Applied partial differential equations. An introduction by Jeffrey A. PDF

By Jeffrey A.

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69) implies that un → u strongly in H01 (Ω). 107) EJDE-2006/MON. 13. Proofs of the main Theorems. In this section we give the definition of a Concrete Palais-Smale sequence, we study the relation between a Palais-Smale sequence and a Concrete Palais-Smale sequence, and we prove that f satisfies the (P S)c for every c ∈ R. 20. 54). 41) implies that the functional f : H01 (Ω) → R ∪ {+∞} defined by f (v) = J(v) + I(v) is lower semi-continuous. To apply the abstract theory, it is crucial to have the following result.

39) we conclude that ∇ξ L (x, uh , ∇uh ) · ∇uh dx {|uh |≤ } ∇ξ L (x, uh , ∇uh ) · ∇uh dx ≤ {|uh |≤kσ} ∇ξ L (x, uh , ∇uh ) · ∇uh dx + K2 (k) ≤ K1 (k) {kσ<|uh |≤(k+1)σ} ≤ K1 (k) νδ + δ ν−δ ∇ξ L (x, uh , ∇uh ) · ∇uh dx {|uh |>(k+1)σ} + K1 (k)K3 (k, δ) + K2 (k) ≤ 31 32 MARCO SQUASSINA EJDE-2006/MON. 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).

43). To do this we will follow the arguments of [33, 36, 112, 132] where the case in which α(s) and β(s) are bounded is studied. In our case we will have to modify the test functions used in these papers in order to get the compactness result. Indeed, here the main difficulty is to find suitable approximations of un that belong to Wun , in order to choose them as test functions. 33) is devoted to find possible improvements of the class of allowed test functions. 9. General setting and main results.

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