Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems

Abstract

In this paper, we deal with the existence, multiplicity and asymptotic behavior of solutions for the following stationary Kirchhoff problems involving the p-Laplacian:−M(∫Ω|∇u|pdx)Δpu=λf(x,u)+|u|p⁎−2uinΩ,u=0 on∂Ω, where Ω is a bounded smooth domain of RN, M:R0+→R0+ is a continuous function satisfying some extra assumptions. By using the mountain pass theorem, the existence of solutions is obtained. By applying an abstract critical point theorem based on the cohomological index, we prove the existence of multiple solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator −‖∇u‖(θ−1)pΔp, where p<N<θp<p⁎. We also investigate the asymptotic behavior of solutions when λ converges to infinity. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero, i.e. the problem is degenerate.