Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity

2017/11/29 by An, Xiaoming, Peng, Shuangjie, Xie, Chaodong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.10655

Abstract

We study the following fractional Schrödinger equation ε^2s(-Δ)^s u + Vu = |u|^{p - 2}u, x∈ ℝ^N. We show that if the external potential V∈ C(ℝ^N;[0,∞)) has a local minimum and p∈ (2 + 2s/(N - 2s), 2^*_s), where 2^*_s=2N/(N-2s), N≥ 2s, the problem has a family of solutions concentrating at the local minimum of V provided that \liminf_{|x|→ ∞}V(x)|x|^2s > 0. The proof is based on variational methods and penalized technique. \textbf Key words: fractional Schrödinger; vanishing potential; penalized technique; variational methods.

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