  The notions of generalized principal eigenvalue for linear second order
elliptic operators in general domains introduced by Berestycki et al.
\cite{BNV,BR0,BR3} have become a very useful and important tool in analysis of
partial differential equations. In this paper, we extend these notions for
quasilinear operator of the form $$\CK_V[u]:=-\Delta_p u +Vu^{p-1},\quad\quad u
\geq0.$$ This operator is a natural generalization of self-adjoint linear
operators. If $\O$ is a smooth bounded domain, we already proved in \cite{NV}
that the generalized principal eigenvalue coincides with the (classical) first
eigenvalue of $\CK_V$. Here we investigate the relation between three types of
the generalized principal eigenvalue for quasilinear operator on general smooth
domain (possibly unbounded), which plays an important role in the investigation
of their asymptotic properties. These results form the basis for the study of
the simplicity of the generalized principal eigenvalues, the maximum principle
and the spectrum of $\CK_V$. We further discuss applications of the notions by
providing some examples.
