Qualitative analysis of multi-peak solutions for Nonlinear Schrödinger equations with nearly critical Sobolev exponents

2025/12/21 by Zhongyuan Liu, Liu, Zhongyuan, Shuying Tian +5
Mathematics · Physics and Astronomy · #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.2512.18663

Abstract

In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schrödinger equations -Δu+V(x)u= u^p-ε, ugt;0, in ℝ^N, where V(x) is a nonnegative continuous function, ε>0, p=(N+2)/(N-2), N≥6. The existence of multi-peak solutions has been obtained by Cao et al. (Calc. Var. Partial Differential Equations, 64: 139, 2025). The main objective in this paper is to establish the local uniqueness and Morse index of the multi-peak solutions in \citeCLl1 provided that V(x) possesses k non-degenerate critical points by using the blow-up analysis based on Pohozaev identities.

Citations

• Non-degeneracy and existence of new solutions for the Schrödinger equations
• The number of positive solutions to the Brezis-Nirenberg problem
• Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Related

• Positive multi-peak solutions for a logarithmic Schrodinger equation
• Multiplicity of positive solutions for a quasilinear Schr "odinger\n equation with an almost critical nonlinearity
• Multiple solutions for a quasilinear Schrödinger equation on ℝ^N
• Studies of normalized solutions to Schrödinger equations with Sobolev critical exponent and combined nonlinearities
• Positive solutions to multi-critical Schrödinger equations