By Prund A. H.
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This better half examines a few matters relating to the phrases realism and naturalism. The advent seeks either to debate the issues within the use of those phrases in terms of past due nineteenth-century fiction and to explain the background of past efforts to make the phrases expressive of yank writing of this era.
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Theorem 7. Assume that ð116Þ g is nonincreasing and that there exists a nondecreasing function YACðRþ 0 ; RÞ; with Yð0Þ ¼ 0; such that f ðr; uÞ XYðuÞ gðuÞ for all rX0 and all If lim sAðsÞ ¼ N s-N u > 0: ð117Þ R. Filippucci / J. Differential Equations 188 (2003) 353–389 385 and Y satisfies the condition Z N H À1 Z t YðtÞ dt À1 dtoN ð118Þ then inequality (105) does not admit any entire positive solution. Proof. Assume by contradiction that (105) admits a positive entire solution u : Rn -Rþ : Since AðsÞps2 AðsÞ for s > 0 by (C1) and gu ðuÞp0 for all uX0; by (105) and (117), we have divðAðjrujÞruÞX gu ðuÞ f ðjxj; uÞ ½AðjrujÞ À jruj2 AðjrujÞ þ gðuÞ gðuÞ X YðuÞ: ð119Þ Consider now the initial value problem ðrnÀ1 Aðjv0 jÞv0 Þ0 ¼ rnÀ1 YðvÞ; vð0Þ ¼ aAð0; uð0ÞÞ; v0 ð0Þ ¼ 0: ð120Þ By Theorem 1 the initial value problem (120) admits a local solution v : ½0; RÞ-Rþ ; where R > 0: Without loss of generality, we may assume that ½0; RÞ is the maximal interval of existence of v: By Theorem 5 we have that RoN: As noted in the proof of Theorem 5 it results that v0 ðrÞ > 0 for rAð0; RÞ: Hence either lim vðrÞ ¼ N or r-RÀ lim v0 ðrÞ ¼ N: r-RÀ Case 1: lim vðrÞ ¼ N: r-RÀ In this case, we can take R1 Að0; RÞ so that vðR1 ÞXmaxfuðxÞ : jxj ¼ R1 g: ð121Þ Deﬁne B1 ¼ fxARn : jxjoR1 g: Then the function v ¼ vðjxjÞ is such that divðAðjrvjÞrvÞ ¼ YðvÞ in B1 and we have that vXu on @B1 : Hence, by Theorem 6 applied with O ¼ B1 ; it results that upv in B1 and this is a contradiction since vð0Þ ¼ aouð0Þ: Case 2: lim v0 ðrÞ ¼ N: r-RÀ 386 R.
Filippucci / J. À1 Z t Z N À1 1 H f ðtÞ dt dt ¼ N; n vð0Þ vð1Þ which contradicts (109). Hence (106) does not admit any global positive solution. & In the sequel, we will also need the weak comparison principle which is due to Pucci, Serrin and Zou (see ), which we state only when the domain OCRn is bounded. Theorem 6 (Pucci et al. [15, Lemma 3]). Let u and v be respective solutions of the differential inequalities: divðAðjrujÞruÞ À YðuÞX0; uX0; ð114Þ divðAðjrvjÞrvÞ À YðvÞp0; vX0 ð115Þ in a bounded domain O of Rn ; nX2: Assume that the function YACðRþ 0 ; RÞ is such that Yð0Þ ¼ 0; Y is nondecreasing in ½0; dÞ; 0odpN: % with uod in O and vXu on @O; then vXu in D.
If lim sAðsÞ ¼ N s-N and Z N H À1 Z À1 t f ðtÞ dt dtoN; ð107Þ where HðsÞ ¼ s2 AðsÞ À AðsÞ; s > 0; ð108Þ then the initial value problem (106) does not admit any global positive solution. Remark. Theorem 5 was proved by Naito and Usami  under stronger regularity assumptions on the operator A and by assuming condition (5) in place of (107). Note that when (107) fails then (106) admits global positive solutions, as shown by Naito and Usami. Thus (107) is a necessary and sufﬁcient condition for the nonexistence of global solutions of (106).