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The existence of positive solutions of some semilinear elliptic equations of the form -Au = Xf(u) is studied. The major results are a nonexistence theorem which gives a A* = X*(f, J2) > 0 below which no positive solutions exist and a lower bound theorem for « max for Q. a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on X, two other nonexistence theorems, and an existence theorem are also proved. 1980 Mathematics subject classificationdoi:10.1017/s144678870003295x fatcat:j64cwrilpfgvrpmycweetpvmfm