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A multiplicity result for a fractional Kirchhoff equation in $ℝ^{N}$ with a general nonlinearity

Dipartimento di Scienze Pure e Applicate (DiSPeA), Università Degli Studi di Urbino ‘Carlo Bo’, Piazza Della Repubblica, 13, 61029 Urbino, Pesaro e Urbino, Italy

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Teresa Isernia

Dipartimento di Matematica e Applicazioni ‘Renato Caccioppoli’, Università Degli Studi di Napoli ‘Federico II’, Via Cintia 1, 80126 Napoli, Italy

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219199717500547Cited by:48 (Source: Crossref)

Abstract

In this paper, we deal with the following fractional Kirchhoff equation

(p + q (1 - s) \iint_{ℝ^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y) {(- Δ)}^{s} u = g (u) in ℝ^{N},

where

s \in (0, 1)

N \geq 2

p > 0

q

is a small positive parameter and

g : ℝ \to ℝ

is an odd function satisfying Berestycki–Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that

q

is sufficiently small.

Keywords:

AMSC: 35A15, 35R11, 49J35

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