Abstract
In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in ℝN:
where N ⩾ 3, s ∈ (0, 1), α ∈ (0, N), \(p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} \frac{{N + a}}{{N - 2s}})and{\kappa _a}(x) = |x{|^{a - N}}\). To get such solutions, we look for critical points of the energy functional
on the constraints
For the value \(p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} ,\frac{{N + a}}{{N - 2s}})\) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c > 0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that, we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any c > 0, there are infinitely many radial critical points of I restricted on S(c).
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11371159 and 11771166).
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Li, G., Luo, X. Existence and multiplicity of normalized solutions for a class of fractional Choquard equations. Sci. China Math. 63, 539–558 (2020). https://doi.org/10.1007/s11425-017-9287-6
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DOI: https://doi.org/10.1007/s11425-017-9287-6