$p$-Kirchhoff type equation with Neumann boundary conditions

TL;DR

This work analyzes a $p$-Kirchhoff type elliptic equation with homogeneous Neumann boundary conditions, focusing on multiplicity of weak solutions under subcritical and critical nonlinearities. The authors recast the problem variationally via the energy functional $I_{b}$ and deploy Brézis–Nirenberg-type linking arguments, together with a strategic space decomposition into $W_{0}$ and constants, to establish two nontrivial solutions in the subcritical regime. In the critical regime, they employ concentration-compactness (Lions) to recover a $(PS)_{c}$ condition and, under the assumption $a>0$, obtain the same two-solution conclusion. Overall, the paper extends known Kirchhoff results by treating the nonlocal coefficient in both growth settings and providing a complete subcritical/critical analysis using variational methods.

Abstract

This paper is concerned with the multiplicity results to a class of $p$-Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by an abstract linking lemma due to Brézis and Nirenberg. We obtain the twofold results in subcritical and critical cases, which is a meaningful addition and completeness to the known results about Kirchhoff equation.

Theorems & Definitions (19)

• Definition 1.1
• Remark 1.2
• Theorem 1.3: Subcritical case
• Theorem 1.4: Critical case
• Definition 2.1: Palais, 1970, Definition 1.3, GrT01
• Lemma 2.2: BN91, Theorem 4
• Lemma 2.3: Lemma 5.1 and Lemma 5.2, GM75 or Lemma 3.3, JCF06
• Lemma 3.1
• proof
• Lemma 3.2