Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

Abstract

Abstract. This paper is concerned with the following periodic Hamiltonian elliptic systemAssuming the potential V is periodic and 0 lies in a gap of σ(−Δ + V ), G(x, η) is periodic in x and asymptotically quadratic in η = (ϕ, ψ), existence and multiplicity of solutions are obtained via variational approach.Mathematics Subject Classification. 35J50, 35J55.

On periodic Hamiltonian elliptic systems with spectrum point zero

“…For example, in 28 the authors considered this case and proved the existence of one ground state solution and an infinite number of geometrically distinct solutions for ( ES ) with under the assumptions that g , f are 1‐periodic in x and superlinear. Later, in 29 they considered the asymptotically linear case and obtained the same results. Very recently, in 27, Zhao and Ding studied the system ( ES ) with gradient terms and periodic or non‐periodic potentials V .…”

“…In recent years, many authors are devoted to study the existence of solutions for Hamiltonian elliptic systems like or similar to ( ES ) via modern variational methods. For example, see 13–15, 19 for the case of a bounded domain, and 16, 18, 21, 23–29 for the case of the whole space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^N$\end{document}. Usually, for the superquadratic case, one needs the following condition due to Ambrosetti‐Rabinowitz 1: which is called the well‐known (AR) growth condition.…”

“…In order to obtain the existence and multiplicity results for ( ES ), we have to overcome some difficulties. Firstly, in view of 0 ∈ σ cont ( A ), the usual working space defined in 27–29 is not complete with respect to the classical norm (see Section 2 in detail). So we have to construct a new working space which is different from the one defined in 27–29, and also different from the one defined in 3.…”

“…Firstly, in view of 0 ∈ σ cont ( A ), the usual working space defined in 27–29 is not complete with respect to the classical norm (see Section 2 in detail). So we have to construct a new working space which is different from the one defined in 27–29, and also different from the one defined in 3. Secondly, since W does not satisfy Ambrosetti‐Rabinowitz growth condition, it is difficult to get the boundedness of ( PS ) (or ( PS *)) sequences.…”

Multiple solutions for asymptotically quadratic and superquadratic elliptic system of Hamiltonian type

“…With the aid of the generalized linking theorem [5], there are some works focused on the existence and multiplicity of solutions for the general periodic problem, see [24,28,29,33,35,[40][41][42] and their references. Additionally, compared to the periodic problem, the nonperiodic problem becomes very complicated since the energy functional has no more translation invariance.…”