Existence and Concentration Results for the General Kirchhoff Type Equations

2022/06/08 by Yinbin Deng, Deng, Yinbin, Shuai, Wei +2
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.03663

openalex publication_date 2022/06/08 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28

Abstract

We consider the following singularly perturbed Kirchhoff type equations -ε² M(ε^2-N∫_\R^N|∇ u|² dx)Δu +V(x)u=|u|^p-2u~\hboxin~\R^N, u∈ H¹(\R^N),N≥ 1, where M∈ C([0,∞)) and V∈ C(\R^N) are given functions. Under very mild assumptions on M, we prove the existence of single-peak or multi-peak solution u_ε for above problem, concentrating around topologically stable critical points of V, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. in 2014 [ARMA,213].

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