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By Frederic Rosen
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Extra resources for The Algebra of Mohammed Ben Musa
Example text
54) for every p ∈ [1, 5). Moreover, if N ≤ 9, then u∗ ∈ L∞ (Ω), f u∗ ∈ L∞ (Ω). 55) Proof . As we have already remarked, for every v ∈ H01 (Ω), Ω ∇u(λ) · ∇v dx = λ λ Ω eu(λ) v2 dx ≤ Ω Ω eu(λ) v dx, |∇v|2 dx. 57) we choose v = e(p−1)u(λ) − 1 ∈ H01 (Ω), for p > 1 arbitrary. We find (p − 1) 2 Ω e(p−1)u(λ) ∇u(λ) dx = λ Ω eu(λ) e(p−1)u(λ) − 1 dx. 57) we put v = e(p−1)/2u(λ) − 1 ∈ H01 (Ω). 59) Qualitative properties of the minimal solution near the bifurcation point 41 Hence λ 2 Ω eu(λ) e(p−1)/2 u(λ) − 1 dx ≤ (p − 1)2 4 2 Ω e(p−1)u(λ) ∇u(λ) dx.
E. e. x ∈ Ω. So, by Fatou’s lemma, lim sup n→∞ Ω G0 x, un dx ≤ Ω G0 (x, u)dx. 110) Thus, E(u) ≤ lim inf n→∞ E(un ). (b) Since λ1 (−Δ − a0 (x)) < 0, there exists φ ∈ H01 (Ω) such that |∇φ|2 dx < [φ=0] a0 φ2 dx. 111) This relation remains valid if we replace φ by φ+ , so we may assume that φ ≥ 0. Next, we show that we can assume φ ∈ L∞ (Ω). Set, for any positive integer k, Ωk := {x ∈ Ω; φ(x) ≤ k }. Then Ω = k≥1 Ωk and for any k ≥ 1, Ωk ⊂ Ωk+1 . Hence Ω |∇φ|2 dx = lim n→∞ Ωk |∇φ|2 dx. 111), there is some positive integer k such that Ωk |∇φ|2 dx < Ωk ∩[φ=0] a0 φ2 dx.
In most situation however, states do not exist for ever, and a more accurate model is given by a decaying state that oscillates at some rate. Eigenvalues are yet another expression of humanity’s narcissist desire for immortality. Our results are related to a certain linear eigenvalue problem. We recall in what follows the results that we need in the sequel. Let Ω be an arbitrary open set in ÊN , N ≥ 3. Consider the eigenvalue problem −Δu = λV (x)u in Ω, u ∈ H01 (Ω). 72) Problems of this type have a long history.