Note that CVX doesn’t distinguish between strict and non-strict inequalities, e.g. between x > 0 and x >=0. The solution with all variables equal to zero is feasible and produces the lowest (given that this is minimization) possible objective value; therefore it is optimal. It appears that your optimization problem formulation is not very good. It is your problem, so you will have to figure out how to improve the formulation.
In the future, it will be helpful if you provide the complete CVX program, as well as solver and CVX output. Minimal-size reproducible programs complete with all input data are always preferable.
MIxed-Integer Geometric Programs can be formulated and solved under CVX if CVXQUAD and its exponential.m replacement for CVX’s is installed (see CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions ), in which case no modifications to your program are required given that you are already using gp mode. Either Mosek or Gurobi can be selected as the solver, however, Mosek is recommended due to the existence of bugs when Gurobi is run under CVX.