As many common problems in mathematical physics can be formulated as equations of Emden-Fowler type, such equations have been an active topic of research in recent years; they have been approaced with a wide variety of methods and techniques. Interesting results can be found in Napoles [22], [21], [25] and the references therein.

This paper is a contribution to the study of the radial equation of Emden-Fowler type with convection term,

(1.1)(|u'|p-2u')'(r)+N-1r|u'|tp-2u'(r)+α|u|q-1u(r)+βr(|u|q-1u)'(r)=0, r>0,

where p>2, q>1, N≥1, α>0 and β>0.

Equation (1.1) has been well studied in the case of p=2. Indeed, with α=1 and β=0, we obtain the classic Emden-Fowler equation studied by Emden [7] and Fowler [9]. In [8], [9] and [10], Fowler showed the existence of, and gave a classification of, global solutions. The case α>0, β=1 and q<1, (1.1) was studied extensively by Hulshof in [16] . The existence of positive solutions in the case α=1, β=0 and N>2 was considered by Aviles [1], Caffarelli, Gidas and Spruck [6], Gidas and Spruck [11], and Lions [18]. Hsu [13], [14] and Hui [15], looked at the case of α>0 and β>0.

In the general case of p>2, equation (1.1) was studied with α=1 and β=0 by Ni and Serrin [23] and [24], M.F. Bidaut [2] and [3], Guedda and Véron [12]. When α>0 and β=1, Leoni [17] studied the existence and the asymptotic behavior of global solutions.

The main feature of this paper is the presence of the convection term that influences the existence and asymptotic behavior of positive solutions of equation (1.1). More precisely, we have improved the result on the asymptotic behavior obtained by M.F. Bidaut [2] by giving equivalent explicit solutions, and their derivatives, at infinity. Our approach is based on the energy methods introduced by M.F. Bidaut [2] and [3], and the oscillation methods of Napoles [20] and [19].

We study equation (1.1) by classical methods developed by Bouzelmate, Gmira and Reyes [4], suitably modified in order to deal with its degenerate character at r=0 as well as at points where u'=0. This is particularly important for local existence, since we are interested in radial solutions and it is natural to impose u'(0)=0. Thus, consider the following ‘initial value problem’

(P)(|u′|p−2u′)′(r)+N−1r|u′|p−2u′(r)+α|u|q−1u+βr(|u|q−1u)′=0,r>0,u(0)=a>0, u′(0)=0,

where p>2,q>1,N≥1,α>0 and β>0.

Existence and uniqueness of solutions of problem (P) has was showed by Bouzelmate and El Hathout in [5]. By reducing the initial value problem (P) to a fixed point problem for a suitable integral operator, we prove that for each a>0, there exists a unique global solution u(.,a,α,β) of problem (P) such that

(1.2)(|u'|p-2u')'(0)=-αaqN.

The behaviou of equation (1.1) depends strongly of the sign of Nβ-α and the comparison of q with the two critical values N(p-1)N-p and N(p-1)+pN-p. The case Nβ-α≥0 was studied in [5]. We now focus on the case Nβ-α<0.

The main theorems are as follows; they will be proved in Sections 3,4, and 5.

Theorem 1. [Existence of non-positive solutions]

Assume that αβ>N. Let u be a solution of problem (P), then u changes sign in the following cases:

(i) q<p-1.

(ii) N≤p.

(iii) N>p and p-1≤q≤N(p-1)N-p.

(iv) N(p-1)N-p<q<N(p-1)+pN-p and αqβ>N(N-p)(p-1)[N(p-1)+p-q(N-p)].

Theorem 2. [Existence of positive solutions]

Assume that αβ>N. Let u be a solution of problem (P). Then u is strictly positive in the following cases:

(i) N-pp-1≥αqβ.

(ii) 0<N-pp-1<αqβ and q≥N(p-1)+pN-p.

Theorem 3. [Behavior of positive solutions near infinity]

Assume that αβ>N>p and q≥N(p-1)+pN-p. Let u be a solution of problem (P). Then

(1.3)limr→∞rpq+1-pu(r)=L

and

(1.4)limr→∞rpq+1-p+1u'(r)=-pq+1-pL

where

(1.5)L=(p−1)N−pp−1−pq+1−p pq+1−pp−1α−qβpq+1−p1q+1−p.

Our paper is organized as follows. In Section 2, we present some basic tools which will be useful to prove the fundamental theorems above. Existence of non-positive solutions of problem (P) is given in Section 3. In Section 4, we prove the existence of positive solutions. In Section 5 we describe the asymptotic behavior of positive solutions. More precisely, we prove that under some assumptions, the positive solution of problem (P) behaves like r-pq+1-p at infinity. Finally, in Section 6 we give conclusions that can be used to study future research work based on the obtained results in this paper.

In this section we will give some basic tools which will be useful for proving the main results.

Lemma 2.1. The solution u of problem (P) is strictly decreasing as long as that it is strictly positive.

Proof. We argue by contradiction. Let r0>0 be the first zero of u'. Since by (1.2) u'(r)<0 for r∼0, there exists, by continuity and the definition of r0, a left neighborhood ]r0-ε,r0[ (for some ε>0) where u' is strictly increasing and strictly negative. That is (|u'|p-2u')'(r)>0 for any r∈]r0-ε,r0[. Hence by letting r→r0 we get (|u'|p-2u')'(r0)≥0. But by equation (1.1), we have (|u'|p-2u')'(r0)=-α|u|q-1u(r0)<0 since u(r0)>0,u'(r0)=0 and α>0, which is a contradiction. This completes the proof.

Proposition 2.2. Let u be a solution of problem (P). If N>1 or N=1 and u is strictly positive, then

(2.1)limr→∞u(r)=limr→∞u'(r)=0.

Proof. We distinguish two cases.

Case 1. N>1. We define the following energy function,

(2.2)E(r)=p-1p|u'(r)|p+αq+1|u(r)|q+1.

According to equation (1.1), we get

(2.3)E'(r)=-ru'2(r)brackN-1r2|u'(r)|p-2+qβ|u(r)|q-1brack.

We show that limr→∞E(r)=0. Since E'(r)≤0 and E(r)≥0 for all r>0, there exists a constant l≥0 such that limr→∞E(r)=l≥0. If l>0, then there exists r1>0, such that

(2.4)E(r)≥l2  for r≥r1.

Now consider the function

D(r)=E(r)+N-12r|u'|p-2u'(r)u(r)+qβ(N-1)2(q+1)|u(r)|q+1.

We get

D'(r)=-qβr|u(r)|q-1(r)u'2(r)-N-12r[|u'(r)|p+Nr|u'|p-2u'u(r)+α|u(r)|q+1].

Since β>0, we have

D'(r)≤-N-12r[|u'(r)|p+α|u(r)|q+1+Nr|u'|p-2u'u(r)].

Recalling that u and u' are bounded (because E is bounded), we have

limr→∞|u'|p-2u'u(r)r=0.

Moreover, by (2.4) we have

|u'(r)|p+α|u(r)|q+1≥p-1p|u'(r)|p+αq+1|u(r)|q+1=E(r)≥l2 for r≥r1.

Consequently, there exist two constants c>0 and r2≥r1 such that

D'(r)≤-cr for r≥r2.

Integrating this last inequality between r2 and r, we get

D(r)≤D(r2)-cln(rr2) for r≥r2.

In particular we obtain limr→∞D(r)=-∞. Since

E(r)+N-12r|u'|p-2u'(r)u(r)≤D(r),

we get limr→∞E(r)=-∞. This is impossible. So limr→∞E(r)=0, giving the conclusion.

Case 2. N=1 and u is strictly positive. Let

(2.5)φ(r)=|u'|p-2u'(r)+βr|u|q-1u(r).

By equation (1.1),

(2.6)φ'(r)=(β-α)|u|q-1u(r).

Since u is strictly positive, it is strictly decreasing. Therefore limr→∞u(r)∈[0,∞[. Suppose that limr→∞u(r)=L>0. Since the energy function E given by (2.2) converges, then necessarily, limr→∞u'(r)=0. Therefore limr→∞φ(r)=∞.

Using L'Hopital's rule, we have

limr→∞φ'(r)=limr→∞φ(r)r.

That is

(β-α)Lq=βLq.

Therefore, -αLq=0, contradicting L>0. Hence, limr→∞u(r)=0.

Lemma 2.3. Assume that αβ>N. Let u be a strictly positive solution of (P).

(i) If N>p, there exists a constant C1>0 such that

(2.7)u(r)≥C1r-N-pp-1 for large r.

(ii) If q>p-1, there exists a constant C2>0 such that

(2.8)u(r)≤C2r-pq+1-p for large r.

Proof.(i) We introduce the following function

(2.9)φ(r)=rN-1|u'|p-2u'(r)+βrNuq(r).

Then by (1.1), we get

(2.10)φ'(r)=(βN-α)rN-1uq(r).

Since, u>0 and Nβ<α, then φ'(r)<0 ∀r>0 and as φ(0)=0, we get φ(r)<0, ∀r>0. Then limr→∞φ(r)∈[-∞,0[. Therefore there exists C>0, such that φ(r)<-C for large r. This gives

(2.11)rN-1|u'|p-2u'(r)<-C for large r.

Consequently

(2.12)u'(r)<-C1p-1r1-Np-1 for large r.

Integrating this last inequality on (r,R) for large r and using the fact that N>p and limr→∞u(r)=0, we deduce by letting R→∞, that there exists a constant C1>0 satisfying (2.7).

(ii) Using the fact that φ(r)<0 and u'(r)<0, ∀r>0, we obtain

(2.13)u'(r)u-qp-1≤-β1p-1r1p-1.

Integrating this last inequality on (0,r) and taking into account q>p-1, we deduce that there exists a constant C2>0 satisfying (2.8).

Now for any c>0, define the function

(2.14)Ec(r)=cu(r)+ru'(r), r>0.

It is clear that

(2.15)rcu(r)'=rc-1Ec(r), r>0.

Hence, using (1.1), we have for any r>0 such that u'(r)≠0,

(2.16)(p−1)|u′|p−2(r) E ′ c(r)=(p−1)(c−N−pp−1)|u′|p−2u′(r)−αr|u|q−1u−qβr2|u|q−1u′(r)=(p−1)(c−N−pp−1)|u′|p−2u′(r)−qβr|u|q−1Eαqβ(r).

Consequently, if Ec(r0)=0 for some r0>0, equation (1.1) gives

(2.17)(p−1)|u′|p−2(r0) E ′ c(r0)=r0|u|q−1u(r0)(qβc−α)+(p−1)cp−1N−pp−1−c|u|p−q−1 (r0 )r0p.

From which the sign of Ec(r) for large r can be obtained.

Lemma 2.4. Assume that αβ>N. Let u be a strictly positive solution of (P).

(i) If N-pp-1≥αqβ, Eαqβ(r)>0 for any r>0.

(ii) If 0<N-pp-1<αqβ, EN-pp-1>0 for any r>0.

Proof.

(i) We distinguish two cases.

Case 1. N-pp-1>αqβ. We have Eαqβ(0)=αqβu(0)>0. Let r0>0 be the first zero of Eαqβ(r). Therefore Eαqβ(r)>0 in [0,r0[, Eαqβ(r0)=0 and E'αqβ(r0)≤0. But using the fact that u(r0)>0 and N-pp-1>αqβ, we have by (2.17), E'αqβ(r0)>0, which is a contradiction.

Case 2. N-pp-1=αqβ.

By (2.16), we have

(2.18)(p-1)|u'|p-2E'αqβ(r)=-qβr|u|q-1Eαqβ(r).

Let r0>0. We introduce the following function

(2.19)f(r)=qβp−1∫r0rs|u′(s)|2−p|u(s)|q−1ds.

By (2.18), we obtain

(2.20)E'αqβ(r)+f'(r)Eαqβ(r)=0.

Hence,

(2.21)ef(r)Eαqβ(r)'=0.

Integrating this last equality from r0 to r, we obtain

(2.22)Eαqβ(r)=Eαqβ(r0)e-f(r)  ∀r>r0.

Since Eαqβ(r0)>0 for any r0>0 close to 0, then Eαqβ(r)>0 for any r>0.

(ii) We will show the result in two steps.

Step 1. Eαqβ(r)≠0 for large r. Assume that there exists a large r0 such that Eαqβ(r0)=0. Using the fact that u>0 and N-pp-1<αqβ, we get from (2.17), E'αqβ(r0)<0 and thereby Eαqβ(r)≠0 for large r.

Step 2. EN-pp-1(r)>0 ∀r>0. We have EN-pp-1(0)>0. Let r0>0 be the first zero of EN-pp-1. Then by (2.17), E'N-pp-1(r0)<0. Therefore EN-pp-1(r)<0 ∀r>r0. On the other hand by Lemma 2.3, we get rαqβu(r)≥C1rαqβ-N-pp-1, hence limr→∞rαqβu(r)=∞. Since Eαqβ(r)≠0 for large r by step 1 , then necessarily Eαqβ(r)>0 for large r. Moreover, by (2.16), we have E'N-pp-1(r)<0 for large r. As EN-pp-1(r)<0 ∀r>r0 we have limr→∞EN-pp-1(r)∈[-∞,0[, which implies that limr→∞ru'(r)∈[-∞,0[ (because limr→∞u(r)=0), but this is impossible since u is positive and bounded. Then EN-pp-1(r)>0 ∀r>0. This completes the proof of Lemma.

Proof. Assume that u is strictly positive. We distinguish seven cases:

Case 1. q<p-1. Since φ(r)<0 and u'(r)<0, for any r>0, we have estimate (2.13), and therefore

u(p-1-q)/(p-1)p-1-qp-1'<-β1/(p-1)rp/(p-1)pp-1'.

Integrating this last inequality twice from 0 to r and letting r→∞, we obtain limr→∞u(p-1-q)/(p-1)=-∞, which contradicts the fact that u is strictly positive.

Case 2. N<p. Since u>0,u'<0,φ(r)<0 and φ'(r)<0, for any r>0, we obtain (2.12). By integrating it on (r0,r) for large r0 and letting r→∞, we obtain limr→∞u(r)=-∞, which is a contradiction with the fact that u is strictly positive.

Case 3. N=p. Since N=p, inequality (2.12) is equivalent to

(3.2)u'(r)<-C1r-1 for large r.

Integrating (3.2) on (r0,r) for large r0 and letting r→∞, we obtain a contradiction with the fact that u is strictly positive.

Case 4. N>p and p-1<q<N(p-1)N-p. In this case we have, 0<N-pp-1<pq+1-p. By Lemma 2.3, we obtain

(3.3)C1rpq+1-p-N-pp-1≤C2 for large r.

Letting r→∞ in this last inequality we obtain a contradiction with the fact that

N-pp-1<pq+1-p.

Case 5. N>p and q=p-1. By (2.13), we obtain

(3.4)u'(r)u(r)<-β1p-1r1p-1.

Integrating in (0,r) for r>0, we get

(3.5)u(r)<u(0)e-p-1pβ1p-1rpp-1.

Then, limr→∞rN-pp-1u(r)=0. But this contradicts (2.7).

Case 6. N>p and q=N(p-1)N-p. Then N-pp-1=pq+1-p. We have by (2.7), u(r)≥C1rp-Np-1=C1r-pq+1-p for large r. Since, Nβ-α<0, we have by (2.10),

(3.6)φ'(r)≤(βN-α)C1qrN-1-pqq+1-p,

Since N=pqq+1-p (because N-pp-1=pq+1-p), then by (3.6)

(3.7)φ'(r)≤(βN-α)C1qr-1  for large r.

Integrating (3.7), we get limr→∞φ(r)=-∞, which implies by (2.9) that

limr→∞rN-1p-1u'(r)=-∞. Using Hopital's rule we obtain, limr→∞rN-pp-1u(r)=∞. But this contradicts the fact that u(r)≤C2r-pq+1-p=C2rp-Np-1.

Case 7. N(p-1)N-p<q<N(p-1)+pN-p and αqβ>N(N-p)(p-1)[N(p-1)+p-q(N-p)]. Using the Pohozaev identity, we put

(3.8)G(r)=rNp-1p|u'|p+α(N-p)Npuq+1+N-pprN-1|u'|p-2u'u, r>0.

Then

(3.9)G'(r)=qβrNuq-1|u'|Eγ(r), r>0

where γ=αqβN(p-1)+p-q(N-p)Np+N-pp.

As αqβ>N(N-p)(p-1)[N(p-1)+p-q(N-p)], we have γ>N-pp-1>0, therefore by Lemma 2.4, Eγ(r)>0 for any r>0, hence G'(r)>0 for any r>0. As G(0)=0 we obtain G(r)>0 for any r>0, then limr→∞G(r)∈]0,∞], which implies that there exists a constant C>0 such that G(r)≥C for large r. This gives by (3.8) and the fact that u' is negative that

(3.10)p-1p|u'|p≥Cr-N-α(N-p)Npuq+1 for large r.

Using (2.8), we obtain

(3.11)p-1p|u'|p≥r-NC-C2q+1α(N-p)NprN-p(q+1)q+1-p: for large r.

Since N-p(q+1)q+1-p=q(N-p)-N(p-1)-pq+1-p<0, we have limr→∞rN-p(q+1)q+1-p=0.

Consequently by (3.11), there exists a constant K>0 such that

p-1p|u'|p≥Kr-N for large r.

This gives, since u'(r)<0, that

(3.12)u'(r)≤-Kpp-11/pr-Np for large r.

Integrating this last inequality on (R,r) and letting R→∞ we see that there exists a constant M>0 such that

(3.13)u(r)≥Mrp-Np for large r.

This gives that limr→∞rpq+1-pu(r)=∞ since pq+1-p>N-pp (because q<N(p-1)+pN-p). But this contradicts the fact that rpq+1-pu(r) is bounded by Lemma 2.3.

We deduce that in the seven cases, u is not strictly positive. Let r0>0 be the first zero of u. Then, u(r)>0, u'(r)<0, for any r∈(0,r0) and u'(r0)≤0. Suppose that u'(r0)=0, hence by (2.9), φ(r0)=0. Since φ'(r)<0 ∀r∈(0,r0), then φ(r0)<φ(r)<φ(0)=0. A contradiction arises, consequently u'(r0)<0 and so u changes sign. This completes the proof of theorem.

The proof requires the following result.

Proposition 4.1. Let u be a solution of problem (P). Assume that the first zero R of u is positive. For 0<k<m, we have

(4.1)∈t0Ruq|u'|ksm-1ds≤q+km-k∈t0Ruq-1|u'|k+1smds.

Proof. By Holder's inequality we have

(4.2)∫0R uq|u′|ksm−1ds≤∫0R uq+k sm−1−k ds1k+1 ×∫0R uq−1 | u′ |k+1 sm dskk+1 .

On the other hand, using the fact that u(R)=0, we obtain

(4.3)∈t0Ruq+ksm-1-k'sds=-∈t0Ruq+ksm-1-kds.

Therefore

(4.4)(q+k)∫0Ru′uq+k−1sm−kds+(m−1−k)∫0R u q+k sm−1−kds=−∫0R u q+k sm−1−kds.

Using the fact that u'<0 in (0,R), we get

(4.5)∈t0Ruq+ksm-1-kds=q+km-k∈t0R|u'|uq+k-1sm-kds.

Applying Holder's inequality again we obtain

(4.6)∫0R u q+ksm−1−kds≤q+km−k∫0R uq+k sm−1−k dskk+1 ×∫0R uq−1 | u′ |k+1 sm ds1k+1 .

Therefore,

(4.7)∈t0Ruq+ksm-1-kds1-kk+1≤q+km-k∈t0Ruq-1|u'|k+1smds1k+1.

Combining (4.2) and (4.7), we obtain easily the estimation (4.1). This completes the proof of proposition.

Now, we turn to the proof of Theorem 2.

Proof. We argue by contradiction and assume that the first zero r0 of u exists and is positive. Then, u(r)>0 ∀r∈[0,r0[, u'(r)<0 ∀r∈(0,r0) and u'(r0)≤0. We distinguish three cases:

Case 1. N-pp-1>αqβ. We have Eαqβ(r)>0 ∀r∈[0,r0[. Indeed, suppose there exists r1∈]0,r0[ such that Eαqβ(r1)=0 (r1 is the first zero because Eαqβ(0)>0). Since u(r1)>0, then u'(r1)<0 and therefore E'αqβ(r1) exists and E'αqβ(r1)≤0. On the other hand, we have by (2.17)

(4.8)(p−1)|u′(r1)|p−2E ′ αqβ(r1)=(p−1)αqβp−1N−pp−1−αqβ|u|p−2u(r1)r1p−1.

Then E'αqβ(r1)>0. This is a contradiction. Hence, Eαqβ(r)>0 ∀r∈[0,r0[. Recall (2.15), this gives rαqβu(r)'>0 in ]0,r0[ and consequently for 0<r<r0, we have rαqβu(r)<r0αqβu(r0)=0 ∀r∈]0,r0[. Which is impossible.

Case 2. N-pp-1=αqβ.

As u>0 and u'<0 on (0,r0), then E'αqβ(r) exists in (0,r0) and we have

(4.9)(p-1)|u'|p-2E'αqβ(r)=-qβruq-1Eαqβ(r) ∀r∈(0,r0).

Since Eαqβ(0)>0, then for r1>0 near to 0, we have Eαqβ(r1)>0 and we obtain by (4.9)

(4.10)Eαqβ(r)=Eαqβ(r1)e-qβp-1∈tr1rs|u'(s)|2-puq-1(s)ds, ∀r∈(r1,r0).

Therefore Eαqβ(r)>0 for all r∈(r1,r0), which is equivalent to rαqβu(r)'>0 on (r1,r0), but this is impossible since u(r0)=0.

Case 3. 0<N-pp-1<αqβ and q≥N(p-1)+pN-p.

Since u>0 and u'<0 on (0,r0), then for any r∈(0,r0),

(4.11) rNp−1p|u′|p+αq+1uq+1+Nq+1 rN−1|u′ |p−2 u′u′=Nq+1−N−pprN−1|u′|p+qβNq+1rNuq|u′|−qβrN+1uq−1 u ′ 2.

Integrating this last inequality on (0,r) for 0<r<r0, we obtain

(4.12)rNp−1p|u′|p+αq+1uq+1+Nq+1rN−1|u′|p−2u′u=Nq+1−N−pp∫0r s N−1 |u′|pds+qβNq+1∫0r s N uq|u′(s)|ds−qβ∫0r s N+1 uq−1 u ′ 2(s)ds.

Since u(r0)=0, then by Proposition 4.1, we have

(4.13)∈t0r0sNuq|u'(s)|ds≤q+1N∈t0rsN+1uq-1u'2(s)ds.

Letting r→r0 in (4.12), we obtain

(4.14)p−1pr0N|u′(r0)|p=Nq+1−N−pp∫0r0 s N−1 |u′|pds+qβNq+1∫0r0 s N uq|u′(s)|ds−qβ∫0r0 s N+1 uq−1 u ′ 2(s)ds.

Then by (4.13) we have

p−1pr0N|u′(r0)|p≤Nq+1−N−pp∫0r0 s N−1 |u′|pds.

As q≥N(p-1)+pN-p, then Nq+1≤N-pp, hence u'(r0)=0 and so φ(r0)=0, where φ is defined by (2.9). But φ'(r)<0 for any r∈(0,r0), this gives φ(r0)<φ(r)<φ(0)=0, which is a contradiction.

In conclusion, u is strictly positive. The proof is complete.