POSITIVE AND OSCILLATORY RADIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS

We prove that the nonlinear partial differential equation 
 Δ u + f ( u ) + g ( | x | , u ) = 0 ,  in   ℝ n , n ≥ 3 , 
with 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> u ( 0 ) > 0 , where f and g are continuous, 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> f ( u ) > 0 and 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> g ( | x | , u ) > 0 
for 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> u > 0 , and 
 0,{\mathrm{ for }}1 lim u → 0 + f ( u ) u q = B > 0 ,  for  1 q n / ( n − 2 ) , 
has no positive or eventually positive radial solutions. For g ( | x | , u ) ≡ 0 , 
when n / ( n − 2 ) ≤ q ( n + 2 ) / ( n − 2 ) the same conclusion holds provided 
 2 F ( u ) ≥ ( 1 − 2 / n ) u f ( u ) , where F ( u ) = ∫ 0 u f ( s ) d s . We also discuss the 
behavior of the radial solutions for f ( u ) = u 3 + u 5 and f ( u ) = u 4 + u 5 in 
 ℝ 3 when g ( | x | , u ) ≡ 0 .


Introduction
In recent years, numerous authors have given substantial attention to the existence of positive solutions of semilinear elliptic equations involving critical exponents (see [2], [5], [9], [10], [12], [13], [14], [15]).We shall consider the solutions of the nonlinear partial differential equation Au w f(u) + g( x ,u) O, in',n_3,   (1.1) where f and g are continuous functions, with f(u) > 0 and g(Ix I,u) > 0 whenever u > 0. Such equations arise in many areas of applied mathematics (see [7], [12]); solutions that exist in n and satisfy u(x)O as Ix Ic are called ground states.96 S. CHEN, J.A. CIMA and W.R. DERRICK By a positive solution we mean a solution u satisfying u(x)> 0 for all x in n.Equation (1.1) is said to involve critical exponents if f(u)= uP+ fo(u), where fo(u) is an algebraic rational function in u with order of growth o(up) at infinity, where p (n + 2)/(n-2 i.s the critical Sobolev exponent.Examples of such fo(u) in U 3 include fo(u) u V'I + 2 and fo(u) ?24/ (1 + u).When g 0 and f(u) up, Equation (1.1) is known to.have a one parameter family of positive solutions.For f(u) uq, q > p and g--0 the authors of [1] have shown that (1.1) has a positive solution provided u(0) > 0.
In this paper we show that the reversal of the inequality in (1.3) together with the assumption that uf(u) > 0 and ug(r, u) > 0 for all u -0, and mild conditions on the order of the growth of f(u) to zero as u---,0, lead to the nonexistence of positive solutions to (.1.2) with u 0 > 0.Moreover, the solutions to (1.2) cannot be eventually positive or eventually negative, but must oscillate about 0 infinitely often.Results of this type can be inferred from careful reading of the literature (see [4], [8], [11]).
However, proofs in the literature are often limited to functions of the form K(Ixl)u p, with assertions that they carry over to more general expressions, and frequently involve deep results about elliptic partial differential equations.Our results are completely elementary and give precise statements of the conditions required for nonexistence of positive and eventually positive or negative solutions.Our paper is organized as follows.
In Section 2 we present some elementary results that will be used in our proofs.
The main tools consist of an "energy function" that was developed in [9] and a modi- fication of an identity due to [14].
Finally, in Section 5 we show that with an additional condition, solutions to (1.5) must oscillate infinitely and converge to 0 as r-*oc.We also discuss the behavior of two such solutions: one with f(u) u 3 + u5, and one with f(u) u 4 + u5.Problems of these types have been studied in [5] and [6].The first of these oscillates to 0 while the second becomes eventually negative and oscillates to 1, as

Elementary Results
In what follows, we shall need some elementary facts concerning solutions of the initial value problems (1.2) and (1.5).Suppose f and g are continuous.
Integration of this equality from 0 to r yields r so that r n-lu'(r) (2.2) 0 Lemma2.1:If f(u) > O and g(r,u) > O for u > O, and u is a positive solution of equation (1.2), then u is strictly decreasing and tends to 0 as Proof: Since the integrand in (2.2) is positive, u'< 0 so the solution is strictly decreasing.Hence, there is a number c > 0 such that u(r) decreases to c as r---.cx3and u'(r)--O as r--.oc.Suppose c > 0, then since f is continuous on the interval [c, u0] it has a minimum fmin > 0 on this interval.Hence, r [f(u(s)) -4-g(s, u(s))]s n -lds > fmin-n-, o implying that u'(r)G -fmin(r/n)---.-cx3.This is a contradiction and hence u tends to zero as r--cx.
Suppose the solution u of (1.2) oscillates about zero a finite number of times and has a local maximum at r 0 for which u(r) > 0 for all r >_ r 0. We call such solutions eventually positive solutions.Then, because u'(ro so that again u is strictly decreasing for r _ r0, and the same proof as above shows that u and u converge to 0 as r-c. (b) g(r,u)--O.By the uniqueness theorem for solutions of initial value problems, a solution of (1.5) cannot satisfy both u'(r) 0 and f(u(r)) O, unless u is constant.Thus, except for such cases, the critical points of any solution of (1.5) are isolated, and are minima whenever f(u(r))< 0 and maxima whenever f(u(r))> O. Let u(r) be a solution of (1.5) and define the "energy-.function of [9]" (2.4) If (1.5) is multiplied by u', one obtains dr 2 + F(u)   n-r l(u') 2 < 0.
(2.5)This implies that the "energy" function Q is strictly decreasing because the critical points of u are isolated.
Lemma 2.2: Suppose that u has a critical point at r O.If u(ro) is a local maximum, then u(r)< U(ro) for all r > to, and if u(ro) is a local minimum, then u(r) > u(ro) for all r > r O.
Proof: Suppose u(rl) u(ro) for r 1 > r 0. Then contradicting the fact that Q is strictly decreasing.We also need the following "energy" version of Pokhozhaev's second identity valid for continuous f and functions u that are C2(n) and radial (see for integers k > 1, and c real.
Positive and Oscillatory Radial Solutions of Semilinear Elliptic Equations 99 Here u u(s) inside the integrals.If Au + f(u) O, then the left side of (2.6) is zero.Verification of this identity is a routine task by using (1.5) instead of Au + f(u).
r 0 Proof: Differentiate (2.7) with respect to r and substitute (1.5) into the resulting An integration yields the desired result.
Lemma 2.4: Let u be a positive solution of the initial value problem (1.5) with u o > O, and suppose that f is continuous, f(u) > 0 for u > O, and lim f(u) B > O for l < q < p (n + 2)/(n-2).
Proof: First assume that u is a positive solution.By Lemma 2.1, u decreases to 0, so some r o exists for which f(u)/u q > g/2 for r >_ r 0.Then, by equation (2.2), Integrating the resulting inequality: o yields an inequality from which the result follows for r >_ 2r 0.
If u is eventually positive, there is an r I > 0 such that u(r) >0 for r > r 1.Then, there is a first maximum of u at r 2 > r I for which (2.3) applies (with g 0) r 1 / s n u'(r) rn_l f(u(s)) -lds.r I 100 S. CHEN, J.A. CIMA and W.R. DERRICK By the comments following the proof of Lemma 2.1, u--.0 as r--c, so some r 0 > r 1 exists for which f(u)/uq> B/2 for r > r 0. The proof then follows by the same argument as the positive solution case.
An identical proof yields: Corollary 2.5: Let u be a solution of (1.5) with u o < O. Suppose f is continuous, f(u) < 0 for u < O, and lim f(u) B > O, for l < q < p. (2.11) If there is an r o such that u(r)< 0 for all r >_ r o (u is eventually negative), then for sufficiently large r t(r) _ cr-2/(q-1). (2.12) 3. Nonexistence of Positive Solutions in the Range 1 < q < n/(n-2) We now show that problem (1.2) does not have positive or eventually positive radial solutions if the order of growth q of f(u) to zero as u0+ is in the interval 1 <q<n/(n--2).
/ Thus, by (3.3), there exist constants c 1 > 0 and r 1 > r 0 such that t(r) >_ elf-fl, for all r >_ r 1. (3.7) We can repeat the process in (3.6) with -/2, obtaining u(r)>_ c2 r-2, for r >_ r 2 > rl, and in general, u(r) >_ cjr-Hi, for all r >_ rj, j <_ k. (3.8) Since /3 k -, we have proved that u(r)>_ ckr -e for all r >_ r k.However, by (2.3) which is a contradiction to Lemma 2.1 and the remark following it.Thus, no local maximum can exist for which the solution is positive thereafter, and consequently no positive solution of (3.2) exists.Corollary 3.2: Let f and g be continuous, uf(u)> 0 and ug(r,u)> 0 for all r and u 5 O, and assume that lim f(u) B > 0 for 1 < q < n _ _ _ _ _ . (3.9) Then the initial value problem (1.2) with u o 5 0 has no positive or negative solutions, nor eventually positive or negative solutions.Proof: Theorem 3.1 yields the case for positive or eventually positive solution.The case for nonnegative or essentially negative solutions follows trivially by setting Nonexistence of Positive Solutions in the Range n/(n-2) _< q < p (, + 9.) In this section we extend Theorem 3.1, for g(r, u)-O, to the range n/(n-2) < q < p, and show that with an additional condition, we again have nonexistence of positive or eventually positive solutions, Theorem 4.1: Let f be continuous, f(u) > 0 for u > O, and assume lim f(u) B > O for l < q < n +-----2 u--+O+ U q n--2" Further, assume that (4.1) (4.2) Then the initial-value problem u" + n ---r lU' + f(u) O, for r > O, u(O) o > 0 u'(O), has no positive solutions.Moreover, if u o < 0 or u(r) becomes negative, there is no r, >_ 0 such that u(r)> 0 for all r > r, >_ O, that is, there is no eventually positive solution.
(4.13) has no eventually positive or eventually negative solutions.Proof: The proof is almost identical with that of Theorem 4.1.

Oscillatory Behavior
Example 4.3 motivates the discussion and results in this section.By the existence theory for initial value problems, we know that (4.13) has a local solution.It is not difficult to show that this solution can be extended to +.Under the conditions listed in Corollaries 3.2 or 4.2, equations (1.2) and (1.5) with u 0 0, respectively, do not have eventually positive or eventually negative solutions.So what is the behavior of the solutions as r---c?Do the solutions converge to some value c, or do they oscillate but have no limit as r---,c?These are some of the questions we f(u) u 4 + u 5 respectively.Observe that for the former, the solution oscillates and converges to zero.In subsequent paper [3], we prove that the order of convergence to zero is u <-cr-2/3, somewhat faster than guaranteed by (5.4).Also, in that paper we prove that the zeros get further apart as For the latter, the solution becomes eventually negative and oscillates about u -1.The numerics show that the oscillations damp to -1 as r--,oc, and we believe that the distance between zeros is bounded from below by a nonzero constant.
The main difference between the two functions f(u) is that the former is restoring, that is uf(u)> 0 for u > 0, while the latter is not since it is positive in -1< u < 0. For large r we can ignore the term (n-1)ul/r in (5.3), so the concavity of the solution changes with its sign for restoring functions, but does not change until u < 1 for the latter function.
It is interesting to speculate what behavior might be expected from the solutions to (5.4) for more general f(u).
Numerical computations show when f(u)-u3(u + 1)(u + 2) that for small u 0 the solution behaves just as if f(u) were restoring, very similar to the behavior of u3+ u5.But for sufficiently large u0, the solution oscillates about u -2, with a behavior similar to that of ud+ u5.Thus, Positive and Oscillatory Radial Solutions of Semilinear Elliptic Equations 107 some value of u 0 is a bifurcation point for these two behaviors. Figure1