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In this paper, we study the p-Laplacian equation with a L p -norm constraint: − Δ p u = λ | u | p − 2 u + μ | u | q − 2 u + g (u) in R N , ∫ R N | u | p d x = a p , where N ⩾ 2, a > 0, 1 < p < q ⩽ p ¯ ≔ p + p 2 N , μ ∈ R , g ∈ C (R , R) and λ ∈ R 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 q < p ¯ and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When q = p ¯ and μ > 0 or q ⩽ p ¯ 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 μ ∈ R , 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. [ABSTRACT FROM AUTHOR]