Existence of solutions to a critical Kirchhoff equation with logarithmic perturbation

Abstract

In this paper, the following Kirchhoff type elliptic equation − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u ln ⁡ | u | 2 + | u | 4 u , which involves a power type nonlinearity with critical Sobolev exponent and a logarithmic type perturbation, is investigated. The fact that the logarithmic term satisfies neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition brings some additional difficulties when one is looking for weak solutions to this problem. By the use of the Mountain Pass Lemma and the concentration compactness principle, it is proved that the problem admits a ground state solution for q ∈ ( 4 , 6 ) and for any 0 $ ]]> λ > 0 . It is worth pointing out that the analysis includes both the nondegenerate case (a>0) and the degenerate case (a = 0).