## Multiplicity and concentration results for a fractional Choquard equation via penalization method.(English)Zbl 1408.35001

The paper is focused on the nonlinear fractional Choquard equation $\varepsilon^{2s}(-\Delta)^su+V(x)u=\varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}} * F(u)\right)f(u),\text{ in }\mathbb{R}^N,\tag{1}$ where $$\varepsilon$$ is a positive parameter, $$s\in (0,1)$$, $$N>2s$$, $$0<\mu<2s$$, the potential $$V:\mathbb{R}^N\to\mathbb{R}$$ is a positive continuous function, the nonlinearity $$f:\mathbb{R}\to\mathbb{R}$$ is a continuous function with subcritical growth, $$F(t)=\int_0^tf(\tau)$$ $$d\tau$$, and the nonlocal operator $$(-\Delta)^s$$ is the fractional Laplacian. Under some assumptions on $$V$$ and $$f$$, the authors study the multiplicity and the concentration of positive solutions of equation $$(1)$$ by using the Ljusternik-Schnirelmann theory and a Moser iteration scheme.

### MSC:

 35A15 Variational methods applied to PDEs 35B09 Positive solutions to PDEs 35R11 Fractional partial differential equations 45G05 Singular nonlinear integral equations
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### References:

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