Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

Abstract

We consider the discrete nonlinear Schro¨dinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave solutions.

Infinitely many homoclinic solutions for sublinear and nonperiodic Schrödinger lattice systems

“…Remark 1.1 To the best of our knowledge, there is no result published concerning the multiplicity of nontrivial solutions for (1.1) with sublinear nonlinearities at both zero and infinity. For the nonperiodic system (1.1), the main differences between our and known results [5,9,15,16,22,23] are as follows:…”

“…In fact, most results are about the periodic Schrödinger lattice systems, such as [2, 4, 12-14, 18, 19, 24]. However, there are only few results about the nonperiodic Schrödinger lattice systems [5,9,15,16,22,23]. In particular, in [3,6,7] the authors recently obtained the existence and multiplicity of homoclinic solutions for a class of Schrödinger lattice systems with perturbed terms.…”

“…(1) The nonlinearities g n (s) in [5,15,16,22,23] are superlinear as |s| → 0 (lim |s|→0 g n (s) s = 0, ∀n ∈ Z), and the nonlinearities g n (s) in [9] are superlinear or asymptotically linear (lim |s|→0…”

Nontrivial solutions of discrete nonlinear equations with variable exponent

“…Under which assumptions the problem has nontrivial solutions? For the case of standard DNLS we refer to [15].…”

On Standing Wave Solutions for Discrete Nonlinear Schrödinger Equations

“…In all these results, the nonlinearity is supposed to be either positive (self-focusing), or negative (defocusing). Pankov [17] studied the DNLS equatifvons with infinitely growing potential and sign-changing nonlinearity (a mixture of self-focusing and defocusing ones). Pankov and Zhang were concerned with the DNLS equations with infinitely growing potential and saturable nonlinearity in [18].…”