Solutions on Asymptotically Periodic Elliptic System with New Conditions

Ground state solutions for Hamiltonian elliptic systems with super or asymptotically quadratic nonlinearity

“…Inspired by the aforementioned works, we continue to study problem (1.1) in this paper and construct two types of ground state solutions, i.e., the least energy solution and the Nehari-Pankov type. 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]. Then we lay emphasis on the asymptotically quadratic case and discuss the existence of ground state solution of Nehari-Pankov type for (1.1).…”

“…On the other hand, the argument in [40,46, see Lemma 4.1] becomes invalid due to the lack of positive assumption onĜ(x, η) (i.e.,Ĝ(x, η) > 0). Motivated by the works [27,33,40,46], we further develop the approach in [29,34,35] to find ground state solution of Nehari-Pankov type for (1.1). Our approach is based on finding a proper subset E + 0 of E \ E -(see (3.4) for the definition) such that, for any η ∈ E + 0 , there exist t = t(η) > 0 and w = w(η) ∈ Esatisfying w + tη ∈ N -, then we can derive a minimizing sequence on the Nehari-Pankov manifold by using the diagonal method, see Lemma 3.9.…”

Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials

“…Another usual way to regain the compactness is by imposing coercive assumption on the potential, see, for instance, [3,8,10,35]. The concentration compactness argument is also well employed to deal with the whole space case provided that the potential and nonlinearity are periodic in the variable x, we refer readers to [6,9,26,31] and the references therein. By using the constrained minimization method [5] and the Nehari manifold method [35,24], and applying the bootstrap argument [9] and some skills related to ordinary differential system [23,45], many authors studied system (1.1) for case that the potentials are nonnegative constants and obtained certain of results on the existence, regularity and uniqueness of ground state solutions.…”

“…More precisely, following super-quadratic condition (SQ) introduced by Liu-Wang [22] and technical condition (DL) introduced by Ding-Lee [13] were used there. Later, by employing the non-Nehari manifold method introduced by Tang [41], these results were improved by Qin, He and Tang [31,32] and generalized to the asymptotically periodic case where the potentials V i (x) and the nonlinearity are allowed to be asymptotically periodic in x. See [26] for related results and [8] for nonexistence result.…”

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

“…Condition (F3) together with | f ( x , u )|= o (| u |) as | u |→0 imply that which is a weaker version of the classic Ambrosetti‐Rabinowitz condition. As pointed out in Qin and Tang, condition was first introduced by Schechter (2.7) or (3.1) for scalar Schrödinger equation, and it was commonly used (see Qin and Tang and Tang) instead of the usual monotonic condition:…”