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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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