Existence, Non-Existence and Degeneracy of Limit Solutions to p-Laplace Problems Involving Hardy Potentials as p→ 1⁺. The Case of A Critical Drift

In this paper we analyze the asymptotic behaviour as p→ 1⁺ p → 1 + of solutions uₚ u p to $$ \{ {array}{rclr} -Δ _pu_p& =& λ |∇ u_p|ᵖ⁻²∇ u_p· x/|x|^2+ f& in Ω ,\\ u_p& =& 0 & on ∂ Ω , {array}. $$ - Δ p u p = λ | ∇ u p | p - 2 ∇ u p · x | x | 2 + f in Ω , u p = 0 on ∂ Ω , where $$Ω $$ Ω is a bounded open subset of $$R^N$$ R N with Lipschitz boundary containing the origin, $$λ ∈ R$$ λ ∈ R , and f is a nonnegative datum in $$LN,∞(Ω )$$ L N , ∞ ( Ω