Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications

Abstract

The main aim of this article is to study non-singular version of Moser-Trudinger and Adams-Moser-Trudinger inequalities and the singular version of Moser-Trudinger equality in the Cartesian product of Sobolev spaces. As an application of these inequalities, we study a system of Kirchhoff equations with exponential non-linearity of Choquard type.In this connection, in 1960's, Pohozaev [24] and Trudinger [26] independently answered the question using the above function with φ(t) = exp(|t| n n−1 ) − 1. Later on,in [20], Moser improved the result by proving the following inequality which is popularly known as the Moser-Trudinger inequality:MT Theorem 1.1. For n ≥ 2, Ω ⊂ R n is a bounded domain and u ∈ W 1,n 0 (Ω),If κ > κ α,n,m then the above supremum is infinite (i.e. κ α,n,m is sharp).

Existence of positive solutions for a class of singular and quasilinear elliptic problems with critical exponential growth

“…Firstly, we recall some important results due to Trudinger-Moser [31,40] and Hardy-Sobolev [25]. A version of Trudinger-Moser inequality for systems can be found in [3]. (Ω) and α > 0, then exp αu…”

Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights

“…Assume that (f4) holds and let {(un, vn)} ∈ H 2 0 (Ω, R 2 ) be a Palais-Smale sequence for Φ, i.e.Φ(un, vn) → c * , Φ (un, vn) → 0 as n → ∞.Proof. Similar to[31, Lemma 3.3], so we delete the proof. Now we are ready to give the proof of our main result.Proof of Theorem 1.1.…”

(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

“…Other references, again in the Euclidean setting, are given by [7][8][9][10][11]. We also refer to the recent paper [12], which contains the proof of a non-singular version of the Moser-Trudinger inequality in the Cartesian product of Sobolev spaces.…”