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

Abstract

ABSTRACT The paper concerns with the existence of positive solutions for the following Kirchhoff problem where a, b>0, 4<p<6, is a parameter and is an exterior domain, that is, Ω is an unbounded domain with nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when is contained in a small ball and 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.