Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in ℝ2

Extremal functions for Moser’s inequality

“…Let Ω be a bounded smooth domain in R n , and u(x) a C 1 function supported in Ω with ∇u q < n. Sobolev's Imbedding Theorem says that if 1 ≤ q < n, then u p ≤ C(n, q), (1) where…”

“…Their result came as a surprise, since it was known at that time that no extremals exist for Sobolev's inequality (1) when Ω is a ball. (See an account of this in the more expository article [10].)…”

“…The key ingredient is the use of n-Green's functions, the singular solutions to the n-Laplacian. As to the solvability of the corresponding Euler equation, see Adimurthi [1] and Struwe [6].…”

On a singular elliptic problem involving critical growth in $${\mathbb{R}^{\bf N}}$$

“…Remark 1.4. The results in this paper were in part motivated by several recent papers on elliptic problems involving critical growth in the Trudinger-Moser case, see [1,2,4,6,9,12,13,15,16,27] and references therein. Here we complement some the results mentioned above by establishing sufficient conditions for the existence of nontrivial solutions for singular case with a ∈ [0, N).…”

“…Here we complement some the results mentioned above by establishing sufficient conditions for the existence of nontrivial solutions for singular case with a ∈ [0, N). For problem (1.1), when a ≡ 0, the existence of nontrivial solutions has been studied on bounded domains by [9] in the semilinear case and by [1,12,23] for the quasilinear equations. For problems in unbounded domains see [6,13].…”