A singularly perturbed Kirchhoff problem revisited. (English) Zbl 1426.35018
Summary: In this paper, we revisit the singularly perturbation problem \[-(\epsilon^2 a + \epsilon b \int\limits_{\mathbb{R}^3} | \nabla u |^2) \Delta u + V(x) u = |u|^{p - 1} u \quad\text{in } \mathbb{R}^3,\] where \(a, b, \epsilon > 0, 1 < p < 5\) are constants and \(V\) is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem \[-(a + b \int\limits_{\mathbb{R}^3} | \nabla u |^2) \Delta u + u = |u|^{p-1} u \quad\text{in } \mathbb{R}^3.\] Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for \(\epsilon > 0\) sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35J20 | Variational methods for second-order elliptic equations |
| 35R09 | Integro-partial differential equations |
| 35J62 | Quasilinear elliptic equations |
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