| import Mathlib | |
| namespace OAI | |
| namespace HenonLaneEmden | |
| abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n) | |
| structure IsSolution (n : ℕ) (p q A B : ℝ) (u v : Space n → ℝ) : Prop where | |
| continuous_u : Continuous u | |
| continuous_v : Continuous v | |
| regular_u : ContDiffOn ℝ 2 u ({0}ᶜ : Set (Space n)) | |
| regular_v : ContDiffOn ℝ 2 v ({0}ᶜ : Set (Space n)) | |
| positive_u : ∀ x, 0 < u x | |
| positive_v : ∀ x, 0 < v x | |
| equation_u : ∀ x, x ≠ 0 → | |
| -(Laplacian.laplacian u) x = Real.rpow ‖x‖ A * Real.rpow (v x) p | |
| equation_v : ∀ x, x ≠ 0 → | |
| -(Laplacian.laplacian v) x = Real.rpow ‖x‖ B * Real.rpow (u x) q | |
| def Subcritical (n : ℕ) (p q A B : ℝ) : Prop := | |
| ((n : ℝ) + A) / (p + 1) + ((n : ℝ) + B) / (q + 1) > (n : ℝ) - (1 + 1) | |
| end HenonLaneEmden | |
| end OAI | |