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Existence and multiplicity results for a nonlinear stationary Schrödinger equation. (English) Zbl 1205.35099

Summary: We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form

$-{\Delta }u+a\left(x\right)u=\lambda b\left(x\right)f\left(u\right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},\phantom{\rule{4pt}{0ex}}u\in {H}^{1}\left({ℝ}^{N}\right),$

where $\lambda$ 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.

##### MSC:
 35J61 Semilinear elliptic equations 35J20 Second order elliptic equations, variational methods 35B45 A priori estimates for solutions of PDE