On a class of Kirchhoff equations involving an anisotropic operator and potential
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Arabian Journal of Mathematics
R E S E A R C H A RT I C L E
Mohammed Massar
On a class of Kirchhoff equations involving an anisotropic operator and potential
Received: 20 July 2020 / Accepted: 3 November 2020 © The Author(s) 2020
Abstract In this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established. Mathematics Subject Classification
35A15, 35D30, 35R09, 35R11
1 Introduction and main result Consider the following fractional Kirchhoff equation 1 + b[u]2α (−x )α u − y u + V (x, y)u = f (u), (x, y) ∈ R N = Rn × Rm , where [u]α =
RN
(1.1)
1 α 2 |(−x ) 2 u|2 + |∇ y u|2 dxdy , α ∈ (0, 1), n, m ≥ 1, (−x )α denotes the fractional
2α u (X, Y ), where α Laplacian in x which is defined via the Fourier transform by (− u is the x ) u(X, Y ) = |X | Fourier transform of u; if u is sufficiently smooth, it can be expressed by u(x, y) − u(z, y) (−x )α u(x, y) = C N ,s P.V. dz, (x, y) ∈ R N , n+2α n |x − z| R
with C N ,α is a normalization constant and P.V. stands for the Cauchy principal value; and V, f are a functions satisfying some conditions which will be specified later. The presence of the nonlocal term [u]2α in (1.1) causes some mathematical difficulties and so the study of such a class of equations is of much interest. Moreover, Eq. (1.1) is a fractional version related to the following hyperbolic equation L 2 2
∂u
∂ 2u ρ0 E
dx ∂ u = 0, ρ 2 − (1.2) + ∂t h 2L 0 ∂ x
∂x2 which was proposed by Kirchhoff [22] as an extension of the classical D’Alembert’s wave equation by considering the changes in the length of the strings produced by transverse vibrations. On the other hand, when b = 0, V ≡ 1 and f (u) = u p , equation (1.1) appears in the study of solitary waves of the generalized Benjamin–Ono–Zakharov–Kuznetsov equation u t + ∂x1 (−x )α u − y u + u p = 0 (x, y) ∈ Rn × Rm , (1.3) M. Massar (B) Department of Mathematics, FTSH, Abdelmalek Essaadi University, Tétouan, Morocco E-mail: [email protected]
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Arab. J. Math.
see [28] for some local and global well-posedness results on this equation with m = n = 1. The anisotropic operator (−x )α u − y u is observed in the study of toy models parabolic equations for which local diffusions occur only in certain directions and nonlocal diffusions. It models diffusion sensible to the direction in the Brownian and Lévy-Itô processes. For some regularity and rigidity properties of this operator, the readers can refer to [6,17]. Recently, Esfahani [15] considered Eq. (1.1) with b = 0. Under suitable assumptions on f ∈ C 1 (R, R), by adapting some arguments developed in [7–9] and using a variant of deformation lemma, the author shows that the equation admits a least energy sign-changing solution. It is interesting to note that the fractional Laplacian (isotropic operator) proble
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