Abstract
This article concerns the Hamiltonian elliptic system: \t\t\t{−Δφ+V(x)φ=Gψ(x,φ,ψ)in RN,−Δψ+V(x)ψ=Gφ(x,φ,ψ)in RN,φ,ψ∈H1(RN).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta \\varphi +V(x)\\varphi =G_{\\psi }(x,\\varphi ,\\psi ) & \\mbox{in } \\mathbb {R}^{N}, \\\\ -\\Delta \\psi +V(x)\\psi =G_{\\varphi }(x,\\varphi ,\\psi ) & \\mbox{in } \\mathbb {R}^{N}, \\\\ \\varphi , \\psi \\in H^{1}(\\mathbb {R}^{N}). \\end{cases} $$\\end{document} Assuming that the potential V is periodic and 0 lies in a spectral gap of sigma (-Delta +V), least energy solution of the system is obtained for the super-quadratic case with a new technical condition, and the existence of ground state solutions of Nehari–Pankov type is established for the asymptotically quadratic case. The results obtained in the paper generalize and improve related ones in the literature.
Highlights
Introduction and main resultsConsider the following Hamiltonian elliptic system:⎧ ⎪⎪⎨– φ + V (x)φ = Gψ (x, φ, ψ) in RN,⎪⎪⎩–φ, ψ + V (x)ψ = ψ ∈ H1(RN ), in RN, (1.1)where φ, ψ : RN → R, V ∈ C(RN, R), and G ∈ C1(RN × R2, R) with gradient ∇G = (Gφ, Gψ )
In [44], the authors studied ground state solutions for a Hamiltonian elliptic system with inverse square potential, and there are other papers concerned with the system in the whole space RN, see [10, 21,22,23, 31, 35, 38,39,40, 45–48] and the references therein
Assuming that V is periodic and positive, nontrivial solutions were obtained by Liao et al in the recent paper [22] under some new super-quadratic conditions
Summary
The following assumptions were introduced in [40, 46] for super and asymptotically quadratic system (1.1): (V) V ∈ C(RN ) are 1-periodic in xi, i = 1, 2, . We first use a technical condition introduced in [29] to consider the super-quadratic case and obtain a least energy solution for (1.1) with the aid of a generalized linking theorem established in [20].
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