Fractional Kirchhoff problem with critical indefinite nonlinearity

Abstract

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form (where Ω ⊂ ℝ is a smooth bounded domain, 0 < < 1 and 1 < < 2. Here is the Kirchhoff coefficient and 2 * = 2 ∕( − 2 ) is the fractional critical Sobolev exponent. The parameter is positive and the ( ) is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results. K E Y W O R D S critical exponent, fractional Laplacian, Kirchhoff type problem M S C ( 2 0 1 0 ) 35A15, 35B33, 35R11

Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

“…The classical Brezis-Nirenberg result was generalized to the case of nonlocal fractional operators through variational techniques. The existence of multiple solutions to the fractional Laplacian equations of Kirchhoff type was considered in [17] and two positive solutions for proper selection of positive parameter λ was obtained.…”

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

“…They addressed the existence of at least two positive solutions depending on the parameters by exploiting the Nehari manifold. do Ó et al [15] studied a fractional stationary Kirchhoff equation involving two nonlinear source terms of different signs:…”

Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

“…Here they proved that if ε > 0 is sufficiently small then there exists a λ * > 0 such that for any λ ∈ (0, λ * ), problem (1.2) has at least two positive solutions, and one of the solution is a ground state solution. We refer to [1,4,7,8,9,20] for Kirchhoff problems involving the classical Laplace operator and p−fractional Laplace operators.…”