0000000000955226
AUTHOR
Chunhua Wang
A singularly perturbed Kirchhoff problem revisited
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.
Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry
Let $1p^{2}+p,a(0)>0$ and $\Omega$ satisfies some geometry conditions if $0\in\partial\Omega$, say, all the principle curvatures of $\partial\Omega$ at $0$ are negative, then the above problem has infinitely many solutions.