imply

Indeed, in the interval [0,62], the function «[Mj,^] provides us with a supersolution of f u" = fuP , i G [0,62], \u(0) = M2, u(b2)= 0, while zero is a subsolution. Thus, this problem has a solution in between zero and «[Mi.bi]- As the solution is unique, (23) holds true. Thanks to (23), for each M > 0, the point-wise limit uM ■■= lim it[MM (24)

f>T°o is well defined in [0,oo). Moreover, um < M. In particular, the set of solutions U[m,6]> b > 0, is bounded in C[0,oo). Now, by a rather standard compactness argument involving the theorem of Ascoli-Arzela, it is apparent that um provides us with a solution of f u" = fvP , t> 0,

By construction, UM{t) > 0 and u'M(t) < 0 for each t > 0. Actually, u'M{t) < 0 for each t > 0, since u'^{t) = f(t)upM(t) > 0. Hence,

tToo

Necessarily L = 0, since / is bounded away from zero at infinity. Consequently, um is a decreasing function approximating zero as 11 00.

We now show that um is the unique solution of (25). The argument proceeds by contradiction. Suppose ui u2 are two (positive) solutions of (25) with iii(io) > u2{to) for some to > 0. Then, there exist t\ e [0, io)

and £2 E (to, oo] such that u\(tj) = U2{tj), j € {1,2}, and ui(t) > U2{t) for each t € (h,t2). Pick tm € (ii, ¿2) such that ui(tm) -u2(tm) = max {ui(i) -u2(t)}. t€[h,te]

Then,

0 > (ui - u2)"(tm) = /(tm)[u?(tm) - up2(tm)} > 0, which is impossible. Therefore, the function um defined through (24) provides us with the unique solution of (25).

Now, due to (23), for each 0 < M < M and b > 0 we have that u[M,b\ < uM,b] and, hence, passing to the limit as b f 00 gives um < u^. Actually, um(0 < for each t > 0, as if there exists io > 0 for which UM(to) = %(<»). necessarily u'M(to) = w'^(io) and, hence, by the uniqueness of the Cauchy problem u' = v, v' = fup, u(i0) = um (io), v(io) = "m^o) , one has that v.M(t) =u^1(t) for each t > 0, which is impossible, because

Subsequently, for each b > 0, we denote by lb the minimal solution of

(u" = fup, 0 < t < b, \ u(0) = 00, u(b) = 00 , whose existence was established in J. Lopez-Gomez14 through the a priori bounds of J. B. Keller11 and R. Osserman22. Fix M > 0 and b > 0. Then, there exists a sufficiently small e > 0 such that

and, hence, £b provides us with a supersolution of f u" = fup, e <t<b-e, \ u(e) = UAf(e), u(b - e) = uM{b - e) , whose unique solution is um- As zero provides us with a subsolution, necessarily um < lb in [e, b — e] and, therefore, for each M > 0 and b > 0, uM < h in [0, b]. (26)

Consequently, by the monotonicity in M, the point-wise limit i := lim um

M too is well defined in [0, oo). By a rather standard compactness argument based on (26), it is apparent that I solves (21). It should be noted that i'(t) < 0 and l"(t) > 0 for each t > 0, and that limtjoo i(t) = 0. Actually, £ provides us with the minimal solution of (21), since any other solution L must satisfy L > I, because L > um for each M > 0. Consequently, we will subsequently denote it by £m¡n. To establish the existence of a maximal solution one can proceed as follows. For each e > 0, let denote the minimal solution of

(u" = fup, t>e, \ u(e) = oo, u(oo) = 0 , whose existence follows easily from the previous results. Now, for each M > 0 and e > 0, let uEM denote the unique solution of

and pick 0 < e < e. As in the interval [e, oo), uEM provides us with a subsolution of (28), we find that ueM < ueM there in, and, hence, passing to the limit as M | oo shows that < £emm in [e, oo). Consequently, the point-wise limit p — ljvn P€ .

is well defined and it provides us with a solution of (21). From the construction itself, it is apparent that ima,x is the maximal solution of (1).

So far, we have established the existence of a minimal and a maximal solution, imin and imax, for (21), in the sense that any other solution £ must satisfy

To complete the proof of the theorem it remains to show that ¿mjn = ¿max-Note that in all previous cases, the limiting solutions are non-negative and, hence, they must be positive by the uniqueness of the associated Cauchy problem. Thus, they cannot admit an interior local maximum in [0, oo) as they are convex.

Let I be any solution of (21) and consider, for each e > 0, the shifted function l(t) := i(t — e), t > e. By (1), for each t > e we have that e"(t) = t"(t - e) = f(t - s)ep(t -e)< f(t)£p(t - e) = f(t)£p(t) and, hence, ¿provides us with a supersolution of (27). Note that

M Too where iemin is the minimal solution of (27) and ueM is the unique solution of (28). Now, we shall show that for each M > 0 and e > 0, i>u*M. (30)

Indeed, since £(0) = oo, (30) holds true for sufficiently small t - e > 0. Thus, if (30) fails to be true, there exist ti > £ and t? £ (t\, oo] such that l(tj)=ueM(tj),j£{l,2},&nd u£M(t)>!(t)=e(t-£), te(ti,i2) -

4(im) - ¿(trn) = max |u^(t) - l(t)} . t€(t l,t2)

Then,

0 > (ueM - i)"(tm) > /(tm)[(«Si)P(im) - ip{tm)] > 0 , which is a contradiction. Thus, (30) holds true and, passing to the limit as M | 00, we find from (29) that for each t > s, iEmin{t)<m = m-e).

Consequently, passing to the limit as e I 0, we find that, for each t > 0, £max(t) = lim^in(i) < £(t).

Therefore, i = fmax. This concludes the proof. □

Subsequently, we shall denote by £(t), t > 0, the unique solution of (21).

Remark 3.1. In the special case when, for some constants (3 > 0 and 7>0, f(t)=/3t\ t> 0, it is easy to see that

ßp- 1 \p-l which is the most common case analyzed in the literature.

The following result collects some useful properties of i(t).

Lemma 3.1. Suppose p > 1 and f £ C[0,oo) satisfies (1). Then, lim £'(t) = —00, lim f(t) = 0, lim [/(i)£p+1 (i)] = 0. (32)

tj.0 tfoo tfoo

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