Periodic problems with double resonance

Nonlinear Differential Equations and Applications NoDEA, Aug 2011

We consider a second order periodic problem with resonance both at infinity and at zero. Combining variational methods together with Morse theory, we produce six nontrivial solutions for the periodic problem.

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Periodic problems with double resonance

Nonlinear Differ. Equ. Appl. Periodic problems with double resonance Giuseppina Barletta Nikolaos S. Papageorgiou We consider a second order periodic problem with resonance both at infinity and at zero. Combining variational methods together with Morse theory, we produce six nontrivial solutions for the periodic problem. - 1. Introduction In this paper, we deal with the following second order periodic problem: −u (t) = f (t, u(t)) a.e. on T = [0, b], u(0) = u(b), u (0) = u (b). ( 1 ) In this problem the reaction term f (t, x) is jointly measurable and C1 in the x-variable. The aim of this work is to prove a multiplicity theorem when the problem is resonant both at infinity and at zero. Such problems are known in the literature as “doubly resonant”. Resonant Dirichlet equations, but with the resonance only at infinity and only with respect to the first two eigenvalues, were studied by Dancer and Gupta [7], Gupta [10], Iannacci and Nkashama [11] and Sanchez [18]. Doubly resonant Dirichlet equations of higher parts of the spectrum were investigated by Su and Li [19] and Zou [22]. For periodic equations, we have the recent work of Su and Zhao [20]. With the exception of Su and Li [19], all the other works produce two nontrivial solutions. Su and Li [19] dealing with Dirichlet equations, produce six nontrivial solutions. Here under more general hypotheses on the reaction f (t, x), we produce six nontrivial solutions for the periodic problem ( 1 ). Our approach combines variational techniques based on the critical point theory together with Morse theory. In the next section, for the convenience of the reader, we recall the main mathematical tools that we will use in this paper. NoDEA 2. Mathematical background-hypotheses Let X be a Banach space and X∗ its topological dual. By ·, · we denote the duality brackets for the pair (X∗, X). Let ϕ : X → R be a C1 function. We say that ϕ satisfies the “Cerami condition” (the “C-condition” for short), if every sequence {xn}n≥1 ⊆ X such that {ϕ(xn)} ⊆ R is bounded and (1 + xn )ϕ (xn) → 0 in X∗as n → ∞, has a strongly convergent subsequence. Using this compactness type condition, we have the following minimax theorem for the critical values of a C1-functional, known in the literature as the “mountain pass theorem”. Theorem 2.1. If X is a Banach space, ϕ : X → R is C1, satisfies the C-condition, x0, x1 ∈ X, 0 < ρ < x1 − x0 , max{ϕ(x0), ϕ(x1)} < inf[ϕ(x) : x − x0 = ρ] = ηρ, and c = inf max ϕ(γ(t)) where Γ := {γ ∈ C0([0, 1], X) : γ(0) = x0, γ( 1 ) = x1}, γ∈Γ t∈[0,1] then c ≥ ηρ and c is a critical value of ϕ. For ϕ ∈ C1(X) and c ∈ R, we introduce the following notation: ϕc = {x ∈ X : ϕ(x) ≤ c}, Kcϕ = {x ∈ Kϕ : ϕ(x) = c}. Kϕ = {x ∈ X : ϕ (x) = 0} and If (Y1, Y2) is a topological pair with Y2 ⊆ Y1 ⊆ X, then for every integer k ≥ 0, by Hk(Y1, Y2) we denote the kth-relative singular homology group for the pair (Y1, Y2) with integer coefficients. The critical groups of ϕ at an isolated critical point x0 ∈ X with ϕ(x0) = c (i.e., x0 ∈ Kcϕ), are defined by Ck(ϕ, x0) = Hk(ϕc ∩ U , ϕc ∩ U \{x0}), for all k ≥ 0, where U is a neighborhood of x0 such that Kϕ ϕc U = {x0} (see [6, 13]). The excision property of singular homology, implies that the above definition is independent of the particular choice of neighborhood U of x0. Assume that ϕ ∈ C1(X) satisfies the C-condition and −∞ < inf ϕ(Kϕ). Let c < inf ϕ(Kϕ). The critical groups of ϕ at infinity are defined by Ck(ϕ, ∞) = Hk(X, ϕc) for all k ≥ 0, (see [5]). The second deformation theorem (see, for example [15, p. 274]), implies that the above definition is independent of the particular choice of the level c < inf ϕ(Kϕ). If Kϕ = {x0}, then we have Ck(ϕ, ∞) = Ck(ϕ, x0) for all k ≥ 0. The next result is useful in computing the critical groups at infinity. It is a slight generalization of a result of Perera and Schechter [17], suitable for functions ϕ ∈ C1(X) which satisfy the C-condition (see [12]). Proposition 2.1. If H is a Hilbert space, {ϕt}t∈[0,1] ⊆ C1(X), ϕt and ∂tϕt both are locally Lipschitz, ϕ0 and ϕ1 satisfy the C-condition, and there exist a ∈ R and δ > 0 such that ϕt(u) ≤ a ⇒ (1 + u ) ϕt(u) ≥ δ for all t ∈ [0, 1] then Ck(ϕ0, ∞) = Ck(ϕ1, ∞) for all k ≥ 0. k≥0 x∈Kϕ q k=0 Also, if mk = x∈Kϕ mk(x) (the Morse-type numbers for ϕ) and βk (the Betti-type numbers for ϕ) are as above, then we have the Morse inequality k≥0 P (t, x) = P (t, ∞) + (1 + t)Q(t) for all t ∈ R. ( 2 ) Remark 2.1. Note that in particular, if there exists R > 0 such that inf [(1 + u ) ϕt(u) : t ∈ [0, 1], u > R] > 0, inf [|ϕt(u) : t ∈ [0, 1], u ≤ R] > −∞, Let X = H be a Hilbert space and let x0 ∈ H be an isolated critical point of ϕ. Let U be a neighborhood of ϕ and assume that ϕ ∈ C2(U ). The “Morse index of x0”, is the supremum of the dimension of the subspaces of H, on which ϕ (x0) is negative definite. The “nullity of x0”, is the dimension of the kernel of ϕ (x0). We say that x0 is “nondegenerate”, if the nullity of x0 is zero (i.e., ϕ (x0) i (...truncated)


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Giuseppina Barletta, Nikolaos S. Papageorgiou. Periodic problems with double resonance, Nonlinear Differential Equations and Applications NoDEA, 2011, pp. 303-328, Volume 19, Issue 3, DOI: 10.1007/s00030-011-0130-5