Existence, multiplicity, perturbation, and concentration by Squassina M.
By Squassina M.
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Additional info for Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems
Example text
60). Then Young inequality implies that, for every ε > 0, we have N +2 N +2 |g(x, s)| ≤ β(ε)(a1 (x)) 4 + ε|s| N −2 + γ(ε, b), where β(ε) and γ(ε, b) are positive constants depending on ε and b. 54). 10. Verification of the key condition. 52). 41), we immediately obtain that J is lower semicontinuous. 3). To this aim, for every k ≥ 1, we define the truncation Tk : R → R at height k, defined as s if |s| ≥ k. 23. 43). Then, for every (u, η) ∈ epi J with J(u) < η, there holds |dGJ |(u, η) = 1. Moreover, if j(x, −s, −ξ) = j(x, s, ξ), ∀ η > J(0)(= 0) it results |dZ2 GJ |(0, η) = 1.
Take 0 < δ < 1 and let ϑδ : R → R be the function defined by setting 0 s − kσ s + kσ ϑδ (s) = −δs + σ + δ(k + 1)σ −δs − σ − δ(k + 1)σ 0 if if if if if if |s| ≤ kσ kσ < s < (k + 1)σ − (k + 1)σ < s < −kσ (k + 1)σ ≤ s < (k + 1)σ + σδ − (k + 1)σ − σδ < s ≤ −(k + 1)σ |s| ≥ (k + 1)σ + σδ . As before, we get ∇ξ L (x, uh , ∇uh ) · ∇ϑδ (uh ) dx + Ω Ds L (x, uh , ∇uh )ϑδ (uh ) dx Ω ≤ g(x, uh )ϑδ (uh ) dx + Ω 1 p p p pp δp wh p −1,p + δ ϑδ (uh ) p 1,p . EJDE-2006/MON. 25), by computations, we deduce that Ds L (x, uh , ∇uh )ϑδ (uh ) dx ≥ 0.
Let (ϑh ) be a sequence in Cc1 (R) with sup ϑh ∞ < ∞, sup ϑh h≥1 ∞ < ∞, h≥1 lim ϑh (s) = 1, lim ϑh (s) = 0. h→∞ h→∞ If jξ (x, u, ∇u) · ∇u ∈ L1 (Ω), the sequence div ϑh (u)jξ (x, u, ∇u) is strongly convergent in W lim h→∞ −1,q (Ω) for every 1 < q < − div ϑh (u)jξ (x, u, ∇u) N N −1 , and + js (x, u, ∇u) = w W −1,q (Ω). in Proof. Let w = − div F with F ∈ L2 (Ω, RN ) and v ∈ Cc∞ (Ω). 81). It results jξ (x, u, ∇u)ϑh (u) · ∇v = − Ω jξ (x, u, ∇u)ϑh (u) · ∇u v − Ω + js (x, u, ∇u)ϑh (u)v Ω F ϑh (u)∇u v + Ω F ϑh (u)∇v.