Abstract
This paper is concerned with the following periodic Hamiltonian elliptic system: \(-\Delta u+V(x)u=H_{v}(x,u,v)\), \(x\in\mathbb{R}^{N}\), \(-\Delta v+V(x)v=H_{u}(x,u,v)\), \(x\in\mathbb{R}^{N}\), \(u(x)\to0\), \(v(x)\to0\) as \(|x|\to\infty\). Assuming the potential V is periodic and 0 lies in a gap of \(\sigma(-\Delta+V)\), \(H(x,z)\) is periodic in x and superquadratic in \(z=(u,v)\). We establish the existence of infinitely many large energy solutions by the generalized variant fountain theorem developed recently by Batkam and Colin.
Highlights
Introduction and main resultsIn this paper, we consider the following Hamiltonian elliptic system:⎧ ⎪⎨– u + V (x)u = Hv(x, u, v), x ∈ RN, ⎪⎩u–(x)v + V (x)v →, =Hu(x, u, v), v(x) → x∈ as |x| RN, → ∞, ( . )where z = (u, v) : RN → R × R, N ≥, V ∈ C(RN, R), and H ∈ C (RN × R, R)
For the case of a bounded domain these systems were studied by a number of authors
Most of them focused on the case V ≡
Summary
U(x) → , as |x| → ∞, has been studied recently in Batkam and Colin [ ]. The variational setting for our problem is more complex and different from the case where V = since the potential V is a general periodic function. Let X = D(|A| / ) be the Hilbert space with the inner product (u, v)X = |A| / u, |A| / v and the corresponding norm u X = (u, v) X/. We introduce the generalized variant fountain theorem, and consider the C -functional λ : E → R defined by λ(u) = I(u) – λJ(u), λ ∈ [ , ]. The following version of the fountain theorem for a strongly indefinite functional is due to Batkam and Colin [ ]. To prove our main result, by the assumption (V) and decomposition of E, we define the following functional on E: λ(z) =. There exists k > such that ξ (x)|zk|p < ε, ∀k ≥ k
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