Exact solutions using power law scalar potential in the Saez-Ballester-K-essence like theory

Abstract: We investigate a K-essence like cosmological model whose scalar-field potential is constructed from a negative power-law Sáez--Ballester potential. By means of a suitable field redefinition from $φ$ to $\varphi$, we show that the resulting field equations acquire a mathematical structure analogous to that of a previously solved Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological model. This correspondence allows us to obtain exact classical solutions for both the scale factor and the scalar field within the Hamiltonian formalism. The resulting cosmological dynamics exhibits a late-time accelerated expansion, with the deceleration parameter approaching the asymptotic value $q\rightarrow -1$, characteristic of a de Sitter phase. At the quantum level, the corresponding Wheeler-DeWitt (WDW) equation is derived and exact quantum solutions are obtained. These results provide a consistent classical and quantum description of the cosmological evolution generated by this class of K-essence models. In this formalism, the scalar field remains as a cosmic background where the universe unfolds, which is glimpsed from the quantum solution perspective.