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Concentration phenomena for fractional magnetic NLS equations
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 2 / April 2022
- Published online by Cambridge University Press:
- 28 May 2021, pp. 479-517
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- April 2022
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Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 23 January 2019, pp. 655-694
- Print publication:
- April 2020
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This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type
where ε > 0 is a parameter, s, t ∈ (0, 1) are such that 2s+2t>3, A:ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ)As is the fractional magnetic Laplacian, V:ℝ3 → ℝ is a continuous electric potential and f:ℝ → ℝ is a C1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough.
$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$
