Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

By G. Autuori, A. Fiscella, Patrizia Pucci. 2014

225 citations recorded by the source corpus.

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problem −M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω, where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63f942f07d85ca7575add35adc7594350e4c277e

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Citations in the corpus

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations (2016, background): “More recently, the results of [17] were extended in [4] to the non-degenerate case, without requiring any longer that M is nondecreasing in R0 .”
Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian (2017, result): “For some recent results in the degenerate case, see for instance [2, 6, 20, 28, 31, 36] and the references therein.”
A fractional Kirchhoff problem involving a singular term and a critical nonlinearity (2017, background): “Concerning fractional Kirchhoffproblems involving critical nonlinearities, we refer to [2, 9, 13, 14, 16, 23] for existence results and to [5, 12, 24, 25, 29] for multiplicity results.”
A multiplicity result for a fractional Kirchhoff equation in ℝN with a general nonlinearity (2016, background): “Subsequently, in [4] the authors investigated the existence and the asymptotic behavior of nonnegative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a critical nonlinearity.”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “Also, see for example [39–42] for non-degenerate Kirchhoff-type problems, and [43,44] for degenerate Kirchhoff-type problems in this direction.”
Kirchhoff–Hardy Fractional Problems with Lack of Compactness (2017, methodology): “As explained in the introduction of [1], the request (1.”
Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations (2016, background): “For instance, M(t) = 2 in [0, 1], M(t) = 3− t in [1, 2], M(t) = 1 in [2, 3], M(t) = t− 2 in [3, 2 √ 3] and finally M(t) = 2 √ 3− 2 in [2 √ 3,∞) is a non monotone function satisfying (M), with θ = 1.”
A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝN (2016, background): “Indeed, the fractional Kirchhoff problems have been extensively studied in recent years; for instance, see [32] about non-degenerate Kirchhoff-type problems and [5, 34] about degenerate Kirchhoff-type problems for the recent advances in this direction.”