Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials

ABSTRACT

We study the following fractional Schrödinger equation ${ϵ}^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$where $s\in (0,1)$, N>2s, p>1 is subcritical and $V\left(x\right)$ is a nonnegative continuous potential. We use penalized technique and variational methods to show that the problem has a family of solutions concentrating at a local minimum of $V\left(x\right)$ as $ϵ\to 0$ provided that $\frac{2s}{N-2}+2<p<\frac{2N}{N-2s}$. The novelty is that V can decay arbitrarily or even be compactly supported.

Keywords:

• Fractional Schrödinger
• local minimum
• compactly supported potential
• variational
• penalized

2010 Mathematics Subject Classification:

• analysis and PDE

Acknowledgments

L. Duan is partially supported by China Scholarship Council (No.201906770024). Y. Peng is supported by the grant NSFC (No. 11501143).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

L. Duan is partially supported by China Scholarship Council (No. 201906770024). Y. Peng is supported by the grant National Natural Science Foundation of China (NSFC) (No. 11501143).