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

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.

Keywords

• Fractional Schrodinger equations
• multiplicity of solutions
• critical growth
• variational method