On a  p   x   -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

Hafid Lebrimchi,Najib Tsouli,Mohamed Talbi,Mohammed Massar

doi:10.1155/2021/4514044

Abstract

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Highlights

Introduction whereWe consider the problem with Navier boundary conditions. 8 >>< ðPλÞ>>: M u= ð ΩΔu jΔuðxÞjpðxÞ pðxÞ = 0 on ∂Ω ! dx Δ2p u =λmðxÞjuðxÞjqðxÞ−2uðxÞ in Ω, ð1Þ p∗2
We study the existence of solutions for nonlocal pðxÞ-biharmonic Kirchhoff-type problem with Navier boundary conditions
It is clear that ðunÞ is bounded in X: there exists v1 ∈ X and a subsequence still denoted by ðunÞ such that un v1 in X: by the Hölder inequality, we get ð mðxÞjunjqðxÞ−2unðun − v1Þdx

Summary

Review Article

Received 7 June 2021; Revised 12 October 2021; Accepted 6 November 2021; Published 30 November 2021. We determine intervals of parameters for which this problem admits at least one nontrivial solution

We assume that the weight m and the Kirchhoff function

ΩmðxÞjujqðxÞ dx kuk

We start with the following auxiliary result

The fact that