Nonexistence and multiplicity of solutions for nonlinear elliptic systems inRN

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

“…Clearly, the concentration compactness argument is no more applicable, moreover, the potential V i considered here is allowed to be sign-changing, which is different from [1,5,10,9,24,33]. Under the local superquadratic condition, we obtain a continuous ground state solution of (1.1) which decays to zero exponentially, the result extends and complements related ones in [10,8]. More precisely, we will prove Theorems 1.1 and 1.2 below by using following local super-quadratic condition instead of (SQ), (S2) there exists a domain Ω ⊂ R N such that lim |z|→∞ F (x,z) |z| 2 = ∞ a.e.…”

“…By assuming symmetry property on the potential and working on the radially symmetric function space, one can recover the compactness of embedding, see, e.g., [23,45]. 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.…”

“…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.…”

“…Remark 1. Condition (V ) has a potential well which is widely used in the literature(see [3,8]), and implies, as a consequence of Molchanov's result [28], that the embedding H :…”

“…Condition like or similar to (1.4) is called nonquadratic at infinity introduced by Costa and Magalhães [10,11] with > (p − 2) min{1, N/2}, and it is widely used in the literature, see [8] with = p where infinitely many nontrivial solutions are obtained by Chen and Ma under some additional conditions besides (V ), (S0), (S1) and (SQ). Particularly, for the scalar equation of (1.1) we deduce from (S1) and (SQ) that…”