Druet [6] proved that if (fγ)γ is a sequence of Moser–Trudinger type nonlinearities with critical growth, and if (uγ)γ solves (1) and converges weakly in H01 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 (see (2)), the loss of compactness (i.e. N positive) 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 (uγ)γ of solutions of (1) having a nontrivial weak limit.
Glueing a peak to a non-zero limiting profile for a critical Moser–Trudinger equation
Mancini G.;
2019-01-01
Abstract
Druet [6] proved that if (fγ)γ is a sequence of Moser–Trudinger type nonlinearities with critical growth, and if (uγ)γ solves (1) and converges weakly in H01 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 (see (2)), the loss of compactness (i.e. N positive) 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 (uγ)γ of solutions of (1) having a nontrivial weak limit.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
