Summary: Using the profile decomposition, we will show the relatively compactness of the minimizing sequence to the critical embeddings between Besov spaces, which implies the existence of minimizer of the critical embeddings of Besov spaces \(\dot{B}^{s_1}_{p_1,q_1}\hookrightarrow \dot{B}^{s_2}_{p_2,q_2}\) in \(d\) dimensions with \(s_1-d/p_1=s_2-d/p_2\), \(s_1>s_2\) and \( 1\le q_1 < q_2 \le \infty\). Moreover, we establish the nonexistence of the minimizer in the case \(\dot{B}^{s_1}_{p_1,q}\hookrightarrow \dot{B}^{s_2}_{p_2,q}\).