We consider the following nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+ V(x)u= a(x)\left| u\right| ^{q-1}u+ f(x), \quad x\in \mathbb {R}^{N}, \end{aligned}$$ - Δ u + V ( x ) u = a ( x ) u q - 1 u + f ( x ) , x ∈ R N , where V is a non-symmetric bounded potential, a is an indefinite weight, $$0<q<1$$ 0 < q < 1 and $$f\ne 0$$ f ≠ 0 is a nonnegative perturbation such that $$f\in L^{2}(\mathbb {R}^{N})\cap L^{\frac{2N}{N+2}}(\mathbb {R}^{N})$$ f ∈ L 2 ( R N ) ∩ L 2 N N + 2 ( R N ) . Using variational methods, we prove the existence of two solutions with negative and positive energies, one of these solutions being nonnegative.