In this paper, we study the p-Laplacian equation with a L p -norm constraint: where N ⩾ 2, a > 0, , , and is a Lagrange multiplier, which appears due to the mass constraint ‖u‖ p = a. We assume that g is odd and L p -supercritical. When and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When and μ > 0 or and μ ⩽ 0, with an additional condition of g, we obtain a positive radial ground state if μ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable , we obtain infinitely many radial solutions for any N ⩾ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N ⩾ 6. In any case mentioned above, the range of μ depends on the value of a: |μ| can be large if a > 0 is small.