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

Abstract:
In this article we consider the multiplicity and concentration behavior of positive solutions for the fractional nonlinear Schrodinger equation
$$\displaylines{ \varepsilon^{2s}(-\Delta)^{s}u + V(x)u= u^{2^*_s-1} + f(u) , \quad x\in\mathbb{R}^N,\cr u\in H^{s}(\mathbb{R}^N), \quad u(x) > 0, }$$
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.

Show me the PDF file (346 KB), TEX file for this article.

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

Return to the EJDE web page