A critical Kirchhoff type problem involving a nonlocal operator

Fractional Kirchhoff equation with a general critical nonlinearity

“…The equation (1.2) is related to the stationary analogue of the Kirchhoff equation u tt − a + b Ω |∇u| 2 dx ∆u = f (x, u) on Ω ⊂ R N bounded, which was proposed by Kirchhoff [13] in 1883 as a generalization the classic D'Alembert's wave equation for free vibrations of elastic strings. Recently, in bounded regular domains of R N , Fiscella and Valdinoci [11] proposed the following fractional stationary Kirchhoff equation…”

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…See [1] and the references therein for a more detailed introduction. Some interesting models involving the fractional Laplacian have received much attention recently, such as the fractional Schrödinger equation (see [2,8,17,22,23]), the fractional Kirchhoff equation (see [18,25]) and the fractional porous medium equation (see [33]). Another driving force for the study of problem (1.1) arises in the study of the following timedependent local Schrödinger equation…”

Entire solutions for critical $p$-fractional Hardy Schrödinger Kirchhoff equations

“…This crucial idea is strongly related to Lemma 2.3 of [10] (see also Lemma 4.3 of [21] and Lemma 6 of [22] for a somehow similar fractional non-degenerate Kirchhoff Dirichlet problem in bounded regular domains). The next lemma indeed is useful to obtain (1.6) and, most importantly, to defeat the lack of compactness due to the presence of a Hardy term and a critical nonlinearity.…”

“…In particular, the paper deals with stationary fractional Kirchhoff pLaplacian equations, involving critical nonlinearities, a topic of great appeal after the publication of the paper [22] due to Fiscella and Valdinoci. We refer e.g.…”

“…For this reason we assume that (1.1) is non-degenerate, that is (1.3) inf Stationary non-degenerate Kirchhoff problems have been extensively studied in the last decades, but usually under the request that M is nondecreasing in R + 0 , as in [22,33,35] and the reference therein. As in [2,10,36] the monotonicity assumption is replaced by (M).…”