High perturbations of critical fractional Kirchhoff equations with logarithmic nonlinearity

Abstract

This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional p-Kirchhoff equations: M([u]s,pp)(−Δ)psu=λ|u|q−2uln|u|2+|u|ps∗−2uinΩ,u=0inRN∖Ω,where Ω⊂RN is a bounded domain with Lipschitz boundary, N>sp with s∈(0,1), p≥2, ps∗=Np∕(N−ps) is the fractional critical Sobolev exponent, and λ is a positive parameter. The main result establishes the existence of nontrivial solutions in the case of high perturbations of the logarithmic nonlinearity (large values of λ). The features of this paper are the following: (i) the presence of a logarithmic nonlinearity; (ii) the lack of compactness due to the critical term; (iii) our analysis includes the degenerate case, which corresponds to the Kirchhoff term M vanishing at zero.