  In this paper, we investigate the limiting absorption principle associated to
and the well-posedness of the Helmholtz equations with sign changing
coefficients which are used to model negative index materials. Using the
reflecting technique introduced in \cite{Ng-Complementary}, we first derive
Cauchy problems from these equations. The limiting absorption principle and the
well-posedness are then obtained via various a priori estimates for these
Cauchy problems. There approaches are proposed to obtain the a priori
estimates. The first one follows from a priori estimates of elliptic systems
equipped general complementing boundary conditions due to Agmon, Douglis, and
Nirenberg in their classic work \cite{ADNII}. The second approach, which is
complement to the first one, is variational and based on the Dirichlet
principle. The last approach, which is complement to the second one, is also
variational and uses the multiplier technique. As a consequence, we obtain new
results on the well-posedness of these equations for which the positivity
conditions concerning the matrix-valued coefficients are imposed partially or
not strictly on the interface of sign changing coefficients. As a result, the
well-posedness can hold even in the case the contrast of the coefficients
across the sign changing interfaces is arbitrary. This allows us to rediscover
and extend largely known results obtained by the integral method, the pseudo
differential operator theory, and the T-coercivity approach. The unique
solution, obtained by the limiting absorption principle, is {\bf not} in
$H^1_{\loc}(\mR^d)$ as usual and possibly {\bf not even} in
$L^2_{\loc}(\mR^d)$. The optimality of our results is also discussed.
