  We prove existence and pathwise uniqueness results for four different types
of stochastic differential equations (SDEs) perturbed by the past maximum
process and/or the local time at zero. Along the first three studies, the
coefficients are no longer Lipschitz. The first type is the equation
\label{eq1} X_{t}=\int_{0}^{t}\sigma
(s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+\alpha \max_{0\leq s\leq t}X_{s}. The
second type is the equation \label{eq2} {l} X_{t} =\ig{0}{t}\sigma
(s,X_{s})dW_{s}+\ig{0}{t}b(s,X_{s})ds+\alpha \max_{0\leq s\leq
t}X_{s}\,\,+L_{t}^{0}, X_{t} \geq 0, \forall t\geq 0. The third type is the
equation \label{eq3} X_{t}=x+W_{t}+\int_{0}^{t}b(X_{s},\max_{0\leq u\leq
s}X_{u})ds. We end the paper by establishing the existence of strong solution
and pathwise uniqueness, under Lipschitz condition, for the SDE \label{e2}
X_t=\xi+\int_0^t \si(s,X_s)dW_s +\int_0^t b(s,X_s)ds +\al\max_{0\leq s\leq
t}X_s +\be \min_{0\leq s \leq t}X_s.
