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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x\in {ℝ}^N$, $u\in H¹\left({ℝ}^N\right)$, where λ is a positive parameter, a and b are positive functions, while $f:ℝ\to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.