Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity

Article type: Research Article

Authors: Deng, Ting^a | Squassina, Marco^{b; *} | Zhang, Jianjun^a | Zhong, Xuexiu^c

Affiliations: [a] College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, PR China | [b] Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, Brescia, Italy | [c] South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, P. R. China

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

Abstract: We are concerned with solutions of the following quasilinear Schrödinger equations −div(φ2(u)∇u)+φ(u)φ′(u)|∇u|2+λu=f(u),x∈RN with prescribed mass ∫RNu2dx=c, where N⩾3, c>0, λ∈R appears as the Lagrange multiplier and φ∈C1(R,R+). The nonlinearity f∈C(R,R) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.

Keywords: Quasilinear Schrödinger equations, normalized solutions, mass critical exponent

DOI: 10.3233/ASY-241908

Journal: Asymptotic Analysis, vol. 140, no. 1-2, pp. 5-24, 2024