We establish that, for n 3, the elliptic equation −∆u = λ|x| µ |u| q−2 u + |x| ν |u| p−2 u on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all λ > 0 provided thatWhen q = 2 and n > 2µ + 6, the same result holds for λ > 0 small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation −∆w − χ |y| 2 w =λ|y| a |w| q−2 w + |y| ν |w| p−2 w, on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all λ > 0 provided that a, ν > −2 and max 2, n + a − √χ − χ √χ < q < n + a √χ withχ = n − 2 2 2 .When q = 2, n > 2a + 6 and 0 < χ <χ − (a + 2) 2 , the same result holds forλ > 0 small.