Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity

Abstract

In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equationα 2 stands for the fractional Laplacian operator, f ∈ C(Ω × R, R) may be sign changing and λ is a positive parameter. We will prove that there exists λ * > 0 such that the problem has at least two positive solutions for each λ ∈ (0 , λ * ). In addition, the concentration behavior of the solutions are investigated.2010 Mathematics Subject Classification. 35J60, 47J30.

Fractional Kirchhoff problem with critical indefinite nonlinearity

“…Using the Nehari manifold technique, authors proved the existence of at least two positive solutions for suitable choice of λ. In the case and similar results for fractional Laplacian has been studied in using the harmonic extension technique introduced by Caffarelli and Silvestre in . In the case and , there is a lot of work addressed by many researchers, see and references therein.…”

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

“…The same results have been obtained by Barrios et al in [2] in the critical case, but in the absence of the Hardy and singularity terms, i.e., when γ = s = 0. Recently, Zhang-Liu-Jiao [25] extended the results of [2] to the problem involving the sign-changing weight function f (x) ∈ C(Ω) with f + ≡ 0, and g(x) ≡ 1. When α = 2, i.e., in the case of the standard Laplacian, problem (1) has been studied extensively in the last decade; see for example [18], [19], [20], [24] and references therein.…”

“…We point out that the non-local problems are still much less understood than their local counterpart. The aim of this paper is to consider the remaining cases and generalize the results of [4] and [25] to the problem involving the Hardy potential, Hardy-Sobolev singularity term, and also signchanging functions f (x) and g(x). More pricesly, we study problem (1) in the sub-critical (i.e., when 2 < p < 2 * α (s)) and critical (i.e., when p = 2 * α (s)) case, separately.…”

Multiple Solutions for a Fractional Laplacian System Involving Critical Sobolev-Hardy Exponents and Homogeneous Term

“…We show that the system (1.1) has at least two positive solutions when the parameters λ, µ and weight functions f , g satisfied some certain conditions. It should be mentioned that in [8,9,10,15,22], some problems involving fractional Laplacian operator were investigated by the Nehari manifold and fibering method. We look for solutions of (1.1) in the Sobolev space From (2.2), we employ the following equivalent norm in X s 0 (Ω):…”