Abstract We prove nonexistence and uniqueness results of solutions for biharmonic equations under the Dirichlet boundary conditions on a smooth bounded domain. We carry on the work in [10] where the Navier boundary conditions were considered, we define the h -starlikeness of Ω with respect to the Dirichlet boundary conditions and a classifying number M * ⁢ ( Ω ) ${M^{*}(\Omega)}$ . This allows us to give a generalized critical exponent for these domains which play the role of the classical critical exponent N + 4 N - 4 ${\frac{N+4}{N-4}}$ . Our approach is based on the Rellich–Pohozaev type identity [20, 23]. We study some examples of Dirichlet h -starlike domains with either rich topology or rich geometry where our results can apply.