A singularly perturbed Kirchhoff problem revisited
By Gongbao Li, Shuangjie Peng, Chang-Lin Xiang. Journal of Differential Equations, 2016
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Abstract In this paper, we revisit the singularly perturbation problem (0.1) − ( ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = | u | p − 1 u in R 3 , where a , b , ϵ > 0 , 1 p 5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + u = | u | p − 1 u in R 3 . Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for ϵ > 0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/662945894afe82de7d26a9ac668967d55a5371d0
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In Applied Mathematics
In Computational Theory and Mathematics
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Citations in the corpus
- Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent (2021, background): “Following [22, 35], u0 is unique since u0 > 0.”