It has been shown by Trudinger and Moser that for normalized functions u of the Sobolev space W 1,N ( ), where is a bounded domain in R N , one has exp(α N |u| N /(N −1) )dx ≤ C N , where α N is an explicit constant depending only on N , and C N is a constant depending only on N and . Carleson and Chang proved that there exists a corresponding extremal function in the case that is the unit ball in R N . In this paper we give a new proof, a generalization, and a new interpretation of this result. In particular, we give an explicit sequence that is maximizing for the above integral among all normalized "concentrating sequences." As an application, the existence of a nontrivial solution for a related elliptic equation with "Trudinger-Moser" growth is proved.

We study the following semilinear biharmonic equation:

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Abstract:We consider a fractional Schrödinger-Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

ABSTRACT. In this paper we study the existence of nontrivial solutions for the following system of two coupled semilinear Poisson equations:where Ω is a bounded domain in R 2 with smooth boundary ∂Ω, and the functions f and g have the maximal growth which allow us to treat problem (S) variationally in the Sobolev spaceWe consider the case with nonlinearities in the critical growth range suggested by the so-called Trudinger-Moser inequality.