Consider the nonlinear difference equations of the form L u = f m ( u ) , m ∈ Z , where L is a Jacobi operator given by L u m = a m u m + 1 + a m − 1 u m − 1 + b m u m for m∈Z, { a m } and { b m } are real valued T-periodic sequences, and f:Z×R→R. Applying critical point theory and a new analytical method, we obtain that the above problem has ground state solutions and infinitely many geometrically distinct solutions under the local superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈ D for some set D⊂Z instead of the global superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈Z.

In this article, we study the quasilinear Schr\”{o}dinger equation \begin{eqnarray*} \begin{array}{ll} \triangle{u}+V(x)u-\triangle(u^2)u=g(x,u), \ x\in\mathbb{R}^N, \end{array} \end{eqnarray*} where the potential $V(x)$ and the primitive of $g(x,u)$ is allowed to be sign-changing. Under more general superlinear conditions on $g$, we obtain the existence of infinitely many nontrivial solutions by using Mountain Pass Theorem. Recent results in the literature are significantly improved.