On nonlinear fractional Schrödinger equations with indefinite and Hardy potentials

Article type: Research Article

Authors: Mi, Heilong^a | Zhang, Wen^{a; b; d} | Liao, Fangfang^{c; *}

Affiliations: [a] College of Science, Hunan University of Technology and Business, 410205 Changsha, Hunan, China | [b] Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, 410205 Changsha, Hunan, China | [c] School of Mathematics and Information, Xiangnan University, Chenzhou, 423000, Hunan, China | [d] Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Correspondence: [*] Corresponding author. E-mail: liaofangfang1981@126.com.

Abstract: This paper is concerned with a class of fractional Schrödinger equation with Hardy potential (−Δ)su+V(x)u−κ|x|2su=f(x,u),x∈RN, where s∈(0,1) and κ⩾0 is a parameter. Under some suitable conditions on the potential V and the nonlinearity f, we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ. Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0.

Keywords: Fractional Schrödinger equation, Hardy potential, Strongly indefinite functional, Ground state solutions

DOI: 10.3233/ASY-221793

Journal: Asymptotic Analysis, vol. 132, no. 3-4, pp. 305-330, 2023