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In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation -div (|&#8711u|^p-2&#8711u) = |u|^p^&#8727-2u + f(x, u), u > 0, x &#8712 &#937 u|_&#915_0 = 0, &#8706u&#8706n |_&#915_1 = 0 is obtained, where &#937 is a bounded smooth domain in R^N, &#8706&#937 = &#915_0 &#8746 &#915_1, 2 &#8804 p < N, p^&#8727 = NpN-p, &#915_0 and &#915_1 are disjoint open subsets of &#8706&#937., number=3, journal=Journal of Partial Differential Equations, year=1992, month=May, pages=61–71