Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity

Abstract

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: −M∫Ω∇updxΔpu=up∗−2u+λup−2u−up−2ulnu2     x∈Ω,                                    u=0                                         x∈∂Ω,                  where Ω⊂ℝN is a bounded domain with a smooth boundary, 2<p<p∗<N, and both p and N are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems.