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We set up the sharp Trudinger–Moser inequality under arbitrary norms. Using this result and the 
$L_{p}$
Busemann-Petty centroid inequality, we will provide a new proof to the sharp affine Trudinger–Moser inequalities without using the well-known affine Pólya–Szegö inequality.
The aim of this paper is to study Hamiltonian elliptic system of the form
0.1
where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane
0.2
$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).
$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$
We consider the semilinear problem
where λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.
We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces 
and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and 
of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.
In this work we study the following class of elliptic systems:
where Ω ⊂ ℝ2 is a smooth bounded domain, H is a C1 function in [0, +∞)×[0, +∞) which is assumed to be in the critical growth range of Trudinger–Moser type and f1, f2 ∈ Lr (Ω), r > 2. Under suitable hypotheses on the functions a, b, c ∈ C(
and using variational methods, we prove the existence of two solutions depending on f1 and f2.
