Scenarios of cosmic string with variable cosmological constant

Exact solutions of the Einstein field equations with cosmic string and space varying cosmological constant, viz., $\Lambda= \Lambda(r)$, in the energy-momentum tensors are presented. Three cases have been studied: where variable cosmological constant (1) has power law dependence, (2) is proportional to the string fluid density, and (3) is purely a constant. Some cases of interesting physical consequences have been found out such that (i) variable cosmological constant can be represented by a power law of the type $\Lambda=3r^{-2}$, (ii) variable cosmological constant and cosmic string density are interdependent to each other according to the relation $\Lambda=-8\pi{\rho}_s$, and (iii) cosmic string density can be scaled by a power law of the type ${\rho}_s=r^{-2}$. It is also shown that several known solutions can be recovered from the general form of the solutions obtained here.