We study the following singularly perturbed nonlocal Schrödinger equationwhere V (x) is a continuous real function on R 2 , F (s) is the primitive of f (s), 0 < µ < 2 and ε is a positive parameter. Assuming that the nonlinearity f (s) has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods. (2010): 35J20, 35J60, 35B33 Mathematics Subject Classifications Introduction and main resultsThe nonlocal elliptic equationthe so-called Choquard equation when N = 3, appears in the theory of Bose-Einstein condensation and is used to describe the finite-range many-body interactions between particles.Here V (x) is the external potential, F (s) is the primitive of the nonlinearity f (s) and the parameters ε > 0, 0 < µ < N . For µ = 1 and F (s) = 1 2 |s| 2 , equation (SN S) was investigated by S.I. Pekar in [42] to study the quantum theory of a polaron at rest. In [28] P. Choquard suggested to use it as approximation to Hartree-Fock theory of one-component plasma. This equation was also proposed by R. Penrose in [36] as a model for selfgravitating particles and it is known in that context as the Schrödinger-Newton equation.Notice that if u is a solution of the nonlocal equation (SN S) and x 0 ∈ R N , then theThis suggests some convergence, as ε → 0, of the family of solutions of (SN S) to a solution u 0 of the limit problem(1.1) This is known as semi-classical limit for the nonlocal Choquard equation and we refer for a survey to [8,9]. The study of semiclassical states for the Schrödinger equationgoes back to the pioneer work [24] by Floer and Weinstein. Since then, it has been studied extensively under various hypotheses on the potential and the nonlinearity, see for example [7,16,17,24,25,26,43,44,46,48] and the references therein. In the study of semiclassical problems for local Schrödinger equations, the Lyapunov-Schmidt reduction method has been proved to be one of the most powerful tools. However, this technique relies on the uniqueness and non-degeneracy of the ground states of the limit problem which is not completely settled for the ground states of the nonlocal Choquard equation(1.3) CC In [33, 15, 37], have been investigated qualitative properties of solutions and established regularity, positivity, radial symmetry and decaying behavior at infinity. Moroz and Van Schaftingen in [38] established the existence of ground states under the assumption of Berestycki-Lions type and for the critical equation in the sense of Hardy-Littlewood-Sobolev inequality. For N = 3, µ = 1 and F (s) = 1 2 |s| 2 , by proving the uniqueness and non-degeneracy of the ground states, Wei and Winter [47] constructed a family of solutions by a Lyapunov-Schmidt type 1 2p.Combining the above estimates with the Hardy-Littlewood-Sobolev inequality and some results due to P.L. Lions, the following existence result was obtained in [6].

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We study the following semilinear biharmonic equation:

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space D 1,2 (R 2 ), completion of smooth compactly supported functions with respect to the Dirichlet norm ∇ ⋅ 2 , and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H 1 (R 2 ) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.

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