Low and high energy solutions of oscillatory non-autonomous Schrödinger equations with magnetic field

Article type: Research Article

Authors: Zhang, Youpei^{a; b} | Tang, Xianhua^{a; *}

Affiliations: [a] School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China. E-mails: [email protected], [email protected] | [b] Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova 200585, Romania

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We are concerned with the mathematical and asymptotic analysis of solutions to the following nonlinear problem −ΔAu=λβ(x)|u|qu+f(|u|)uin Ω,u=0on ∂Ω, where ΔAu is the magnetic Laplace operator, Ω⊂RN is a smooth bounded domain, A:Ω↦RN is the magnetic potential, u:Ω↦C, λ is a real parameter, β∈L∞(Ω,R) is an indefinite potential, q is nonnegative, and f:[0,+∞)↦R is a reaction that oscillates either in a neighborhood of the origin or at infinity. We analyze two distinct cases, in close relationship with the oscillatory growth of the reaction. Additionally, we give asymptotic estimates for the norm of the solutions in related function spaces.

Keywords: Magnetic Laplace operator, blow-up, low- and high-energy solutions, oscillation, variational methods

DOI: 10.3233/ASY-201629

Journal: Asymptotic Analysis, vol. 123, no. 3-4, pp. 203-236, 2021