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Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal \(L^2\)-critical or \(L^2\)-supercritical perturbation. (English) Zbl 1443.81033

Summary: In this paper, we study the existence and asymptotic properties of solutions to the fractional Schrödinger equation \((-\Delta)^\sigma u=\lambda u+|u|^{q - 2}u+\mu \left(I_\alpha * |u|^p\right)|u|^{p-2}u\) under the normalized constraint \(\int_{\mathbb{R}^N}u^2=a^2\), where \(N \geq 2\), \(\sigma\in(0,1)\), \(\alpha \in(0,N)\), \(q\in\left( 2+ \frac{4\sigma}{N}, \frac{2N}{N-2\sigma}\right]\), \(p\in\left[1+ \frac{2 \sigma+\alpha}{N}, \frac{N+\alpha}{N-2\sigma}\right)\), \(a,\mu>0\), \(I_\alpha(x)=|x|^{\alpha-N}\), and \(\lambda\in\mathbb{R}\) appears as a Lagrange multiplier. By using a refined version of the min-max principle, we show that the above problem admits a mountain pass type solution \(\widehat{u}_\mu\) for some \(\widehat{\lambda}<0\) under suitable assumptions on the related parameters. In particular, we can prove that \(\widehat{u}_\mu\) is a ground state if \(p\leq \frac{q}{2}+\frac{\alpha}{N}\). Furthermore, we give some asymptotic properties of the solutions. We mainly extend the results in the work of S. Bhattarai [J. Differ. Equations 263, No. 6, 3197–3229 (2017; Zbl 1515.35307)] and B. Feng et al. [J. Math. Phys. 60, No. 5, 051512, 12 p. (2019; Zbl 1414.35206)] concerning the above problem from the \(L^2\)-subcritical setting to \(L^2\)-critical and \(L^2\)-supercritical settings with respect to \(p\), involving the Sobolev critical case \(q = \frac{2N}{N-2\sigma}\) especially.
©2020 American Institute of Physics

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35R11 Fractional partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35Q40 PDEs in connection with quantum mechanics
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