This paper deals with the existence of many solutions a degenerateweighted elliptic equation which its the number of solutions dependon degeneracy term $a$ i.e., the number of subdomains of$\Omega\setminus a^{-1}(0)$ 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$C^{1,\alpha}(\Omega).$