We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions $N_f$. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $epsilon$ expansion. For small $N_f$, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large $N_f$, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate $N_f$, including the graphene case ($N_f = 2$) no stable and physically admissible fixed point exists. Graphenes phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.
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