The paper is concerned with some problems in the \(L_p\) Brunn-Minkowski theory of convex bodies. Several mixed \(L_p\) geominimal surface areas are introduced for multiple convex bodies for all \(-n\neq p\in \mathbb R\). Properties like the affine invariance are put forward. Related Alexander-Fenchel-type inequalities, Santaló-type inequalities, affine isoperimetric inequalities, and cyclic inequalities are established. An equivalent formula for the mixed \(p\)-affine surface area is provided. Relations between mixed \(L_p\) geominimal surface areas and the mixed \(p\)-affine surface area are discussed. The \(i\)th mixed \(L_p\) geominimal surface areas and related isoperimetric inequalities are put forward.