Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

We are concerned with Dirichlet problems of the form $$ div(|D u|^{p-2}Du)+f(u)=0 mbox{ in }Omega,qquad u=0 mbox{ on }partialOmega, $$ where $Omega$ is a bounded domain of $R^n$, $nge 2$, $10$ small enough, there exists a unique solution of the Dirichlet problem in the domain $Omega=Omega^Gamma_e={(x_1,x_2)inR^2 : distig((x_1,x_2),Gammaig)2$ and $Omega$ is, for example, a domain of the type $$ Omega=widetildeOmega^Gamma_{e,s}={(x_1,x_2,y) : (x_1,x_2)inOmega^Gamma_e, yinR^{n-2}, |y