CLASSIFICATION OF SUPERSOLUTIONS AND LIOUVILLE THEOREMS FOR SOME NONLINEAR ELLIPTIC PROBLEMS

. In this paper we consider positive supersolutions of the elliptic equation − ∆ u = f ( u ) |∇ u | q , posed in exterior domains of R N ( N ≥ 2), where f is continuous in [0 , + ∞ ) and positive in (0 , + ∞ ) and q > 0. We classify su- persolutions u into four types depending on the function m ( R ) = inf | x | = R u ( x ) for large R , and give necessary and suﬃcient conditions in order to have su- persolutions of each of these types. As a consequence, we also obtain Liouville theorems for supersolutions depending on the values of N , q and on some in- tegrability properties on f at zero or inﬁnity. We also describe these questions when the equation is posed in the whole R N .

1. Introduction and results. The purpose of the present paper is to analyze the existence and nonexistence of supersolutions of the elliptic problem where N ≥ 2, q > 0 and B R0 stands for the ball with radius R 0 centered at zero. The nonlinearity f is a continuous function defined in [0, +∞) and positive in (0, +∞).
In particular, we are interested in obtaining Liouville type theorems for (1). Nonlinear Liouville theorems go back to the pioneering reference [25], where the model problem with f (t) = t p , p > 1 was considered. Later, some other works have dealt with the same problem, either considering alternative proofs to that in [25] (see [16]) or obtaining similar results for more general nonlinearities (cf. [32]) and operators (see [35] for the context of the p-Laplacian).
A natural question related to (2) is to know how the nonexistence results are modified when the equation is perturbed in some way, for instance with the introduction of a gradient term. Of course, this can be done in multiple ways, but a problem that has been somewhat studied is where q > 0. This problem has been considered in the reference case f (u) = u p , p > 0, in [17] and later in [34] and [37] (cf. also the extension to the p−Laplacian setting considered in [22], [23]). In most of these works the study was restricted to radial solutions, but this restriction was later dropped in the works [2], [3] and [1], where the case q ≥ 1 was analyzed, obtaining nonexistence results for general positive supersolutions (see also [36] for a survey on the problem). Our intention in this paper is to analyze (2) when it is perturbed with a gradient term in a different way. Thus we will restrict to the study of problem (1) in the rest of the paper. We will perform a complete analysis of the positive supersolutions of (1), inspired by the results in [4]. Let us mention that this precise problem has been studied in [13] for some more general operators when 0 < q < 1, but only the case f (t) = t p is dealt with there. Also, in [21], the more general problem −div(h(x)g(u)A(|∇u|)∇u) ≥ f (x, u, ∇u) in R N was analyzed, but the nonlinearity f (x, u, ∇u) is essentially like u p |∇u| q , so that again only the power case is known.
Before stating precise results, let us clarify that we will always be dealing with continuous weak supersolutions, that is, functions u ∈ H 1 loc (R N \B R0 )∩C(R N \B R0 ) verifying Type 4: m(R) is increasing and lim R→+∞ m(R) = +∞. In order to simplify the exposition, we will assume throughout the paper that whenever a positive supersolution u of (1) exist, the radius R 0 has been chosen so that m(R) is monotone for R > R 0 .
We are intentionally excluding from our classification all constant solutions of (1). Also, it is important to observe that the singular nature of the problem allows the existence of supersolutions which are not constant, but are eventually constant in the sense that they are constant for |x| > R 1 for some R 1 > R 0 , at least when 0 < q < 1 (see Remark 1 in Section 2). Thus these supersolutions are also excluded from our classification. Let us mention in passing that this phenomenon does not seem to be possible for q ≥ 1.
It is perhaps interesting to note that the presence of supersolutions of types 2 and 3 is in contrast with problems (2) and (3), where only supersolutions of types 1 and 4 are possible. This is related to the fact that all constants are indeed solutions of (1).
The existence of each type of supersolution depends first of all on the dimension N . The cases N ≥ 3 and N = 2 -as in multiple well-known situations-are genuinely different, as can be seen for instance from the fact that supersolutions of type 4 are never possible when N ≥ 3, while for N = 2, the only types which can arise are 2 and 4.
Let us begin with the case of higher dimensions N ≥ 3. Since f is assumed to be positive in (0, +∞), the existence of supersolutions of types 2 and 3 does not really depend on f , and it is only related to the relative values of q and N . However, when considering supersolutions of type 1, the relevant condition is for some δ > 0, where This condition resembles the one found in [4] for the case q = 0 (and actually reduces to that one in this particular case). Our results for problem (1) in the case N ≥ 3 can be summarized as follows. (c) For 0 < q < 1, there never exist supersolutions of (1) of types 2 and 3, while supersolutions of type 1 exist if and only if (4) holds. Moreover, positive supersolutions of type 4 never exist in this case.
As a consequence of the statements above, we have a Liouville theorem for (1). It is worthy of mention that the next result is optimal in view of Theorem 1.1.
Corollary 1 (Liouville theorem). Assume N ≥ 3 and f ∈ C([0, +∞)) is positive in (0, +∞). If q < 1 and (4) does not hold, then every positive supersolution of problem (1) is eventually constant.  Of particular interest in (1) is the special case where f is a power, The above theorems directly apply to this case to obtain for instance: when N ≥ 3, if 0 < q < 1 and 0 < p ≤ N −q(N −1) , then every positive supersolution of (7) is eventually constant; for N = 2, the nonexistence of not eventually constant positive supersolutions holds if 0 < q < 2. Both results are sharp, and coincide with those in [21] when the equation is considered in R N (for N ≥ 3).
Observe that all the nonexistence results stated above apply equally to the equation in (1) when it is posed in the whole R N , namely where N ≥ 3 (it is well known that the only positive, superharmonic functions in R 2 are constants, so that the case N = 2 is uninteresting). However, it is to be stressed that the maximum principle implies m(R) = inf |x|≤R u(x) for all positive supersolutions, so that the function m(R) is always strictly decreasing, unless u is constant. Therefore, only supersolutions of types 1 and 3 are possible, and it also follows that eventually constant supersolutions which are not constant do not exist. On the other hand, if u is a positive supersolution of (1) of one of these types, then it is easily checked that the functionũ = min{u, m(R 1 )} is a weak H 1 loc supersolution of (8) when R 1 is large enough. Therefore, To conclude the discussion of our results, let us mention that the very interesting question of existence and nonexistence of positive solutions of (8) seems to be very delicate for general functions f . However, as a consequence of the uniqueness theorems for ode's, the only radially symmetric, positive solutions of (8) are constants if f is locally Lipschitz in (0, +∞) and q ≥ 1. The case 0 < q < 1 seems more difficult to deal with, even with radial symmetry. We refer the reader to [27] for an Emden-Fowler analysis in the particular case f (t) = t p , p > 0.
Finally, let us say a word about our methods of proof. As far as the nonexistence goes, the idea in higher dimensions (N ≥ 3) is similar as the one in [4]. It is possible to show that if there exists a positive supersolution u of types 1, 2 or 3, then there exists a positive, radially symmetric solution v which is of the same type as u. Thus the problem is reduced to a radial one, and a further change of variables transforms it into a one-dimensional, singular, sort-of initial value problem. The analysis of the latter is then performed along similar lines as in [4]. As for the case N = 2, the previous reduction is not possible. Therefore, we only perform it in annulus whose outer radius goes to infinity, which involves some extra estimates for the obtained radially symmetric solutions.
With regard to the existence results, they are obtained either by explicit construction in some cases, or by employing Schauder's fixed point theorem for an integral operator in a suitable space of functions.
The rest of the paper is organized as follows: in Section 2 we gather some preliminaries, which have to do with the classification of positive supersolutions of (1) and with the reduction to the radial case. Sections 3 and 4 are dedicated to the cases N ≥ 3 and N = 2, respectively.

2.
Preliminaries. In this section we consider several questions related to the classification of positive supersolutions of (1) and the reduction to the radial setting. We begin with an extension of a result in [4] (see also [2]), which gives sense to the classification introduced in the previous section. Remember that, for a positive supersolution u of (1) we always denote in the weak sense. Then, there exists R 1 > R 0 such that the function m(R) is either strictly increasing, or strictly decreasing or constant in (R 1 , +∞).
Proof. According to Lemma 3 in [4] , there exists R 0 ≥ R 0 such that m(R) is monotone for R > R 0 . As we have observed in the introduction, we can always Suppose first that m is nondecreasing. Let us show that if there exists an interval the function m(R) attains an interior minimum in the annulus A(R, R 3 ) = {x ∈ R N : R < |x| < R 3 }. Thus by the maximum principle u has to be constant in A(R, R 3 ), contradicting that m(R 3 ) > m(R 1 ). We deduce that either m is constant for R > R 0 or m is strictly increasing in (R 0 , +∞).
A similar argument shows that, if m is nonincreasing and it is constant in an interval [R 1 , R 2 ], then m = m(R 1 ) if R ∈ (R 0 , R 1 ). Assume such an interval exists (if not m is strictly decreasing and we are done) and m is not constant for R > R 2 . Choose R 2 with the property that the interval [R 1 , R 2 ] is the maximal interval where m is constant. Then m is decreasing for R > R 2 . If not, there would exist But this would imply m is constant for R < R 4 , a contradiction. Therefore either m is strictly decreasing for R > R 2 or m is constant for R > R 2 . This concludes the proof.
Remark 1. As mentioned in the introduction, if q < 1 there always exist positive nonconstant supersolutions of (1) which are constant for large |x|. Indeed, fix λ > 0 and denote where r = |x|, as usual. Since (R 0 + δ − r)/r ≤ δ/R 0 , the previous inequality can always be achieved if δ is chosen small enough and then A is large enough. Finally, the function will be a positive C 1 supersolution of (1).
Our next step is to make a little more precise the classification of supersolutions. Depending on the dimension N , only some types of supersolutions are possible. This is the content of the following lemma. Proof. Assume u is a positive, not eventually constant supersolution of (1) and consider initially N ≥ 3. Choose R 2 > R 1 > R 0 and define the function Now assume that u is of type 4, that is, lim R→+∞ m(R) = +∞. Letting R 2 → +∞ in (10) with fixed R and R 1 we arrive at a contradiction.
Next we analyze the case N = 2. We consider the function which is obtained by replacing in the definition of Φ the power 2 − N by a logarithm, that is for R ∈ (R 1 , R 2 ). Assume u is not of type 4, so that lim R→+∞ m(R) = ∈ [0, +∞). We can let R 2 → +∞ in (12) to obtain m(R) ≥ m(R 1 ). Since R and R 1 are arbitrary, it follows that m is nondecreasing. Therefore u has to be of type 2. The proof is concluded.
Finally, we show that when N ≥ 3, and for a suitable range of q, it is always possible to reduce problem (1) to a radial setting.
If there exists a positive supersolution u of (1), then there exists R 1 > R 0 and a positive, radially symmetric solution v of (1) in R N \ B R1 which is of the same type as u.
Proof. For R 2 > R 1 > R 0 , consider the function Φ given by (9) in A(R 1 , R 2 ). To stress the dependence of Φ on R 2 we will temporarily denote it by Φ R2 . As before, It is clear that u is a supersolution of (13) while Φ R2 is a subsolution. Therefore, by the method of sub and supersolutions there exists a minimal solution v R2 of (13), which is radially symmetric (see for instance the Appendix in [4]; only the case q = 0 is dealt with there, but the proofs are easily adapted to cover the full range q ∈ (0, 2] with the aid of Theorem 2.1 in [14] and the standard estimates in Section 4.3 of [31]). Now we would like to pass to the limit as R 2 → +∞. We have the inequalities 0 < v R2 ≤ u, so this gives local bounds for the set {v R2 } R2>R1 . Since q ≤ 2, we can also obtain bounds for the gradient of the solutions using the results in Section 4.3 of [31]. Then it is standard to get local C 1,α bounds (cf. [26]), so that by means of a diagonal procedure we obtain a sequence R 2,n → +∞ such that v R2,n → v in C 1 loc (R N \ B R1 ). In particular, we obtain that v is a radially symmetric weak solution of the equation where = lim R→+∞ m(R), which is finite by Lemma 2.2. Now reaching the desired conclusion is easy: if u is of type 1, then = 0, so that v(r) → 0 as well, therefore v is decreasing for large r and it is of type 1. If u is of type 2 then > 0 and v(r) < m(r) < for r > R 1 , and we obtain lim r→+∞ v(r) = by (14). Hence v is of type 2. Finally, when u is of type 3 we have m(R 1 ) > , so that again by (14) v > and lim r→+∞ v(r) = ; thus v is of type 3.

Remark 2.
When N = 2 and there exists a positive supersolution of (1), it is equally possible to obtain a positive, radially symmetric solution v as in Lemma 2.3. Unfortunately, it is not possible to show with the same methods that v is of the same type as u. Actually, it can be shown in some cases that v is not! 3. The case N ≥ 3. The purpose of the present section is to prove Theorem 1.1.
For the sake of clarity, we split the proof into a series of lemmas, dealing in turn with each type of supersolutions. Of course we will assume throughout the section that N ≥ 3 and f ∈ C([0, +∞)) is positive in (0, +∞). We begin by considering the case q ≥ 2.
Proof. Since we are looking for radially symmetric solutions, we assume u(x) = v(r), r = |x|, so that we need to solve the equation With the change of variables s = r 2−N /(N − 2), v(r) = w(s), the previous equation Since f is continuous and ν ≥ 0, it follows by Cauchy-Peano's theorem that there exists at least a local solution w of this problem, defined in an interval [0, s 0 ] for some small positive s 0 . This solution is in addition positive if s 0 is small enough. Thus there exists a positive, radially symmetric solution u of (1) in the complement of a ball. Finally, observe that u is of type 1 when λ = 0, µ > 0, of type 2 when λ > 0, µ < 0 and of type 3 if λ > 0, µ > 0. The proof is concluded.
We next turn to the subquadratic case 0 < q < 2, and consider supersolutions of type 1. The proof of our next lemma is an adaptation of that of Theorem 6 in [4].
for some δ > 0, where Proof. Assume first that (16) does not hold, and there exists a positive supersolution of type 1 of (1). By Lemma 2.3 there exists R 1 > R 0 and a positive, radially symmetric solution v of (1) in R N \ B R1 of type 1. We make the change of variables s = r 2−N /(N − 2), w(s) = v(r) in the ordinary differential equation satisfied by v.
Then the function w is nondecreasing and verifies, for some c > 0, s 0 > 0, We claim that w > 0 in (0, s 0 ). Indeed, if w (s 1 ) = 0 for some s 1 ∈ (0, s 0 ), then since w ≤ 0, we would obtain w (s) = 0 for s ∈ (s 1 , s 0 ), so that w would be constant in (s 1 , s 0 ), against our assumptions. Thus w > 0. Also, the mean value theorem gives w(s) = w (ξ)s ≥ w (s)s, where ξ is some point in the interval (0, s). Hence The monotonicity of w implies that w(s) ≥ C 0 s for some C 0 > 0 and every s ∈ (0, s 0 ). We divide the equation in (17) by (w ) q−1 and integrate in (s, s 0 ) to arrive at t θ w (t)dt for every s ∈ (0, s 0 ). Hence, using (18): for s ∈ (0, s 0 ). Taking into account that w(s) ≥ C 0 s, (19) implies We deduce, since w(0) = 0 and (16) does not hold, that lim s→0 w(s)/s = +∞. Thus, we can diminish s 0 to ensure that w(s) ≥ s in (0, s 0 ). We will reach a contradiction by iterating the use of (19). The inequality w(s) ≥ s in (0, s 0 ) gives: Taking this inequality again in (19) we have the improved inequality where we have used H(w(s 0 )) = 0. It is possible to iterate this procedure to obtain two sequences {a k } ∞ k=1 and {b k } ∞ k=1 given by It is not hard to obtain an explicit expression for a k :

From this expression, it easily follows that
for some positive constant C 1 , which in turn gives for b k the inequality b k ≤ for k ≥ 1. Iterating this inequality from k = 1 we see that for k ≥ 1. The sum in the last exponent is an arithmetic-geometric sum, hence we can explicitly evaluate it: and it follows finally from (22) and (23) for some C 2 > 1.
Letting k → +∞, and observing that θ > 2 − q, we reach a contradiction, so that there are no positive supersolutions of (1) of type 1 when condition (16) does not hold.
Next, let us prove the converse implication and assume that (16) holds. For λ > 0, consider the Cauchy problem Observe that any positive solution of (25) gives rise, with the change of variables s = r 2−N /(N − 2), w(s) = v(r), to a positive, radially symmetric solution of (1) of type 1. Therefore our proof is reduced to show that there actually exists a positive solution of (25) when λ is small enough. Denote z λ (s) = λs. In the Banach space X = {z ∈ C 1 [0, s 0 ] : z(0) = 0} endowed with the standard C 1 norm |z| C 1 = max{|z| ∞ , |z | ∞ }, consider the set B = {z ∈ X : |z − z λ | C 1 ≤ λ 2 }, which is closed and convex, and for z ∈ B define the integral operator We claim that T is well-defined, maps B into B and is compact. To show the first two assertions, notice that for every z ∈ B, we have the inequalities λ 2 s ≤ z ≤ 3λ 2 s, in [0, s 0 ], taking λ small enough (observe that θ + q − 1 > 1 since N ≥ 3 and 0 < q < 2). It follows in a similar way that |T z(s) − z λ | ≤ λ 2 . Thus T is well defined and maps B into B.
To show that T is compact, let {z n } ∞ n=1 be an arbitrary sequence and denote w n = T z n . It follows by (26) that {w n } ∞ n=1 is uniformly bounded in [0, s 0 ], so that {w n } is equicontinuous and uniformly bounded, and we may assume that w n → w uniformly in [0, s 0 ], for some w ∈ C[0, s 0 ]. We claim that w ∈ C 1 [0, s 0 ] and w n → w uniformly in [0, s 0 ].
Observe that |w n (s)| = s −θ f (z n (s)) is uniformly bounded in compacts of (0, s 0 ]. Hence by means of Arzelá-Ascoli's theorem and a diagonal argument we may assume that also w n →w uniformly in compacts of (0, s 0 ] for somew ∈ C(0, s 0 ]. We readily get that w ∈ C 1 (0, s 0 ] andw = w . But the convergence is indeed uniform in [0, s 0 ] (defining w (0) = λ). To prove it, take ε > 0. By (26): and the same is true for w(s) by passing to the limit. Hence provided s ∈ [0, δ] for δ < s 0 small enough. Since w n → w uniformly in [δ, s 0 ], we also have |w n (s) − w (s)| ≤ ε for s ∈ [δ, s 0 ] if n is large enough. Thus w n → w uniformly in [0, s 0 ], that is, w n → w in X and T is compact. The continuity of T is shown by using a similar argument. Hence we can apply Schauder's fixed point theorem to obtain a fixed point w ∈ B of T , which is a solution of (25) with w > 0 in (0, s 0 ]. As already observed, this concludes the proof of the lemma. Remark 3. It can be easily deduced from the above proof that condition (16) is necessary and sufficient for existence of solutions even if f is not defined at zero.
We conclude our preliminary lemmas with supersolutions of type 2. Proof. Assume first q < 1. Let u be a positive supersolution of (1) of type 2. By Lemma 2.3, there exists a positive, radial solution v of (1) of the same type defined in R N \ B R1 for some R 1 > R 0 . Thus v verifies (15) in Lemma 3.1, while v ≥ 0 for large r and lim r→+∞ v(r) = > 0. We perform the same change of variables s = r 2−N /(N − 2) and w(s) = v(r). Then, for some small positive s 0 and some c > 0: where now w ≤ 0. Arguing as in Lemma 3.2 we can show that w < 0 in (0, s 0 ). Letting z = − w, we have that Observe also that z > 0. Then, since f is strictly positive in a neighborhood of , we obtain for some positive constants C and D, after dividing the previous equation by (z ) q and integrating between s and s 0 for s ∈ (0, s 0 ): We arrive at a contradiction by letting s go to zero, since θ > 1 in this case. When q ≥ 1, it is easy to construct a positive supersolution of (1) with radial symmetry. Indeed, we can look for u in the form u(x) = − e −α|x| , where α > 0 is chosen suitably and > 0 is arbitrary. It is not hard to check that u will be a supersolution of (1) provided that for large |x|. Denoting M = sup t∈[0, ] f (t), the previous inequality is a consequence of This inequality can be easily achieved for |x| ≥ R 0 by choosing α large enough. The proof is concluded.
We finally turn to the proof of our main result in the case N ≥ 3.
Proof of Theorem 1.1. First of all observe that, by Lemma 2.2, there do not exist positive supersolutions of type 4 when N ≥ 3. Also, part (a) is a direct consequence of Lemma 3.1 when q ≥ 2.
On the other hand, all the assertions dealing with supersolutions of types 1 and 2 are already proved in Lemmas 3.2 and 3.3, respectively. Thus, only the statements regarding supersolutions of type 3 need to be shown. But observe that if u is a supersolution of type 3 and = lim R→+∞ m(R) ∈ (0, +∞), then v = u − is a positive supersolution of type 1 for the problem 4. Positive supersolutions in two dimensions. In this final section we deal with positive supersolutions of (1) in the planar case N = 2. It is worthy of mention that the reduction to a radial setting as in Lemma 2.3 is not possible, so a slightly different approach is needed. In most proofs we will actually reduce the problem to a radial one, but in a finite interval (R 1 , R 2 ). This implies that some further estimates are needed before we can let R 2 → +∞.
By Lemma 2.2 we know that only supersolutions of types 2 and 4 are possible. Let us begin with the former ones. Proof. The proof is similar to that of Lemma 3.3 in Section 3. Indeed, the proof that there are positive supersolutions of (1) of type 2 when q ≥ 1 is exactly the same as in that lemma, just setting N = 2.
Thus we assume in what follows that q < 1 and show that no positive supersolutions of (1) of type 2 exist. Assume for a contradiction that there is one such u. Choose R 1 > R 0 and for R 2 > R 1 , consider the problem where Ψ is given by (11) in Lemma 2.2 (recall that Ψ is harmonic in A(R 1 , R 2 ) and Ψ(R i ) = m(R i ) for i = 1, 2). By the same arguments as in the proof of Lemma 2.3, there exists a radially symmetric solution v ∈ C 1 [R 1 , R 2 ] of this problem verifying Ψ ≤ v ≤ u in A(R 1 , R 2 ) (the solution v depends of course on R 2 , but we are not making this dependence explicit for brevity). By the maximum principle, and since m is an increasing function, we have v ≥ m(R 1 ). Moreover, since v is radially symmetric, v(r) ≤ m(r) ≤ m(R 2 ) ≤ lim R→+∞ m(R) =: if R 1 ≤ r ≤ R 2 , where ∈ (0, +∞). It also follows that v attains its maximum at R 2 .
Let γ = min t∈[m(R1), ] f (t) > 0. We deduce that v verifies With the change of variables s = log r, w(s) = v(r), we arrive at the problem Let us see that w > 0 in (s 1 , s 2 ). Indeed, if we had w (s) = 0 for somes ∈ (s 1 , s 2 ), using the monotonicity of w we would arrive at w (s) ≤ 0 for every s ∈ (s, s 2 ). Since w attains its maximum at s 2 , this would imply that w(s) = m(R 2 ) if s ∈ (s, s 2 ). This in turn yields v(r) = m(R 2 ) for r ∈ (es, R 2 ), which leads to m(r) = m(R 2 ) if r ∈ (es, R 2 ), which is not possible.
Thus w > 0. Dividing by (w ) q and integrating between s 1 and s 2 , we have Next, we claim that w (s 1 ) ≤ C for a positive constant C independent of R 2 . Indeed, choose a small δ > 0, and let M = sup t∈[m(R1), ] f (t). Arguing in a similar way as before, we have Dividing by (w ) q−1 and integrating between s 1 and s we obtain: If we assume the right-hand side is positive (otherwise there is nothing to prove), we can raise to the power 1 2−q and integrate in (s 1 , s 1 + δ) to have This shows (30).
Coming back to (29), we have: Letting R 2 → +∞, we obtain a contradiction, which shows that no positive supersolutions of (1) of type 2 exist. The proof is concluded.
The analysis of supersolutions of type 4 is slightly different since there are necessary and sufficient conditions for their existence only when q ∈ (0, 2). Let us deal with this case next.
where s i = log R i , i = 1, 2. It is equally proved that w > 0 in (s 1 , s 2 ). Hence we may divide the equation by (w ) q−1 and integrate between s 1 and s 2 to get With the aid of (30), this inequality gives s2 s1 e (2−q)t f (w(t))w (t) dt ≤ C 2−q 2 − q .
Next, we claim that w(s) ≤ Ks for large s ∈ (s 1 , s 2 ), where K does not depend on R 2 . Indeed, since w ≤ 0, we deduce using (30): where K does not depend on R 2 . This shows the claim. Coming back to (33), since s ≥ 1 K w(s), we see that Letting R 2 → +∞ we obtain that (31) holds, against the assumption. Thus no positive supersolutions of (1) of type 4 can exist.
To conclude the proof, let us assume that (31) holds. Our intention is to show that the problem admits a positive solution. The change of variables s = log r, w(s) = v(r) will then provide with a positive, radially symmetric solution of (1) which is of type 4. For this sake, we use again Schauder's fixed point theorem, but with some important differences with respect to the proof of Lemma 3. Let us prove that T maps B into B if λ is chosen large enough. To begin with, assume λ ≥ 2(2 − q)/a. Taking into account that z(s) ≥ λ 2 s, λ 2 ≤ z (s) ≤ 3λ 2 in [s 0 , +∞) for every z ∈ B: if λ is chosen large enough, since 0 < q < 2. Integrating this inequality we also obtain: for every s > s 0 , hence T is well-defined and T ( B) ⊂ B.
Let us finally show that T is compact (as before, the continuity of T is shown by arguing in a similar way). Take an arbitrary sequence {z n } ∞ n=1 ⊂ B and let w n = T z n . By Arzelá-Ascoli's theorem and a diagonal argument, since the sequences {T z n }, {(T z n ) } and {(T z n ) } are locally uniformly bounded, we may assume w n → w, w n → w uniformly on compact sets for some function w ∈ C 1 [s 0 , +∞).
Let us show that the convergence w n → w is actually uniform in [s 0 , +∞). Indeed, observe that, if we fix s 1 > s 0 , take s > s 1 , and argue as in (35) we arrive at e at f (t)dt.
A similar equality holds for w, by passing to the limit. Hence for every s > s 1 . Next take ε > 0. Choosing s 1 large enough we have the last term less than ε 2 . Taking n large enough we also have |w n (s 1 ) −w(s 1 )| ≤ ε 2 , hence |w n (s) −w(s)| ≤ ε if s > s 1 . Since this inequality also holds in [s 0 , s 1 ] for large enough n, we obtain that w n → w uniformly in [s 0 , +∞). Hence |w n (s) − w(s)| s ≤ 1 s s s0 |w n (t) − w (t)|dt → 0 uniformly in [s 0 , +∞), as n → +∞. This shows that T is compact in B. Therefore we can apply Schauder's fixed point theorem to obtain that T has a fixed point w in B, which is a solution of (34) verifying λ 2 s ≤ w(s) ≤ 3λ 2 s. Observe that this implies that w is positive and lim s→+∞ w(s) = +∞. As remarked above, this function provides with a positive, radially symmetric solution of (1) of type 4. The proof is concluded.
We finally consider the only left case where q ≥ 2. It is worth mentioning that, although positive supersolutions can always be constructed, they can be of different types, depending on the behavior at infinity of their derivatives. However, we are not exploring this distinction further.
We can argue in a completely similar way as in (35), except that now the exponent in the exponential is negative, to obtain that, for λ ≤ 2/(3a): e −at f (t)dt.
This difference can be made less than or equal to λ 2 provided λ is chosen small enough, since q > 2. The rest of the proof in this case is essentially the same as that of Lemma 4.2 and therefore will be omitted.
To conclude the proof, consider the case where f does not satisfy (38). In particular, f verifies lim t→+∞ F (t) = +∞. Now let w be the solution of (36) with q = 2, obtained at the beginning of the proof. Since lim s→+∞ w (s) = 0, it follows that −w = f (w)(w ) 2 ≥ e −(q−2)s f (w)(w ) q , s ≥ s 0 , if s 0 is large enough, so that w is a positive supersolution of (36). The proof is concluded.
Proof of Theorem 1.2. It is immediate with the use of Lemmas 4.1, 4.2 and 4.3.