2018/07/26 by Gabriele Mancini, Mancini, Gabriele, Pierre-Damien Thizy +1
Computer Science · Mathematics · #35B33 #35B44 #35J15 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1807.10098
openalex publication_date 2018/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Druet [6] proved that if (fγ)γ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if (uγ)γ solves \begincases amp;Δu =fγ(x,u) ,~~ ugt;0 in Ω ,
amp;u =0 on ∂Ω , \endcases and converges weakly in H10 to some u_∞, then the Dirichlet energy is quantified, namely there exists an integer N≥ 0 such that the energy of uγ converges to 4πN plus the Dirichlet energy of u_∞. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities, the loss of compactness (i.e. N>0) implies that u_∞≡ 0. In contrast, we prove here that there exist sequences (fγ)γ of Moser-Trudinger type nonlinearities which admit a noncompact sequence of solutions (uγ)γ having a nontrivial weak limit.