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By using the penalization method and the Ljusternik-Schnirelmann theory, weinvestigate the multiplicity of positive solutions of the following fractionalSchr\"odinger equation $$ \e^{2s}(-\Delta)^{s} u + V(x)u = f(u) \mbox{ in }\mathbb{R}^{N} $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$,$(-\Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuouspotential with local minimum, and $f$ is a superlinear function withsubcritical growth. We also obtain a multiplicity result when$f(u)=|u|^{q-2}u+\lambda |u|^{r-2}u$ with $2<q<2^{*}_{s}\leq r$ and$\lambda>0$, by combining a truncation argument and a Moser-type iteration.

Multiplicity of positive solutions for a class of fractional Schr\"odinger equations via penalization method