Trace operator and a nonlinear boundary value problem in a new space

We develop a new function space and discuss trace operator on the same genealogical spaces. We also prove that the nonlinear boundary value problem with Dirichlet condition: −Δu=f(|u|)sgnu in the given domain, u=0 on the boundary, possesses only a trivial solution if f obeys the slope condition: α′(x)>2nn−2α(x)x, where α is the anti-derivative of f with α(0)=0.