Elliptic equations with decaying cylindrical potentials and by Badiale M., Guida M., Rolando S.

By Badiale M., Guida M., Rolando S.

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Example text

By (60) and the definition of Ωε we have f (u) ≤ C1 (N −2)(p−2)−α |x| u u α ≤ε α |x| |x| A A α ≥ α |y| |x| and for almost every x ∈ Ωε . Hence, by definition of weak solution, one obtains ∇u · ∇h dx = Ωε Ωε f (u) h dx − A = − (A − ε) Ωε uh dx |x|α Ωε uh dx ≤ ε |y|α Ωε uh dx − A |x|α Ωε uh dx |x|α for every h as in the statement of the lemma. In order to develope our comparison argument, define 1−α/2 vε (x) := e−β ε |x| for all x ∈ RN . Lemma 36 vε ∈ D1,2 (RN ) and for all nonnegative h ∈ D01,2 (Ωε ) one has Ωε ∇vε · ∇h dx ≥ − (A − ε) Ωε vε h dx .

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Then, for every ψ ∈ Cc (Λ2 ), the product ϕ (r) ψ (t) belongs to Cc∞ (Λ1 × Λ2 ) and we have ψ (t) Λ2 u ˜ (r, t) ϕ (r) dr dt = u ˜ (r, t) ϕ (r) ψ (t) drdt Λ1 ×Λ2 Λ1 ∂u ˜ (r, t) ϕ (r) ψ (t) drdt ∂r ∂u ˜ ψ (t) = − (r, t) ϕ (r) dr dt ∂r Λ2 Λ1 = − Λ1 ×Λ2 ˜ (r, t) ϕ (r) dr = − Λ1 ∂∂ru˜ (r, t) ϕ (r) dr for alagain by Lemma 37. Hence Λ1 u most every r ∈ Λ2 , whence, upon multiplying by t(m−1)/2 and integrating over Λ2 , one readily deduces that u ¯ is the weak derivative of u ¯ on Λ1 . 2 For any α > 0, we now define v (r) := rk−1−α/2 u ¯ (r) for all r > 0.

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