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Additional info for Aspects of positivity in functional analysis: proceedings of the conference held on the occasion of H.H. Schaefer's 60th birthday, Tubingen, 24-28 June 1985
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Moreover, in this case the solution is unique. 14). The following result asserts that such a solution exists for small values of the real parameter A, up to a certain range. 14) has a blow-up boundary solution for any value of A. We observe that our assumptions on the potential p allow that it may vanish on 852. This is the most interesting case, because it corresponds to the "competition" between a vanishing potential and a blow-up nonlinearity. 6 Assume that f satisfies conditions (f 2) and (f 3).
3, we obtain the following theorem. 15). Proof Let 4D(t) := pof (t) - ((a((,,t - b, t > 0, where po = info p > 0 and b := supo b + 1 > 0. (t) = 0. 1 we deduce that limt_,,, 4)(t)/ f (t) = po > 0. 2. Hence, there exists a positive blow-up solution of Av = 4b(v + r) in Q. U+b=pof(U) in 52, and blows up at the boundary of 52. 16) with 0 - n. 17, un < un+l < U in fl. It follows that u(x) := limn. 15). 15) satisfies un < v in 52, which yields u < v in fl. 15). This concludes the proof. The case of vanishing potentials We assume in this section that p vanishes in 52.
11 is a KellerOsserman-type condition around the origin, which establishes that, to guarantee the maximum principle for the nonlinear operator -Au + f (u), it suffices that f has at least a linear decay rate near the origin. 19). The particular case p = 1 corresponds to the linear setting, as described in Stampacchia's maximum principle. 19) is not fulfilled. Moreover, Pucci and Serrin have extended the maximum principle to nonhomogeneous differential operators. 26) where Sl is a bounded in 1R^'.