Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity
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Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity K. Saoudi1,2 · A. Ghanmi3 · S. Horrigue4 Received: 14 June 2020 / Revised: 6 July 2020 / Accepted: 10 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this work, we study the existence and the multiplicity of non-negative solutions for the following problem ⎧ ⎨ Lu = a(x)u q + λb(x)u p in , (Pλ ) ⎩ u = 0, in Rn \ , where ⊂ Rn (n ≥ 2) , is a bounded smooth domain, λ, p, q are positive real numbers, s ∈ (0, 1), a, b are continuous functions, and L is a nonlocal operator defined later by (1.1). We establish the existence and we give a multiplicity of solutions by constrained minimization of the Euler-Lagrange functional corresponding to the problem (Pλ ), on suitable subsets of Nehari manifold and using the fibering maps. Precisely, we show the existence of λ0 > 0, such that for all λ ∈ (0, λ0 ), problem (Pλ ) has at least two non-negative solutions. Keywords Non-local operator · Fractional Laplacian · Multiple solutions · Sign-changing weight functions · Nehari manifold · Fibering maps
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A. Ghanmi [email protected] K. Saoudi [email protected] S. Horrigue [email protected]
1
College of Sciences at Dammam, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of Saudi Arabia
2
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441 Dammam, Saudi Arabia
3
Faculté des Sciences de Tunis, LR10ES09 Modélisation mathématique, analyse harmonique et théorie du potentiel, Université de Tunis El Manar, Tunis 2092, Tunisie
4
Faculté des Sciences de Tunis, Université de Tunis El Manar, Tunis 2092, Tunisie
K. Saoudi et al.
Mathematics Subject Classification 34B15 · 37C25 · 35R20
1 Introduction The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past four decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. For more details, we refer to [3,11–13,24]. Recently the study of existence and multiplicity of solutions for fractional elliptic equations attracts a lot of interest in nonlinear analysis such as in [7–9,14,15,17– 22] and references therein. Caffarelli and Silvestre [6] gave a new formulation of fractional Laplacian through Dirichlet-Neumann maps, this formulation transforms problems involving the fractional Laplacian into a local problem which allows one to use the variational methods. In the local setting, s = 1, in the papers [2] and [23] , are the starting point on semilinear elliptic problems with positive nonlinearities. Moreover, there is a large literature on polynomial type nonlinearity with sign-changing weight functions using Nehari manifold and fibering map analysis see [5,10,25] and references therein. The aim of this work is to study the existence and multiplicity of non-negative solutions
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