Existence and decay property of ground state solutions for Hamiltonian elliptic system

Abstract

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term −∆u + b(x) • ∇u + u + V (x)v = f (x, |z|)v, x ∈ R N , −∆v − b(x) • ∇v + v + V (x)u = f (x, |z|)u, x ∈ R N , where z = (u, v) ∈ R 2. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.

Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

“…We would like to point out that the global super-quadratic (SQ) is indispensable in verifying the linking geometry structure and constructing Cerami sequence by the diagonal method and linking argument, see [37,40]. Unfortunately, in the present paper we have no global information on the nonlinearity like (SQ), then the non-Nehari manifold method seems not work to our problem under the local super-quaratic condition.…”

“…Since gµ(x, s) is increasing in s on ( , +∞) due to (f ), we can obtain Gµ(x, t) ≤ for t ≥ by using some arguments in [37,40]. So, we get the rst conclusion from the above formula.…”

“…Later, this result has been extended to more general nonlinearity model by Liao et al [15]. An asymptotically periodic case was considered in [40], and some properties of ground state solutions were obtained by constructing linking levels and analyzing behavior of Cerami sequence. The existence of least energy solution for the non-periodic super-quadratic case was studied in [30].…”

“…For further related topics including the Hamiltonian systems, we refer the reader to [3,11,12,21] and their references. When b ≠ , as we all know, there are a few works devoted to the existence and multiplicity of solutions of system (1.1), see [15,30,32,36,40]. For this case, since the appearance of the gradient term in system itself, system (1.1) has some di erences and di culties comparing to system (1.2).…”

Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

“…see [24,35,39,40,43]. For this case, since the appearance of the gradient term in system, system (1.1) has some di erences and di culties compared with the case b(x) = .…”