POSITIVE AND OSCILLATORY RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS

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1 Jounal of Applied Mathematics and Stochastic Analysis, 0: (997), POSITIVE AND OSCILLATORY RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS SHAOHUA CHEN and WILLIAM R. DERRICK Univesity of Montana, Depatment of Mathematics Missoula, MT 5982 USA JOSEPH A. CIMA Univesity of Noth Caolina, Depatment of Mathematics Chapel Hill, NC USA (Received Septembe, 995; Revised Febuay, 996) We pove that the nonlinea patial diffeential equation Au+f(u)+g(Ixl,u)--O, inn, n>_3, with u(0)> 0, whee f and g ae continuous, f(u)> 0 and g(ix I,u)> 0 fo u > 0, and lim u =B>0, f(u) fo l<q<n/(n-2), t q has no positive o eventually positive adial solutions. Fo g(lx l, u) O, when n/(n- 2) _< q < (n + 2)/(n- 2) the same conclusion holds povided 2F(u) >_ (-2/n)uf(u), whee F(u)-ff(s)ds. We also discuss the 0 behavio of the adial solutions fo f (u) u 3 + u 5 and f (u) u 4 + u 5 in N3 when g( x,u) O. Key wods: Semilinea Elliptic Equations, Positive Radial Solutions. AMS subject classifications: 35J65, 34C5. u. Intoduction In ecent yeas, numeous authos have given substantial attention to the existence of positive solutions of semilinea elliptic equations involving citical exponents (see [2], [5], [9], [0], [2], [3], [4], [5]). We shall conside the solutions of the nonlinea patial diffeential equation Au w f(u) + g( x,u) O, in,n_3, (.) whee f and g ae continuous functions, with f(u) > 0 and g(ix I,u) > 0 wheneve u > 0. Such equations aise in many aeas of applied mathematics (see [7], [2]); solutions that exist in n and satisfy u(x)o as Ix Ic ae called gound states. Pinted in the U.S.A. ()997 by Noth Atlantic Science Publishing Company 95

2 96 S. CHEN, J.A. CIMA and W.R. DERRICK By a positive solution we mean a solution u satisfying u(x)> 0 fo all x in n. Equation (.) is said to involve citical exponents if f(u)= up+ fo(u), whee fo(u) is an algebaic ational function in u with ode of gowth o(up) at infinity, whee p (n + 2)/(n-2 i.s the citical Sobolev exponent. Examples of such fo(u) in U 3 include fo(u) u V I +2 and fo(u)?24/ ( + u). When g 0 and f(u) up, Equation (.) is known to. have a one paamete family of positive solutions. Fo f(u) u q, q > p and g--0 the authos of [] have shown that (.) has a positive solution povided u(0) > 0. In this pape we will deal only with adially symmetic solutions of (.). Hence, using pola coodinates, we need only conside the singula initial-value poblem u" + n- + f(u) + g(,u) O, fo > O, u(0)-u00-u (0), whee f and g ae continuous, with f(u) > 0 and g(, u) > 0 wheneve u > 0. Obseve that the function u()- A/(B+ 2)/(q-), A,B > 0, is a positive solution of the poblem _ u" + n- lu + f(u) O, u(o) A/B q -, u (O) O, with 4qB f(u) + 2(q(n- 2)- n)uq. (q )2A2(q )u2q (q )2Aq Fo both tems of f(u) to have nonnegative coefficients, we must have q >_ n/(n- 2). In addition, the authos of [] have shown, fo g _= 0, that positive solutions to (.2) exist wheneve u 0 > 0, f is Lipschitz, f(0) 0, f(u) > 0 fo u > 0 and 2F(u) (-)uf(u), fo u >_ 0, (.3) whee - F(u) f(s)ds. 0 Thus, the existence of positive solutions to (.2) depends on the ode of gowth of f(u) fo small u > 0 and on popeties of the antideivative of f(u). In this pape we show that the evesal of the inequality in (.3) togethe with the assumption that uf(u) > 0 and ug(, u) > 0 fo all u 0, and mild conditions on the ode of the gowth of f(u) to zeo as u---,0, lead to the nonexistence of positive solutions to (..2) with u 0 > 0. Moeove, the solutions to (.2) cannot be eventually positive o eventually negative, but must oscillate about 0 infinitely often. Results of this type can be infeed fom caeful eading of the liteatue (see [4], [8], []). Howeve, poofs in the liteatue ae often limited to functions of the fom K(Ixl)up, with assetions that they cay ove to moe geneal expessions, and fequently involve deep esults about elliptic patial diffeential equations. Ou esults ae completely elementay and give pecise statements of the conditions equied fo nonexistence of positive and eventually positive o negative solutions. Ou pape is oganized as follows. In Section 2 we pesent some elementay esults that will be used in ou poofs. The main tools consist of an "enegy function" that was developed in [9] and a modi-

3 Positive and Oscillatoy Radial Solutions of Semilinea Elliptic Equations 97 fication of an identity due to [4]. In Section 3 we pove that if f and g ae continuous, f(u)> O, g(,u)> 0 fog u > 0, and lim f(u)/uq= B>0 fo < q < n/(n-2), then the initial-value u--*o + poblem (.2) with u 0 > O, has no positive solutions. In Section 4 we shall assume that g( u) = O, so ou esults apply to solutions of u"+ n-!-u + f(u) O, fo > 0, u(0) u0:26_ 0 u (0). (.5) If f is continuous, f(u) > 0 fo u > Ohn + f(u)/u q B > 0 fo some q in < q < p, and 2F(u) > (-2g)uf(u), (.6) fo u > 0, then equation (.5) has no positive solutions o eventually positive solutions. Finally, in Section 5 we show that with an additional condition, solutions to (.5) must oscillate infinitely and convege to 0 as -*oc. We also discuss the behavio of two such solutions: one with f(u) u 3 + u 5, and one with f(u) u 4 + u 5. Poblems of these types have been studied in [5] and [6]. The fist of these oscillates to 0 while the second becomes eventually negative and oscillates to, as 2. Elementay Results In what follows, we shall need some elementay facts concening solutions of the Suppose f and g ae continuous. initial value poblems (.2) and (.5). (a) g(, u) O. Equation (.2)can be ewitten as Integation of this equality fom 0 to so that ( n lu ) n l[f(u) -4- g(, u)]. yields n- lu () / [f(u(s)) + g(s, u(s))]sn- lds, (2.) u () 0 n- [f(u(s)) + 9(s, u(s))]s n- ds. (2.2) 0 Lemma2.: If f(u) > O and g(,u) > O fo u > O, and u is a positive solution of equation (.2), then u is stictly deceasing and tends to 0 as Poof: Since the integand in (2.2) is positive, u < 0 so the solution is stictly deceasing. Hence, thee is a numbe c > 0 such that u() deceases to c as ---.cx3 and u ()--O as --.oc. Suppose c > 0, then since f is continuous on the inteval [c, u0] it has a minimum fmin > 0 on this inteval. Hence, o [f(u(s)) -4- g(s, u(s))]s n -lds > fmin-n-, implying that u ()G -fmin(/n)---.-cx3. This is a contadiction and hence u tends

4 98 S. CHEN, J.A. CIMA and W.R. DERRICK to zeo as --cx. Suppose the solution u of (.2) oscillates about zeo a finite numbe of times and has a local maximum at 0 _ fo which u() > 0 fo all >_ 0. We call such solutions eventually positive solutions. Then, because u (o)- O, u () [f(u(s)) q- g(s, u(s))]s n lds, (2.3) n 0 so that again u is stictly deceasing fo 0, and the same poof as above shows that u and u convege to 0 as -c. (b) g(,u)--o. By the uniqueness theoem fo solutions of initial value poblems, a solution of (.5) cannot satisfy both u () 0 and f(u()) O, unless u is constant. Thus, except fo such cases, the citical points of any solution of (.5) ae isolated, and ae minima wheneve f(u())< 0 and maxima wheneve f(u())> O. Let u() be a solution of (.5) and define the "enegy-.function of [9]" Q(u()) + F(u()). (2.4) If (.5) is multiplied by u, one obtains d 2 + F(u) n- l(u ) 2 < 0. (2.5) This implies that the "enegy" function Q is stictly deceasing because the citical points of u ae isolated. Lemma 2.2: Suppose that u has a citical point at O. If u(o) is a local maximum, then u()< U(o) fo all > to, and if u(o) is a local minimum, then u() > u(o) fo all > O. Poof: Suppose u(l) u(o) fo > 0. Then Q(U(l))- u (l)2 F(u(o) Q(u(o)), + F(U(l)) _ contadicting the fact that Q is stictly deceasing. We also need the following "enegy" vesion of Pokhozhaev s second identity valid fo continuous f and functions u that ae C2(n) and adial (see [4])" + f(u))(su + cu)skds k + Q(u()) + aku()u () + (n k) k lu2() (2.6) + (2n- 3 k- 2a) / Q(u(s))sCds- a(n- k)(k )/ u2 sk 2 (s) -2ds [auf(u) 2(n c)(u)]skds, fo integes k >, and c eal.

5 Positive and Oscillatoy Radial Solutions of Semilinea Elliptic Equations 99 Hee u u(s) inside the integals. If Au + -- f(u) O, then the left side of (2.6) is zeo. Veification of this identity is a outine task by using (.5) instead of Au + f(u). Lemma 2.3" Let u be a solution of (.5) and let J(, u) nu 2() + (n 2) n lt()tt() 2nF(u). (2.7) If u (o) 0 fo some o >_ O, then fo all >_ to, g(, u) / [2nF(u(s)) (n 2)u(s)f(u(s))]sn- lds + 2F(u(o) ). 0 Poof: Diffeentiate (2.7) with espect to and substitute (.5) into the esulting equation to obtain _ dj cl- [2nF(u) (n- 2)uf(u)] n- An integation yields the desied esult. Lemma 2.4: Let u be a positive solution of the initial value poblem (.5) with u o > O, and suppose that f is continuous, f(u) > 0 fo u > O, and lim f(u) B > O fo l < q < p (n + 2)/(n- 2). (2.9) u---+o-b q Then, fo sufficiently lage, thee is a constant c :> 0 such that t() c- 2/(q- ). (2.0) The esult also holds fo eventually positive solutions. Poof: Fist assume that u is a positive solution. By Lemma 2., u deceases to 0, so some o exists fo which f(u)/u q > g/2 fo >_ 0. Then, by equation (2.2), o n- f(u(s)) lds du d n- f(u(s)) ds 0 0 " 2 n- uq(s) lds < Integating the esulting inequality: f (o) 2n- 0 0 u-"du<_f [--n+l]d o yields an inequality fom which the esult follows fo >_ 2 0. If u is eventually positive, thee is an I > 0 such that u() >0 fo >. Then, thee is a fist maximum of u at 2 > I fo which (2.3) applies (with g 0) / s n u () n_l f(u(s)) -lds. I

6 00 S. CHEN, J.A. CIMA and W.R. DERRICK By the comments following the poof of Lemma 2., u--.0 as --c, so some 0 > exists fo which f(u)/uq> B/2 fo _ > 0. The poof then follows by the same agument as the positive solution case. An identical poof yields: Coollay 2.5: Let u be a solution of (.5) with u o < O. Suppose f is continuous, f(u) < 0 fo u < O, and f(u) lim B > O, fo l < q < p. (2.) If thee is an o such that u()< 0 fo all >_ o (u is eventually negative), then fo sufficiently lage t() c-2/(q-). (2.2) 3. Nonexistence of Positive Solutions in the Range < q < n/(n- 2) We now show that poblem (.2) does not have positive o eventually positive adial solutions if the ode of gowth q of f(u) to zeo as u0+ is in the inteval <q<n/(n--2). Theoem 3.: If f and g ae continuous, f(u) > O, g(, u) > 0 fo u > O, and then the initial-value poblem lim f( U -B>O fol<q< u O -t- U q n n--2 (3 ) u",, + n lu + f(u) + g(, u) O, fo >0, u(0) u0>0 u (0), (3.2) has no positive solutions. Moeove, if u o < 0 o u() becomes negative thee is no o >_ 0 fo which u()> 0 fo all >_ to, that is (3.2) has no eventually positive solutions. Poof: Suppose a local maximum of the solution exists at 0 such that u() > 0 fo all >_ 0. By Lemma 2. and the emak following its poof, it follows that u0 as oc. Since n > (n- 2)q select 0 < c < /q and an intege k > 0 such that both of the following inequalities hold" and ] n (n 2)q-(n-2-c)(q-) >0 qk_ n_2_(n-2-e)(q-)>>e_ qk_ 2 l [n (n 2)q l]q k By L Hopital s ule, povided the ight side s limit exists, fo any/3 > 0. () --, cxz oo fi- In paticula, let 3 n- 2. constant c o > 0 exists so that lim _-=lim =lim cxz f [f(u(s)) + g(s, u(s))]s n fi n- 2 lds (3.3) Then, since the integal in (3.3) is positive, some

7 Positive and Oscillatoy Radial Solutions of Semilinea Elliptic Equations 0 u() k Cot- (" -2), fo all k 0. (3.4) Define/3j + fljq- 2 + with/3 0 n- 2. It is easy to pove by induction that /j- (n- 2- c) ( qj-l)+c, qk---_ By hypothesis, f(u)/u q >_ -B u,, u() < u, fo all > >_ 0. Using (3.3) with/3 [3, obseve that j<k. (3.5) fo all u _< and since u-+o as -+c, we can assume B uq(8)8n ld8 -Bcof 8 f [f(u(s)) + g(s u(s))]s n Ids q n -(n 2)qd, 3 0 fln- 2- fln- 2- fln- 2 f 2 )q (3.6) )--,cx, as --cx. / Thus, by (3.3), thee exist constants c > 0 and > 0 such that t() >_ elf-fl, fo all >_. (3.7) We can epeat the pocess in (3.6) with -/2, obtaining u()>_ c2-2, fo >_ 2 _ > l, and in geneal, u() >_ cj-hi, fo all >_ j, j <_ k. (3.8) Since /3 k -, we have poved that u()>_ ck -e fo all >_ k. Howeve, by (2.3) and (3.6) / /2Bc/ u () [f(u(s)) + g(s, u(s))]s n- lds >_ s n qds n- n- 0 k C I -eq - -,c, as -c, which is a contadiction to Lemma 2. and the emak following it. Thus, no local maximum can exist fo which the solution is positive theeafte, and consequently no positive solution of (3.2) exists. Coollay 3.2: Let f and g be continuous, uf(u)> 0 and ug(,u)> 0 fo all and u 5 O, and assume that f(u) lim B > 0 fo < q < n. (3.9) Then the initial value poblem (.2) with u o 5 0 has no positive o negative solutions, no eventually positive o negative solutions. Poof: Theoem 3. yields the case fo positive o eventually positive solution. The case fo nonnegative o essentially negative solutions follows tivially by setting

8 02 S. CHEN, J.A. CIMA and W.R. DERRICK Nonexistence of Positive Solutions in the Range n/(n- 2) _< q < p (, + 9.) In this section we extend Theoem 3., fo g(, u)- O, to the ange n/(n- 2) < q < p, and show that with an additional condition, we again have nonexistence of positive o eventually positive solutions, Theoem 4.: Let f be continuous, f(u) > 0 fo u > O, and assume Futhe, assume that Then the initial-value poblem lim f(u) B > O fo l < q < n u--+o+ U q n--2" (4.) 2F(u) >_ (-)uf(u) > 0 fo u > O. (4.2) u" + n--- lu + f (u) O, fo > O, u(o) o > 0 u (O), has no positive solutions. Moeove, if u o < 0 o u() becomes negative, thee is no, >_ 0 such that u()> 0 fo all >, >_ O, that is, thee is no eventually positive solution. Poof: As a consequence of Theoem 3., we only need to pove the conclusions fo n/(n- 2) <_ q < p, o, equivalently 2q/(q- ) _< n < 2(q + )/(q- ). Suppose that the conclusions ae not tue. Then thee exists a point, _> 0 such that u() > 0 fo all >, >_ 0. By Lemma 2.4, we have u() _< c Using (2.3) with g(, u)-0 and (4.4), we get 2 q- i, fo all _> 0 _>,. (4.4) <,,C 2Bc / q n i + n s,._i } f(u(s)) n -lds 0 n-l-2q/(q-) C -(q+l)/(q-) s d8 < + Cl whee we choose o >_. such that 0 < f(u())u/uq() <_ 2B fo v > 0. q > n/(n- 2), it follows that n- > (q + )/(q- ). Thus, fom (4.5), (4.5) Since By L Hopital s ule, we have fom (4.) -() f f(u)du lim F(u()) lia o (q A- )/(q it () --< c2 ), fo lage. (4.6) ---cx u q + l( --x) u q + l( f(u()) -lin (q + )uq() q + " (4.7) Now, using (4.4), (4.6), and (4.7) and a 2(q + )/(q- )- n > 0, we get B

9 _ Positive and Oscillatoy Radial Solutions of Semilinea Elliptic Equations 03 < :/(q_ + < and both n- lu() u () c- c, (4.9) (4.0) fo lage. Let be the fist maximum point such that u()> 0 fo all _>. Then u (l)- 0. Using Lemma 2.3 and (4.2) we get I:( U) nl,a,() 2 _ + ( 2)n () () + 2F() ( + )c i4.) and 2F(t(l) (4.2) 2F(U(l) > O. Letting --,c, we see that (4.) contadicts (4.2), so the theoem is poved. Coollay 4.2: Let f be continuous, uf(u) > 0 fo u : O, and assume that and J(, t) / [2F(t(s)) ( 2)t(8)f(u(s))]sn- lds -- lim f (u) B > O, <.q < n + 2 _.Oululq-- n-2 2F(u) _> (-2)uf(u), fo all u. u" + n- u, + S(u) O, fo > O, u(o) u o # 0 u (O),. (4.3) has no eventually positive o eventually negative solutions. Poof: The poof is almost identical with that of Theoem 4.. Example 4.3: The function f(u) u 5 + u3v/ + u 2 in 3 satisfies the hypotheses of Coollay 4.2. Thus, (4.3) has no eventually positive o eventually negative solutions fo this function. The function f(u)- uh+ u4/( / u)in 3 satisfies the hypotheses of Theoem 4., but f(u)> 0 in - < u < 0. Thus, (4.3) has no eventually positive solution, but may have an eventually negative solution. 5. Oscillatoy Behavio Example 4.3 motivates the discussion and esults in this section. By the existence theoy fo initial value poblems, we know that (4.3) has a local solution. It is not difficult to show that this solution can be extended to +. Unde the conditions listed in Coollaies 3.2 o 4.2, equations (.2) and (.5) with u 0 0, espectively, do not have eventually positive o eventually negative solutions. So what is the behavio of the solutions as ---c? Do the solutions convege to some value c, o do they oscillate but have no limit as ---,c? These ae some of the questions we addess in this section. Fo simplicity, we will assume that g(, u) O.

10 04 S. CHEN, J.A. CIMA and W.R. DERRICK Fist we show that with a stonge condition than that in Coollay 4.2, we can pove that the solution conveges to zeo as Lemma 5.: Let f(u) be continuous, and satisfy Futhe, assume that Suppose u is lim If(u) B > O, fo < q < n / 2 (5 ) lul q (n- 2)uf(u) >_ 2F(u) >_ (-)uf(u) > 0 fo u =/= O. (5.2) a solution of u" / n- u, / f(u) O, fo > O, u(o) u o = O, u (O) O. Then 2 lu() _c q /, fo lage. (5.4) Poofi Pokhozhaev s fist identity [4] with a n- 2 can be ewitten as 2F(u()) + /2[u () + (n- 2)u()] 2 + / [(n- 2)u(s)f(u(s)) 2F(u(s))]s ds 0 (n- 2)2u2(0) 2 Since the second and thid tems on the left side of this equation ae nonnegative, we have by (5.2) (n 2)u()f(u()) < 2F(u()) < (n 2)2u2(0) O< n 2 Thus, F(u(v))O as v, and by the fist inequality, F is only zeo at u- O. Hence uo as o. Then U q + f(u)) < n(n 2)u2(0) and using (5.) the conclusion in (5.4) follows. Example 5.2: The function f(u)- u5+ u 3 in R3 satisfies the hypotheses of Lemma 5. with q-3. Hence, the solution to (5.3) with this f will oscillate infinitely and decay as lu() <_ c -/2 as ---,oo. On the othe hand, f(u)- (u/( + u2)) 3 satisfies only the second inequality in (5.2) fo u # 0, so the conclusion (5.4) cannot be assumed. The situation fo functions f(u) that ae not negative fo u < 0 is moe complicated. The following lemma can be extended to moe geneal functions, and shows that they too oscillate, but now about a negative numbe. Lemma 5.3: Let u() be a solution of (5.3) with f(u) u 4 / u5. Then u() oscillates infinitely about, and tends to, as ---<x. Poof: Suppose othewise, and let be the last exteme point of u. Then u() + does not change sign fo > l. Since u (l)- 0, by (2.) and L Hopital s ule with/ _< n- 2, lia u()/ lia --cx) u () f ---,x -/ lim / [u4(s) + u5(s)]s n- lds. (5.5) --,o fl-f-2+n

11 Positive and Oscillatoy Radial olutions of Semilinea Elliptic Equations 05 If fl n- 2, we have + -lim u()_(n_2) -clim / [ + u(s)]lu(s)[4s -- n-lds : O, I and the limit exists o is infinite. Hence, thee is a constant c I > 0, and an 2 > such that u(s) + C8 -(n 2), /(8) < 0, and 4 tt(s) fo s 2. (5.6) Obseve that / [ + u(s)]u4(s)s n- lds / ]u(s)+ ] ]u(8)]48 n- ld8 (5.7) I 2 sds, as. 2 If n- 2 >, take n- 3 in (5.5), and we have fom (5.7) + Clf sds So thee exists a constant c2 > 0 such that ]u()+ c -(n-3), fo. If n-3>l, epeat these steps until n-kl. Then, u()+l cn_2-, fo 2. On the othe hand, ] ()] 4 n- f ( )]Sn- n_l u(s)[ + -lds /2, [u(s)] ] + S ds ffdl -2ds Co > 0, 2 2 fo > 2 +, which is a contadiction because u ()O, as. Thus, the solution u must oscillate infinitely about -. Since u ()O, as, it follows that u()-. Remak 5.4: We now study the solutions of the initim-value poblem (5.3) numeically. We can ewite (5.3) as the nonautonomous system v = _(n- )v f(u), v(o) O. The numeical solution is vey sensitive to the singulaity - 0. To decide what is appopiate thee, we apply L Hopital s ule: Hence (n-)v -o limv -lim( --,o f(u)) -(n ) -olimv - f(uo). limv --+0 f(u) t (5.0) Applying the Runge-Kutta method to (5.8) except at - 0, whee (5.0) is used, we obtain the gaphs in Figues and 2 fo the solutions when f(u)-u3+u 5 and

12 06 S. CHEN, J.A. CIMA and W.R. DERRICK f (u) u 4 + u 5 espectively. u Figue Figue 2 Obseve that fo the fome, the solution oscillates and conveges to zeo. In subsequent pape [3], we pove that the ode of convegence to zeo is u <-c-2/3, somewhat faste than guaanteed by (5.4). Also, in that pape we pove that the zeos get futhe apat as Fo the latte, the solution becomes eventually negative and oscillates about u -. The numeics show that the oscillations damp to - as --,oc, and we believe that the distance between zeos is bounded fom below by a nonzeo constant. The main diffeence between the two functions f(u) is that the fome is estoing, that is uf(u)> 0 fo u > 0, while the latte is not since it is positive in -< u < 0. Fo lage we can ignoe the tem (n-)ul/ in (5.3), so the concavity of the solution changes with its sign fo estoing functions, but does not change until u < fo the latte function. It is inteesting to speculate what behavio might be expected fom the solutions to (5.4) fo moe geneal f(u). Numeical computations show when f(u)- u3(u + )(u + 2) that fo small u 0 the solution behaves just as if f(u) wee estoing, vey simila to the behavio of u3+ u5. But fo sufficiently lage u0, the solution oscillates about u -2, with a behavio simila to that of ud+ u5. Thus,

13 Positive and Oscillatoy Radial Solutions of Semilinea Elliptic Equations 07 some value of u 0 is a bifucation point fo these two behavios Figue 3 Pefeences [] [2] [3] [4] [6] [8] [0] Ill] [2] [3] Beestycki, H., Lions, P.L. and Peletie, L.A., An ODE appoach to the existence of posi-tive solutions fo semilinea poblems in n, Indiana Univesity Math. J. 30 (98), Bezis, H. and Nienbeg, L., Positive solutions of nonlinea elliptic equations involving ci-tical Sobolev exponents, Comm. Pue and Appl. Math. 36 (983), Deick, W.R., Chen, S. and Cima, J.A., Oscillatoy adial solutions to semilinea elliptic equations, submitted. Ding, W.-Y. and Ni, W.-M., On the existence of positive entie solutions of a semilinea elliptic equation, Ach. Rational Mech. Anal. 9 (986), Jones, C.K.R.T., Radial solutions of a semilinea elliptic equation at a citical point, Ach. Rational Mech. Anal. 04 (988), Jones, C. and Kiippe, T., On the infinitely many solutions of a semilinea elliptic equation, SIAM J. Math. Anal. 7 (986), Li, Y., Remaks on a semilinea elliptic equation on n, j. Diff. Eq. 74 (988), Li, Y. and Ni, W.-M., On the existence and symmety popeties of finite total mass solutions of the Matakuma equation, the Eddington equation, and thei genealizations, Ach. Rational Mech. Anal. 08 (989), McLeod, D., Toy, W.C. and Weissle, F.B., Radial solutions of Au + f(u)- 0 with pescibed numbes of zeos, J. Diff. Eq. 83 (990), Ni, W.-M., Uniqueness of solutions of nonlinea Diichlet poblems, J. Diff. Eq. 50 (983), Ni, W.-M., On the elliptic equation Au+K(x)u (n +2)/(n-2)- 0, its genealizations and applications to geomety, Indiana Univ. Math. J. 3 (982), Ni, W.-M. and Yotsutani, S., Semilinea elliptic equations of Matukuma-type and elated topics, Japan J. Applied Math. 5 (988), -32. Pan, X., Positive solutions of the elliptic equation Au+u(n+2)/(n-2)+

14 08 S. CHEN, J.A. CIMA and W.R. DERRICK K(x)u q -0 in 3 and in balls, ]. Math. Anal. Appl. 72 (993), [4] Pokhozhaev, S.I., On the asymptotics of entie adial solutions of quasilinea elliptic equations, Soviet Math. Dokl. 44 (992), [5] Yamagida, E., Uniqueness of positive adial solutions of Au+g()u+ h()u p -0 in 3, Ach. Rational Mech. Anal. 5 (99),

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics