Existence of extremals for Trudinger–Moser inequalities involved with a trapping potential

In this paper, we establish the existence of extremals for the Trudinger–Moser functional under the weighted Sobolev norm involved with the trapping potential. Since the trapping potential is non-radial, possible extremals of the Trudinger–Moser functional do not have to be radially symmetric. As a result, one can’t expect to apply the symmetrization argument to obtain the existence of extremal functions for these inequalities. Thus, the standard blow-up analysis relying on the symmetrization argument no longer works. In this paper, we will develop a new symmetrization-free approach to establish the existence of an extremal (Theorem 1.1) We first show that if the maximizing sequence does not converge strongly to an extremal function, then it must be a concentration or vanishing sequence through excluding the dichotomy phenomenon (see Lemma 2.9). Since the vanishing or concentration sequence may not be radially symmetric, this causes much challenge to exclude the vanishing or concentration phenomenon of the maximizing sequence. Then the elimination of the vanishing phenomenon can be done by comparing the supremums of the Trudinger–Moser functional under the constraint of the classical Sobolev norm (i.e., S_∞(α)) and weighted Sobolev norm associated with the potential (i.e., S(V, α)) respectively. For the elimination of the concentration phenomenon of the maximizing sequence, we will develop the blow-up analysis procedure for the non-radial maximizing sequence. Furthermore, we also study the existence of extremals for the perturbed Trudinger–Moser inequality in R² involved with trapping potential (see Theorem 1.3). While an extremal function of the classical Trudinger–Moser and the perturbed Trudinger–Moser inequality does not exist in some situations, surprisingly, the supremum of the Trudinger–Moser and the perturbed Trudinger–Moser functional involved with the trapping potential can always be achieved.