We provide answers to some questions raised in a recent work by H. Brezis, J. Van Schaftingen and P.-L. Yung [2] , [3] concerning the Gagliardo semi-norm | u W s q computed at = 1 when strong L is replaced weak . In particular, we address generalization of results for general domain non-smooth functions.

In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations:    −∆u = v , in Ω, −∆v = f(u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R . We assume that 0 < p < 2 N−2 , and the function f is superlinear and with no growth restriction (for example f(s) = s e); then the system has a nontrivial (strong) sol...