On critical Kirchhoff problems driven by the fractional Laplacian

Luigi Appolloni,Simone Secchi,Giovanni Molica Bisci

doi:10.1007/s00526-021-02065-8

Abstract

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the Sobolev embedding sense. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.

Highlights

The equation that goes under the name of Kirchhoff equation was proposed in [17] as a model for the transverse oscillation of a stretched string in the form ρh ∂t2t u − Eh p0 + 2L L |∂x u|2 d x∂x2x u + δ ∂t u + f (x, u) = 0 (1)for t ≥ 0 and 0 < x < L, where u = u(t, x) is the lateral displacement at time t and at position x, E is the Young modulus, ρ is the mass density, h is the cross section area, L the length of the string, p0 is the initial stress tension, δ the resistance modulus and g the external force
In the recent [21], Liu, Squassina and Zhang studied ground state solutions for the Kirchhoff equation plus a potential with a non linear term asymptotic to a power with critical growth in low dimension. It is worth mentioning [22] where Mingqi, Radulescu and Zhang proved the existence of nontrivial radial solutions in the non-degenerate and degenerate cases for the non local Kirchhoff problem in which the fractional Laplacian is replaced by the fractional magnetic operator
Even if some of the results we are going to prove are known, we present a proof based on a adaptation to the fractional case due to Palatucci and Pisante ([25]) of the Lions second concentration-compactness principle; for the original version of the lemma we refer to [20], as well as [19]

Summary

Introduction

The equation that goes under the name of Kirchhoff equation was proposed in [17] as a model for the transverse oscillation of a stretched string in the form ρh ∂t2t u −. See [14], Faraci, Farkas and Kristály studied Eq (4) with g(x, u) = 0 and under suitable assumptions on the parameters a and b they proved that the functional associated to the problem is sequentially weakly lower semicontinuous, satisfies the Palais–Smale condition and is convex. In the recent [21], Liu, Squassina and Zhang studied ground state solutions for the Kirchhoff equation plus a potential with a non linear term asymptotic to a power with critical growth in low dimension It is worth mentioning [22] where Mingqi, Radulescu and Zhang proved the existence of nontrivial radial solutions in the non-degenerate and degenerate cases for the non local Kirchhoff problem in which the fractional Laplacian is replaced by the fractional magnetic operator.

Semicontinuity and the validity of the Palais–Smale condition

Application to a perturbed Kirchhoff problem