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07 ∇ξ L (x, uh , ∇uh ) · ∇uh dx + K ≤ε ,ε , {|uh |> } where we have fixed δ > 0 in such a way that K1 (k) νδ+δ ν−δ ≤ ε. 29) relaxes [112, condition (9)]. 12. Let c ∈ R. Then each (CP S)c -sequence for f is bounded in W01,p (Ω). Proof. e. x ∈ Ω and all s ∈ R qG(x, s) ≤ sg(x, s) + a0 (x). Now, let (uh ) be a (CP S)c -sequence for f and let for all v ∈ Cc∞ (Ω) ∇ξ L (x, u, ∇u) · ∇v dx + w, v = Ds L (x, u, ∇u)v dx − Ω Ω g(x, uh )v dx. 11, for each ε > 0 we have − wh −1,p uh 1,p ∇ξ L (x, uh , ∇uh ) · ∇uh dx + ≤ Ω Ds L (x, uh , ∇uh )uh dx − Ω ∇ξ L (x, uh , ∇uh ) · ∇uh dx + ≤ Ω g(x, uh )uh dx Ω Ds L (x, uh , ∇uh )uh dx Ω −q G(x, uh ) dx + Ω a0 dx Ω ∇ξ L (x, uh , ∇uh ) · ∇uh dx + ≤ (1 + ε) {|uh |>R } L (x, uh , ∇uh ) dx + qf (uh ) + −q Ds L (x, uh , ∇uh )uh dx Ω Ω a0 dx + KR ,ε .

1) possess a sequence (um ) of weak solutions in H01 (Ω, RN ) under suitable assumptions, including symmetry, on coefficients ahij and G. To prove this result, we looked for critical points of the functional f0 : H01 (Ω, RN ) → R defined by n 1 f0 (u) = 2 N ahij (x, u)Di uh Dj uh dx − G(x, u) dx.

55). Then every (CP S)c -sequence {un } for f is bounded in H01 (Ω). Proof. 109). 3]. 42. 41. We can now state the following result. 43. 55). Then the functional f satisfies the (P S)c condition at every level c ∈ R. EJDE-2006/MON. 07 ON A CLASS OF QUASI-LINEAR ELLIPTIC PROBLEMS 53 Proof. Let {un } ⊂ dom(f ) be a concrete Palais-Smale sequence for f at level c. 41 it follows that {un } is bounded in H01 (Ω). 40 f satisfies the Concrete Palais-Smale condition. 39 implies that f satisfies the (P S)c condition.

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Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems by Squassina M.


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