Multi-bump Solutions for a Strongly Indefinite Semilinear Schrödinger Equation Without Symmetry or convexity Assumptions

2008/05/18 by Shaowei Chen, Chen, Shaowei
Computer Science · Mathematics · #35J20 #35J70 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.0805.2711

openalex publication_date 2008/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the following semilinear Schrödinger equation with periodic coefficient: -\triangle u +V(x)u=f(x,u), u∈ H¹(ℝ^N). The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term f(x,t) satisfies some superlinear growth conditions and need not be odd or increasing strictly in t. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are isolated, then we can show that this equation has infinitely many m-bump solutions for any positive integer m≥ 2.

Citations

• Periodic nonlinear Schrödinger equation with application to photonic crystals

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