On the critical $p$-Kirchhoff equation

TL;DR

This paper investigates a nonlocal p-Kirchhoff elliptic equation with critical Sobolev exponent, establishing conditions for the Palais-Smale condition and proving existence and multiplicity results using Morse theory and cohomological tools.

Contribution

It provides new sufficient conditions for the Palais-Smale condition and applies Morse theory with cohomological indices to obtain multiple solutions for the p-Kirchhoff problem.

Findings

Established conditions for the Palais-Smale condition.

Proved existence of solutions under certain conditions.

Demonstrated multiplicity of solutions using Morse theory.

Abstract

We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for the (PS) condition to hold. Then we prove some existence and multiplicity results using tools from Morse theory, in particular, the notion of a cohomological local splitting and eigenvalues based on the Fadell-Rabinowitz cohomological index.