Existence Results for a Nonhomogeneous Weighted p-Laplacian Problem with Critical Sobolev Growth

In this paper, we study a class of nonlinear elliptic Dirichlet problems involving a weighted p-Laplacian operator and critical Sobolev growth. More precisely, we consider

-div(α(x)|∇u|^(p−2)∇u) = |u|^(p*−2)u + f in Ω,
u = 0 on ∂Ω,

where Ω ⊂ ℝ^N is a bounded smooth domain, 1 < p < N, p* = Np/(N−p) is the critical Sobolev exponent, α(x) is a positive continuous weight, and f ∈ W^(−1,p′)(Ω) is a nontrivial perturbation.

The presence of the critical exponent leads to a lack of compactness in the associated Sobolev embedding, which makes the variational analysis highly nontrivial. The weight function α(x) introduces additional analytical difficulties and breaks the homogeneity of the problem, affecting the compactness properties of Palais–Smale sequences.

Our approach is based on variational methods and mountain pass arguments in the spirit of Brezis–Nirenberg, combined with concentration–compactness techniques. The problem is motivated by the seminal work of Brezis and Nirenberg, further extended by Tarantello to nonhomogeneous perturbations, and later generalized to weighted p-Laplacian frameworks by Benhamida and Yazidi.

We establish existence results under suitable assumptions on f and α(x), and we discuss how the interaction between the weight, the geometry of Ω, and the critical exponent influences the variational structure of the problem.