Infinitely many periodic solutions for a class of fractional Kirchhoff problems

Abstract

We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schrödinger operator with periodic boundary conditions and involving different types of nonlinearities.2010 Mathematics Subject Classification. 34K13, 35R11, 35A15, 35B33.

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

“…Other types of periodic problems related to the spectral fractional Laplacian can be found in [2][3][4][5][6]. Now, we will give our first existence result in the case of attractive singularity.…”

Finer analysis of the Nehari set associated to a class of Kirchhoff-type equations

“…and H k ðuÞ ¼ 0: \l à ðkÞ which is a contradiction. Therefore a\0, so by taking t ¼ jaj 1 2 , it follows that tu is a nonnegative solution of problem (1). h Now we look for the second solution when ðk; lÞ belong to the extremal curve.…”

“…In the work of D'Ancona and Spagnolo [8] it was proved global solvability results in the degenerate case, that is p 0 ¼ 0 (see the references therein for the non-degenerate case). We note that (1) is the stationary version of (2) and we refer the reader to some works and their references: Alves et al [1], Ambrosio [2], Ambrosio and Isernia [3], Autuori et al [4], Corrêa and Figueiredo [6], Fiscella and Valdinoci [9], Santos Junior and Siciliano [12]. These works concerns various kinds of hypothesis and models related to Kirchhoff-type equations.…”