By Yanheng Ding

This detailed ebook makes a speciality of severe aspect concept for strongly indefinite functionals in an effort to take care of nonlinear variational difficulties in components corresponding to physics, mechanics and economics. With the unique elements of Lipschitz walls of solidarity of gage areas (nonmetrizable spaces), Lipschitz normality, and enough stipulations for the normality, in addition to existence-uniqueness of move of ODE on gage areas, the ebook provides for the 1st time a deformation conception in in the community convex topological vector areas. It additionally bargains pleasant variational settings for homoclinic-type recommendations to Hamiltonian structures, Schrödinger equations, Dirac equations and diffusion platforms, and describes contemporary advancements in learning those difficulties. The recommendations and techniques used open up new issues valuable of in-depth exploration, and hyperlink the topic with different branches of arithmetic, comparable to topology and geometry, offering a viewpoint for extra stories in those components. The analytical framework can be utilized to address extra infinite-dimensional Hamiltonian structures.

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We need the additional assumption: (Φ+ ) there exists ζ > 0 such that u < ζ PY u for all u ∈ Φ0 . 1. Let S0 ⊂ S be any countable dense subset with associated family P0 of semi-norms. 1. Indeed, let (un )n in Φa which P0 -converges to u ∈ E, and write un = xn + yn , u = x + y ∈ X ⊕ Y . Then yn − y → 0, hence yn is bounded. It follows from (Φ+ ) that xn , hence un is bounded. This implies that un weakly hence P-converges to u. Now (Φ0 ) implies that u ∈ Φa , and Φ (un )v → Φ (u)v for all v ∈ E. 5) with 0 ≤ a ≤ b.

D + L is selfadjoint on L2 (R, R2N ) with domain D(A) = Note that A = − J dt H 1 (R, R2N ). Let σ(A) and σc (A) denote, respectively, the spectrum and the continuous spectrum. Set µe := inf{λ : λ ∈ σ(A) ∩ [0, ∞)}. 4) Throughout the book by | · |q we denote the usual Lq -norm, and (·, ·)L2 the usual L2 -inner product. 1. Assume (L0 ) is satisfied. Then 1◦ A has only absolute continuous spectrum : σ(A) = σc (A); 2◦ σ(A) ⊂ R \ (−λ0 , λ0 ); 3◦ if (L1 ) also holds, σ(A) is symmetric : σ(A) ∩ (−∞, 0) = −σ(A) ∩ (0, ∞); and µe ≤ Λ 0 .

Results of this type are needed for the existence of multiple critical points. 5 ([Bartsch and Ding (2006I)]). Consider a, b ∈ R with a < b, I := [a, b], such that Φ : (Φba , TP ) → R is upper semi-continuous, and Φ : (Φba , TP ) → (E ∗ , Tw∗ ) is continuous. 4. 5) is satisfied, then the conclusion as in a) holds. Proof. Again we shall only prove b) because the proof of a) is similar and somewhat simpler. Fix c ∈ (a, b) and σ > 0. Since A is a (C)I -attractor there exists α > 0 with (1 + u ) · Φ (u) ≥ 2α if u ∈ Φba \ Uσ/3 .

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