This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of n a1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C1; ( ):
