Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation

The paper concerns with the existence of positive solutions for the following Kirchhoff problem [GRAPHICS] where a, b > 0,4 < p < 6, epsilon > 0 is a parameter and Omega subset of R-3 is an exterior domain, that is, Omega is an unbounded domain with R-3\Omega nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when R-3\Omega is contained in a small ball and epsilon > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309-330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.