This paper is dedicated to studying the semilinear Schrödinger equation
$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\ u\in H^{1}({\mathbb {R}}^{N}), \end{array}\right. \end{aligned}$$
where
$$V\in \mathcal {C}(\mathbb {R}^N, \mathbb {R})$$
is sign-changing and either periodic or coercive and
$$f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})$$
is subcritical and local super-linear (i.e. allowed to be super-linear at some
$$x\in \mathbb {R}^N$$
and asymptotically linear at other
$$x\in \mathbb {R}^N$$
). Instead of the common condition that
$$\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x, s)\mathrm {d}s}{t^2}=\infty $$
uniformly in
$$x\in \mathbb {R}^N$$
, we use a local super-quadratic condition
$$\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x,s)\mathrm {d}s}{t^2}=\infty $$
a.e.
$$x\in G$$
for some domain
$$G\subset \mathbb {R}^N$$
to show the existence of nontrivial solutions for the above problem.