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In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth − a + b ∫ R 3 | ∇ u | 2 Δ u = u 5 , u > 0 in R 3 , where a , b > 0 are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equations. [ABSTRACT FROM AUTHOR]

We study L ∞ -variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]