Positive ground state solutions for generalized quasilinear Schrödinger equations with critical growth

2023/03/20 by Xin Meng, Meng, Xin, Shuguan Ji +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2303.10830

openalex publication_date 2023/03/20 · openalex created_date 2023/03/23 · openalex updated_date 2026/07/28

Abstract

This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrödinger equations in ℝ^N with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space H¹(ℝ^N). We use the method of Nehari manifold for the modified equation, establish the minimax characterization, then obtain each Palais-Smale sequence of the associated energy functional is bounded. By combining Lions's concentration-compactness lemma together with some classical arguments developed by Brézis and Nirenberg \citebn, we establish that the bounded Palais-Smale sequence has a nonvanishing behaviour. Finally, we obtain the existence of a positive ground state solution under some appropriate assumptions. Our results extend and generalize some known results.

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