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In this paper, we explore the existence of multiple solutions to the following p-Laplacian type of equation with supercritical Sobolev-exponent: p u + |u| r-2 u = γ |u| s-2 u in , u = 0 on ∂ , where is a smoothly bounded open domain in R n (n > p ≥ 2), r > p * , p * np n-p is a critical Sobolev exponent. We prove that if 1 < s < p < n and γ ∈ R + , the above equation possesses infinitely many weak solutions. Furthermore, if 1 < s = p < n and λ m < γ ≤ λ m+1 , there exists at least m-pair nontrivial solutions, where λ m is the m-eigenvalue value defined in (2.2).