This paper deals with the existence of many solutions a degenerateweighted elliptic equation which its the number of solutions dependon degeneracy term ie, the number of subdomains of whose boundary is made by submanifoldswith 1-codimension. Moreover, by the Ljusternik–-Schnirelmanprinciple, we would study the corresponding eigenvalue problem andobtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on
