Ground state and multiple normalized solutions of quasilinear Schrödinger equations in the L2-supercritical case and the Sobolev critical case

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 27; no. 3; p. 84
• Main Authors: Gao, Qiang, Zhang, Xiaoyan
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.09.2025
• Subjects: Asymptotic properties; Critical point; Ground state; Lagrange multiplier; Perturbation methods; Schrodinger equation
• ISSN: 2730-5422
• DOI: 10.1007/s11784-025-01233-z

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation -Δu-uΔu2+λu=|u|p-2uinRN,N=3,4where λ appears as a Lagrange multiplier and p∈(4+4N,2·2∗]. The solutions correspond to critical points of the energy functional subject to the L2-norm constraint ∫RN|u|2dx=a2>0. In the Sobolev critical case p=2·2∗, the energy functional has no critical point. As for L2-supercritical case p∈(4+4N,2·2∗): on the one hand, taking into account Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions for the non-radial case; on the other hand, we get the existence of infinitely many normalized solutions in Hr1(RN). Moreover, our results cover several relevant existing results. And in the end, we get the asymptotic properties of energy as a tends to +∞ and a tends to 0+.