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In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrödinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a nontrivial least energy solution is considered for the nonlinear fractional Schrödinger-Poisson systems (−Δ)su + V(x)u + ϕu = |u|p−1u and (−Δ)tϕ = u2 in R 3 , where (−Δ)α is the fractional Laplacian for α = s, t ∈ (0, 1) with s < t and 2s + 2t > 3. Under some appropriate assumptions on the non-constant potential V(x), we prove the existence of a nontrivial least energy solution when 2 < p < 2 s * − 1 = (3 + 2 s) / (3 − 2 s) by using some new analytical skills and the Nehari-Pohožaev type manifold. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical fractional Schrödinger equation: ( − Δ ) s u + V ( x ) u = | u | 2 s * − 2 u + λ f ( x , u ) , x ∈ R N , where λ > 0, 0 < s < 1, N > 2s, 2 s * = 2 N N − 2 s , (−Δ)s denotes the fractional Laplacian of order s, and f is a continuous superlinear but subcritical function. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by the Nehari method. [ABSTRACT FROM AUTHOR]

In the present paper, it is proved that there are many quasi-periodic solutions of a class of space fractional nonlinear Schrödinger equations with the Riesz fractional derivative by means of KAM (Kolmogorov-Arnold-Moser) theorem. [ABSTRACT FROM AUTHOR]

This paper considers the fractional Schrödinger-Poisson system in R³. We prove that the problem has m-bump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multi-bump solutions as ϵ → 0. [ABSTRACT FROM AUTHOR]

This paper is a direct offspring of the work of Garbaczewski and Stephanovich ["L'evy flights and nonlocal quantum dynamics," J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed. [ABSTRACT FROM AUTHOR]