2^*_s-1} + f(u) ,
 \quad  x\in\mathbb{R}^N,\cr
 u\in H^{s}(\mathbb{R}^N), \quad u(x) > 0,
where $\varepsilon$ is a positive parameter, $s \in (0,1)$, N >2s and $2^*_s= \frac{2N}{N-2s}$ is the fractional critical exponent, and f is a $\mathcal{C}^{1}$ function satisfying suitable assumptions. We assume that the potential $V(x) \in \mathcal{C}(\mathbb{R}^N)$ satisfies $\inf_{\mathbb{R}^N} V(x)>0$, and that there exits k points $x^j \in \mathbb{R}^N$ such that for each j=1,...,k, $V(x^j)$ are strictly global minimum. By using the variational method, we show that there are at least k positive solutions for a small $\varepsilon$>0. Moreover, we establish the concentration property of solutions as $\varepsilon$ tends to zero.

Submitted March 10, 2017. Published February 12, 2019.
Math Subject Classifications: 35A15, 58E05.
Key Words: Fractional Schrodinger equations; multiplicity of solutions; critical growth; variational method.

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Xudong Shang
School of Mathematics
Nanjing Normal University
Taizhou College
225300, Jiangsu, China
email: xudong-shang@163.com
Jihui Zhang
Jiangsu Key Laboratory for NSLSCS
School of Mathematical Sciences
Nanjing Normal University
Nanjing 210023, China
email: zhangjihui@njnu.edu.cn

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