  We consider a one dimensional Euclidean network which is grown using a
preferential attachment. Here the $j$th incoming node gets attached to the
$i$th existing node with the probability $\Pi_i \propto k_i {{l}}_{ij}^\alpha$,
where ${l}_{ij}$ is the Euclidean distance between them and $k_i$ the degree of
the $i$th node. This network is known to have a static phase transition point
at $\alpha_c \simeq 0.5$. On this network, we employ three different searching
strategies based on degrees or distances or both, where the possibility of
termination of search chains is allowed. A detailed analysis shows that these
strategies are significantly affected by the presence of the static critical
point. The distributions of the search path lengths and the success rates are
also estimated and compared for the different strategies. These distributions
appear to be marginally affected by the static phase transition.
