Existence of Solutions for p-Kirchhoff Problem of Brézis-Nirenberg Type with Singular Terms

Atika Matallah,Hayat Benchira,M El Mokhtar Ould El Mokhtar,Serena Matucci

doi:10.1155/2022/7474777

Atika Matallah, Hayat Benchira + Show 2 more

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https://doi.org/10.1155/2022/7474777

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Abstract

In this paper, we prove the existence of positive solution for a p-Kirchhoff problem of Brézis-Nirenberg type with singular terms, nonlocal term, and the Caffarelli-Kohn-Nirenberg exponent by using variational methods, concentration compactness, and maximum principle.

Highlights

We prove the existence of positive solution for a p-Kirchhoff problem of Brézis-Nirenberg type with singular terms, nonlocal term, and the Caffarelli-Kohn-Nirenberg exponent by using variational methods, concentration compactness, and maximum principle
Where Ω is a bounded smooth domain in RN, 1 < p < N, 0 ∈ Ω, KðuÞ = akukq + b,a, b, q > 0,0 ≤ α < ðN − pÞ/p, α ≤ β < α + 1,0 ≤ γ < p,0 ≤ μ < μ ≔ 1⁄2ðN − ðα + 1ÞpÞ/pp,λ > 0, p∗ = pN/1⁄2N − pð1 + α − βÞ is the critical Caffarelli-Kohn-Nirenberg exponent corresponding to the noncompact embedding of DαðΩÞ into Lp∗ ðΩ, jxjβp∗ Þ, where DαðΩÞ is the closure of C∞ 0 ðΩÞ with respect to the norm ð j∇ujp
It is natural for us to consider the quasilinear Brézis-Nirenberg problem in [10] with nonlocal term and singular weights, ðp > 1, a ≠ 0 and ðα, β, γ, μÞ ≠ ð0, 0, 0, 0ÞÞ: The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of DαðΩÞ into Lp∗ ðΩ, jxjβp∗ Þ prevent us from using the variational methods in a standard way

Summary

Introduction and Main Result

We consider the following p-Kirchhoff problem of Brézis-Nirenberg type with singular terms. In the case p = 2 and α = β = γ = μ = 0, it is analogous to the stationary version of equations that arise in the study of string or membrane vibrations, namely, utt − KðuÞΔu = gðx, uÞ ð3Þ where u denotes the displacement and gðx, uÞ is the external force Equations of this type were first proposed by Kirchhoff in 1883 [1] to describe the transversal oscillations of a stretched string. It is natural for us to consider the quasilinear Brézis-Nirenberg problem in [10] with nonlocal term and singular weights, ðp > 1, a ≠ 0 and ðα, β, γ, μÞ ≠ ð0, 0, 0, 0ÞÞ: The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of DαðΩÞ into Lp∗ ðΩ, jxjβp∗ Þ prevent us from using the variational methods in a standard way.

Variational Framework and Preliminary Results

Conclusion 1