Existence and nonexistence of radial positive solutions of superlinear elliptic systems

Publ. Mat. 45 (2001), 399–419 EXISTENCE AND NONEXISTENCE OF RADIAL POSITIVE SOLUTIONS OF SUPERLINEAR ELLIPTIC SYSTEMS Abdelaziz Ahammou Abstract The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic system (S+)   −∆pu=f(x, u, v)inΩ, −∆qv=g(x, u, v)inΩ, u=v=0 on∂Ω, where Ω is a ball in RNand f,gare positive continuous functions satisfying f(x, 0,0) = g(x, 0,0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use the topological degree theory combined with the blow up method of Gidas and Spruck. When Ω = RN, we give some sufficient conditions of nonexistence of radial positive solutions for Liouville systems. 1. Introduction and main results We are concerned with the existence of radial positive solutions of the problem (S+)     −∆pu=a(x)u|u|α−1+b(x)v|v|β−1in Ω, −∆qv=c(x)u|u|γ−1+d(x)v|v|δ−1in Ω, u=v=0 on∂Ω, where Ω := BRis the ball in RNcentered at zero and radius R>0. Here as usual for m>1(m=p, q), ∆mdenotes the m-Laplacian operator. 2000 Mathematics Subject Classification. 35J25, 35J60. Key words. Blow up argument, topological degree theory. 400 A. Ahammou During the last years the problem of existence for (S+) has been studied by many authors, see for example, [1], [2], [7], [8], [13], [14], [15], [17]. In particular in [16], Souto proved the existence of a positive solution taking Ω a smooth general bounded domain in RNand p= q= 2. In the case p=q, we mention the recent results of Boccardo, Fleckinger and de Th´elin [2] where the authors prove the existence of solutions of the following problem:      −∆pu=a(x)u|u|α−1+b(x)v|v|β−1+h1(x)inΩ, −∆qv=c(x)u|u|γ−1+d(x)v|v|δ−1+h2(x)inΩ, u=v=0 on∂Ω, when p−1≥α, q −1≥δ and (p−1)(q−1) >βγ with Ω a smooth general bounded domain in RN, and h1∈Lp(Ω), h2∈Lq(Ω). We remark that if h1and h2are identically zero, the solution (u, v) can be a trivial solution. Our goal is to find sufficient conditions on exponents α,β,γ, and δin order to have a radial positive solution of (S+) when Ω is a ball. At this point, we introduce two subclasses of systems (S+): 1) System (S+) is strongly coupled if i) β≥qβ +p(q−1) pγ +q(p−1)αand ii) γ≥pγ +q(p−1) qβ +p(q−1)δ. 2) System (S+) is weakly coupled if j) β<qβ +p(q−1) pγ +q(p−1)αor jj) γ<pγ +q(p−1) qβ +p(q−1)δ. Similarly, the weakly coupled definition has been introduced in De Figueiredo [6] when j) and jj) are both satisfied. It is interesting to note that our definition includes clearly this case. The existence and nonexistence for quasilinear systems have been studied by several authors by using different approaches; recent results can be seen in [4], [5], [16]. For the scalar case see [3], [9] and [10]. Now, let us make the following assumptions max(p, q)<N,(H1) a, b, c, d ∈C0([0,+∞[)(H2) Radial Positive Solutions 401 and satisfy inf r∈[0,+∞[(a(r),b(r),c(r),d(r)) >0, (p−1)(q−1) <βγ.(H3) We suppose that (S+) is a superlinear system, i.e., p−1<α and q−1<δ.(H4) The main results are the following. Theorem 1.1. We assume that the system S+is strongly coupled and that the hypotheses (H1),(H2),(H3)and (H4)hold. We suppose furthermore that max qβ+p(q−1) γβ−(p−1)(q−1) −N−p p−1;pγ+q(p−1) γβ−(p−1)(q−1) −N−q q−1≥0,(Hs) is satisfied. Then, the problem (S+)has a solution (u, v)in C1(BR)∩ C2(BR\{0}), such that u>0and v>0in BR. Theorem 1.2. We assume that the system (S+)is weakly coupled and that the hypotheses (H1),(H2),(H3), and (H4)hold. We suppose furthermore that N(p−1) N−p>α and N(q−1) N−q>δ,(Hw) is satisfied. Then the problem (S+)has a solution (u, v)in C1(BR)∩ C2(BR\{0}), such that u>0and v>0in BR. We will adapt rather classical techniques of the mapping degree: we will consider the solution operator S1associated to a problem (S+) acting in a suitable functional space. Then, we will look for solutions of the problem as fixed points of S1. As it is usual in this setting, the main difficulty will be to obtain existence of a priori bounds of positive solutions. This paper is organized as follows. Section 2 contains notations and some definitions of functional spaces and of operators Sλand Tτassociated to the problem (S+). In Section 3 we treat the nonexistence of radial positive solution for the Liouville problem (S+ ∞) associated to (S+). This is the goal of Theorem 3.1. In Section 4 we get a priori estimates for the solutions of the system in the strongly coupled case and weakly coupled case. Finally in Section 5 we apply our results to obtain the proof of Theorems 1.1 and 1.2. 402 A. Ahammou 2. Notations Besides fixing notations, in this section we recall the results that we use throughout the paper. Let Rbe a positive number; we consider the following space: χ:= {(u, v)∈C0([0,R]) ×C0([0,R]) such that u(R)=v(R)=0} endowed with the norm (u, v)=u∞+v∞, which makes it a Banach space. Let Sλand Tτ:χ→χbe the operators defined by Sλ(u, v)=(S1(u, v); S2(u, v)) and Tτ(u, v)=(T1(u, v); T2(u, v)) such that S1(u, v)(r):=λ1 p−1R rt1−Nt 0 sN−1(a(s)|u(s)|α+b(s)|v(s)|β)ds1 p−1 dt, S2(u, v)(r):=λ1 q−1R rt1−Nt 0 sN−1(c(s)|u(s)|γ+d(s)|v(s)|δ)ds1 q−1 dt and T1(u, v)(r):=R rt1−Nt 0 sN−1(a(s)|u(s)|α+b(s)|(v(s)+τ)|β)ds1 p−1 dt, T2(u, v)(r):=R rt1−Nt 0 sN−1(c(s)|u(s)|γ+d(s)|v(s)|δ)ds1 q−1 dt. It is well known that, for all λ∈[0,1] and for all τ∈[0,∞[, Sλand Tτ are completely continuous operators on χ. From the Maximum Principle this implies that Sλ(χ)⊂χand that the problem (S+) is equivalent to find some non trivial positive fixed point (u, v)∈χof the operator (S1) (by taking λ= 1) such that u(0) = v(0) = 0. The main difficulty will be to obtain suitable a priori estimates to guarantee that we are in the conditions of the fixed point theorem. To study this question, we use a Blow up argument like in Gidas-Spruck paper [9]. This technique transforms the problem (S+) into a problem (S+ ∞)     −∆pu=A∞(y)u|u|α−1+B∞(y)v|v|β−1in RN, −∆qv=C∞(y)u|u|γ−1+D∞(y)v|v|δ−1in RN, u>0 and v>0inRN, where A∞,B∞,C∞and D∞are positive continuous functions on [0,+∞[. Radial Positive Solutions 403 A Liouville type Theorem proves that (S+ ∞) has no radial positive solutions. For the case A∞=D∞=0,p=q=2,β>1 and γ>1 this kind of theorems are obtained in [12] under the hypothesis 1 β+1+1 γ+1 >N−2 2. 3. Liouville’s Theorem In this section we prove the nonexistence of radial positive solutions of the following Liouville’s system (S+ ∞): −∆pu≥a(x)u|u|α−1+b(x)v|v|β−1in RN,(3.1) −∆qv≥c(x)u|u|γ−1+d(x)v|v|δ−1in RN.(3.2) We will need the following fundamental lemmas given in [5]. Lemma 3.1. Let w∈C1([0,R]) ∩C2(]0,R]),w≥0, satisfying −d dr rN−1 dw dr (r) m−2dw dr (r)≥0on [0,R],(3.3) with N>m>1. Then, for any r∈]0,R 2[we have: w(r)≥CN,mr dw dr (r) (3.4) where CN,m =m−1 N−m(1 −2m−N m−1).(3.5) Lemma 3.2. Let be a positive function w∈C0([0,+∞[) ∩C2(]0,+∞[) satisfying −d dr rN−1 dw dr (r) m−2dw dr (r)≥0in ]0,+∞[(3.6) with N>m>1. We suppose that for some real r0≥0we have w(r0)>0. Then we obtain, w(r)>0for all r≥r0and there exists CN,m >0such that rN−m m−1w(r)≥CN,m for any r≥r0.(3.7) 404 A. Ahammou Theorem 3.1. We assume (H1),(H3),(H4)and one of the following assumptions: max qβ+p(q−1) βγ−(p−1)(q−1) −N−p p−1;pγ+q(p−1) βγ−(p−1)(q−1) −N−q q−1≥0(Hs) N(p−1) N−p>α or N(q−1) N−q>δ.(Hw) Then the only radial positive solution of (S+ ∞)is (0,0). Proof: We assume that the hypotheses (H1), (H3) and (H4) hold: 1st case: If (Hs) is satisfied, in [5] and [11] the authors prove the nonexistence of radial positive solutions of −∆pu≥b(x)v|v|β−1in RN,(3.8) −∆qv≥c(x)u|u|γ−1in RN.(3.9) Applying this result, we deduce the nonexistence of radial positive solutions of (S+ ∞). 2nd case: Let u,vbe a radial positive solution of the system (S+ ∞). We can rewrite it as −(rN−1|u(r)|p−2u(r))=rN−1a(r)|u(r)|α+b(r)|v(r)|β,(3.10) −(rN−1|v(r)|q−2v(r))=rN−1c(r)|u(r)|γ+d(r)|v(r)|δ (3.11) u(0) = v(0) = 0.(3.12) Integrating (3.10) and (3.11) on (0,r) and taking into account that u(r), v(r)<0, for r>0, we find −u(r)≥a(0) N1 p−1 r1 p−1(u(r)) α p−1 (3.13) −v(r)≥d(0) N1 q−1 r1 q−1(v(r)) δ q−1.(3.14) Thus, from Lemma 3.1 we deduce that there exists some constant C>0 depending only of (N, p, q, a, d) such that u(r)≥−CN,pru(r)≥Cr p p−1(u(r)) α p−1 (3.15) v(r)≥−CN,qrv(r)≥Cr q q−1(v(r)) δ q−1 (3.16) for all r∈]0,+∞[. Radial Positive Solutions 405 Thus, by Lemma 3.2, there exists some constant C>0 depending only of (N, p, q, a, d) and r0>0 such that for all r≥r0,wehave 1≥Cr p p−1−N−p p−1 α−p+1 p−1 (3.17) and 1≥Cr q q−1−N−q q−1 δ−q+1 q−1.(3.18) Consequently if (Hw) is satisfied, we get p p−1−N−p p−1 α−p+1 p−1>0,(3.19) or q q−1−N−q q−1 δ−q+1 q−1>0.(3.20) Hence a contradiction. 4. A priori bounds for positive solutions of (S+) In this section we will study a priori bounds for radial positive solutions of the system (S+), in the strongly coupled and weakly coupled cases. For that, we need the following lemma: Lemma 4.1. We assume that there is a sequence {(˜un,˜vn)}in (C1([0.Rn]) ∩C2(]0,R n]))2of positive solutions of the following system (˜ Sn): −d dy yN−1 d˜un dy (y) p−2d˜un dy (y)=yN−1Fn(˜un(y),˜vn(y)),(4.1) −d dy yN−1 d˜vn dy (y) q−2d˜vn dy (y)=yN−1Gn(˜un(y),˜vn(y)),(4.2) d˜un dy (0) = d˜vn dy (0) = ˜un(Rn)=˜vn(Rn)=0(4.3) where Fn(˜un(y),˜vn(y)) = An(|y|)|˜un(y)|α+Bn(|y|)|˜vn(y)+τ n|β, Gn(˜un(y),˜vn(y)) = Cn(|y|)|˜un(y)|γ+Dn(|y|)|˜vn(y)|δ, 406 A. Ahammou where {(An,B n,C n,D n)}are sequences of positive functions in (C0 loc ([0,+∞[))4which converge to (A∞,B ∞,C ∞,D ∞)in (C0 loc ([0,+∞[))4 . If, lim n→∞ τ n=0,lim n→∞ Rn=+∞,(4.4) and 0<(˜un(0),˜vn(0))≤1for all n∈N, then there exists a subsequence {(˜unk,˜vnk)}of {(˜un,˜vn)}converging in (C0 loc([0,+∞[))2and whose limit (˜u, ˜v)is a positive radially symmetric solution of the following Liouville’s system (S+ ∞)−∆pu=A∞(|y|)u|u|α−1+B∞(|y|)v|v|β−1in RN, −∆qv=C∞(|y|)u|u|γ−1+D∞(|y|)v|v|δ−1in RN. Proof: We argue as in [5]. We first note that, since (˜un(0),˜vn(0))≤1 there exists a subsequence {(˜unk(0),˜vnk(0))}which converges in R2to (˜u0,˜v0) such that (˜u0,˜v0)≤1. Next we will prove that the restriction of {(˜un,˜vn)}to [0,R ]is equicontinuous in (C0([0,R ]))2. In fact, multiplying (4.1) by d˜un dy and (4.2) by d˜vn dy we obtain the following equations: p−1 p d dy  d˜un dy (y) p+N−1 y d˜un dy (y) p (4.5) +Fn(˜un(y),˜vn(y))d˜un dy (y)=0 q−1 q d dy  d˜vn dy (y) q+N−1 y d˜vn dy (y) q (4.6) +Gn(˜un(y),˜vn(y))d˜vn dy (y)=0. From (4.5) and (4.6) it follows that p−1 p d dy  d˜un dy (y) p+an d˜un dy (y)≤0(4.7) q−1 q d dy  d˜vn dy (y) q+bn d˜vn dy (y)≤0(4.8) where an= 2 max s∈[0,R]{An;Bn}and bn= 2 max s∈[0,R]{Cn;Dn}.(4.9) Radial Positive Solutions 407 Since the sequence {(An,B n,C n,D n)}converges to (A∞,B ∞,C ∞,D ∞) in (C0([0,R ]))4, then there exist a1>0 and b1>0 such that an<a 1 and bn<b 1for all n∈N. Hence, from (4.7), (4.8) and by integrating from 0 to y∈[0,R ] we find that p−1 p d˜un dy (y) p +a1y 0 d˜un dy (s)ds ≤0(4.10) q−1 q d˜vn dy (y) q +b1y 0 d˜vn dy (s)ds ≤0(4.11) and hence that  d˜un dy (y) ≤C1 (4.12)  d˜vn dy (y) ≤C2 (4.13) uniformly in n, which imply the equicontinuity of the sequence {(˜un,˜vn)}. Thus, by the Ascoli-Arzel`a Theorem, there exists a subsequence {(˜unk,˜vnk)}, such that {(˜unk,˜vnk)}converges to (˜u, ˜v) when k→∞, in (C0([0,R ]))2. Now from (4.1) and (4.2) we have that {(˜unk,˜vnk)} satisfy ˜unk(0)−˜unk(y)= y 01 tN−1t 0 sN−1Fnk(˜unk(s),˜vnk(s))ds1 p−1 dt(4.14) ˜vnk(0)−˜vnk(y)= y 01 tN−1t 0 sN−1Gnk(˜unk(s),˜vnk(s))ds1 q−1 dt.(4.15) By the Dominated Convergence Theorem we obtain that (˜u, ˜v) satisfy ˜u0−˜u(y)=y 01 tN−1t 0 sN−1F∞(˜u(s),˜v(s))ds1 p−1 dt(4.16) ˜v0−˜v(y)=y 01 tN−1t 0 sN−1G∞(˜u(s),˜v(s))ds1 q−1 dt(4.17) where F∞(˜u(s),˜v(s)) = A∞(s)|˜u(s)|α+B∞(s)|˜v(s)|β, G∞(˜u(s),˜v(s)) = C∞(s)|˜u(s)|γ+D∞(s)|˜v(s)|δ, 414 A. Ahammou where Fn(¯un(y),¯vn(y)) = An(y)|¯un(y)|α+Bn(y) ¯vn(y)+ τn σk n β,(4.65) Gn(¯un(y),¯vn(y)) = Cn(y)|¯un(y)|γ+Dn(y)|¯vn(y)|δ (4.66) and An(y)=aytn σntp nσn−p+l(α−p+1) Bn(y)=bytn σntp nσn−p−l(p−1)+kβ, (4.67) Cn(y)=cytn σntq nσn−q+k(q−1)+lγ Dn(y)=dytn σntq nσn−q+k(δ−q+1), (4.68) Rn=σn tn R.(4.69) By choosing the positive numbers land kas in (4.37) we obtain An(y)=aytn σntp nσnlα−kβ,B n(y)=bytn σntp n,(4.70) Cn(y)=cytn σntq n,D n(y)=dytn σntq nσnkδ−lγ .(4.71) 1st case: lim n→+∞¯un(0) >0. By choosing tn=σ −lα+kβ p n, as in Lemma 4.1, we obtain that the first equation of the system (S+ n) converges to the equation −∆p¯u=a(0)¯u|¯u|β−1in RN,(4.72) where ¯u>0 is the limit of ¯un. Thus, from (Hw) and Liouville’s Theorem 3.1, the equation (4.72) has no radial positive solution. Hence the contradiction follows. Radial Positive Solutions 415 2nd case: lim n→+∞¯un(0) = 0. We recall that ¯un(0) = ˜un(0) and ¯vn(0) = ˜vn(0) where (˜un,˜vn) are the functions defined in the proof of Proposition 4.2. Then from our claim, we deduce that lim n→+∞¯vn(0) >0 and γ<pγ +q(p−1) qβ +p(q−1)δ. Consequently, in this case it suffices to choose tn=σ −kδ+lγ q n. Then, we obtain that {¯vn}converges to ¯vand ¯vsatisfies −∆q¯v=d(0)¯v|¯v|δ−1in RN. Hence, as in the 1st case, we get a contradiction. Proposition 4.3. Assume that (H1),(H2),(H3), and (H4)hold. Moreover, assume that one of the assumptions (Hs)or (Hw)holds. Then, the family of operators (Tτ)τ≥0satisfies the following properties: (P1)∃τ0>0such that, if Tτhas a fixed point (u, v)τfor some τ≥0 then τ≤τ0. (P2)∃C0>0such that, if (u, v)τis a fixed point of Tτwith τ≤τ0+1, then (u, v)τ≤C0. Proof: First, from the Maximum Principle, it follows that the problem (u, v)=Tτ((u, v))(4.73) is equivalent to find positive solutions u,vof the following system −(rN−1|u(r)|p−2u(r))=rN−1a(r)|u(r)|α+b(r)|v(r)+τ|β,(4.74) −(rN−1|v(r)|q−2v(r))=rN−1c(r)|u(r)|γ+d(r)|v(r)|δ,(4.75) u(0) = v(0) = u(R)=v(R)=0.(4.76) It follows that 0 ≤u(r), 0 ≤v(r) and by integrating on [0,r] we obtain −u(r)≥Cr 1 p−1(v(r)+τ)β p−1,(4.77) −v(r)≥Cr 1 q−1(u(r)) δ q−1.(4.78) Thus from (4.77) −u(r)≥Cr 1 p−1τβ p−1.(4.79) 416 A. Ahammou By integrating (4.79) from 0 to R, we obtain that u(0) ≥CR p p−1τβ p−1.(4.80) Now we suppose that (P1) is not true. Then there is a sequence {τn} such that lim n−→ +∞τn=+∞(4.81) and such that for each τnthere exists a solution (un,v n) of (4.74)–(4.76). Then from (4.80) and (4.81), we have lim n−→ +∞(un,v n)=+∞.(4.82) Hence, respectively if (Hs) [or (Hw)] holds then from Proposition 4.1 [Proposition 4.2] we obtain a contradiction. We argue similarly if (P2) is not true. Then for all n∈Nthere exists τn≤τ0and a solution (un,v n)of (4.74)–(4.76) such that (un,v n)>n, this implies that lim n−→ +∞(un,v n)=+∞. Hence, from Proposition 4.1, and Proposition 4.2 we deduce a contradiction and Proposition 4.3 is proved. 5. Proof of the theorems Proposition 5.1. We assume (H1),(H2)and (H3). Then there exists aρ1>0such that: ∀ρ∈[0,ρ 1[and ∀λ∈[0,1], we have that (0,0) is the only fixed point of Sλ,inB(0,ρ). Proof: Let us take λ∈[0,1] and (u, v)∈χsuch that (u, v)=Sλ(u, v)(5.1) with (u, v)=ρ>0. Notice that by the definition of Sλwe get u≤0, v≤0in[0,R]. By integrating on [0,R]wehave u(0) =λ1 p−1R 0t1−Nt 0 sN−1(a(s)|u(s)|α+b(s)|v(s)|β)ds1 p−1 dt,(5.2) v(0) =λ1 q−1R 0t1−Nt 0 sN−1(c(s)|u(s)|γ+d(s)|v(s)|δ)ds1 q−1 dt.(5.3) Radial Positive Solutions 417 Hence (u, v)=u(0) + v(0). Thus, from (H3) there exist two numbers l>0 and k>0 such that β p−1>l k>q−1 γ.(5.4) Denote σ=(u(0))1 l+(v(0)) 1 k,(5.5) then (u, v)<σ l+σkand from (5.2) and (5.3) we deduce, (u(0))1 l≤Cλ 1 l(p−1) [σlα +σkβ]1 l(p−1) ,(5.6) (v(0))1 k≤Cλ 1 k(q−1) [σlγ +σkδ]1 k(q−1) .(5.7) Summing up (5.6) and (5.7), we deduce that σsatisfies (5.8) 1 ≤Cλ 1 l(p−1) [σl(α−p+1) +σkβ−l(p−1)]1 l(p−1) +Cλ 1 k(q−1) [σlγ−k(q−1) +σk(δ−q+1)]1 k(q−1) . Hence, there exist m1>0, m2>0 and C>0 such that C<λ m1σm2≤σm2.(5.9) On the other hand, by definition of σ, there exists m3>0 such that σ≤2ρm3.(5.10) Then, from (5.9) and (5.10) it suffices to choose ρ1such that 21+ 1 m3ρ1= C1 m2m3, to complete the proof of Proposition 5.1. Proof of Theorem 1.1: From Proposition 4.3, it follows that for ρ2= C0+ 1, the equation (u, v)=Tτ((u, v)) with (u, v)∈∂B(0,ρ 2) has no solution for τ∈[0,τ 0+ 1]. Then, deg(I−Tτ,B(0,ρ 2),0) is well defined and by the property of topological degree we get that deg(I−Tτ,B(0,ρ 2),0) = constant,∀τ∈[0,τ 0+1].(5.11) Moreover, since for τ1=τ0+ 1, the operator Tτ1has no a fixed point in B(0,ρ 2), we have deg(I−Tτ1,B(0,ρ 2),0) = 0.(5.12) Then, it follows from (5.11), that deg(I−T0,B(0,ρ 2),0) = deg(I−Tτ1,B(0,ρ 2),0) = 0.(5.13) Moreover, from Proposition 5.1, for ρ1>0 sufficiently small we have deg(I−Sλ,B(0,ρ 1),0) = constant ∀λ∈[0,1].(5.14) 418 A. Ahammou Hence deg(I−S1,B(0,ρ 1),0) = deg(I−S0,B(0,ρ 1),0) = +1.(5.15) Then, since S1=T0and from (5.13) and (5.15), by the excision property we obtain deg(I−S1,B(0,ρ 2)\B(0,ρ 1),0) =0.(5.16) So, there is a fixed point (u, v)ofS1. Hence we obtain the conclusions in Theorem 1.1. Proof of Theorem 1.2: Using Proposition 4.3, the proof of Theorem 1.2 is similar as the proof of Theorem 1.1. Acknowledgement. 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D´epartement des Math´ematiques et Informatique Facult´e des Sciences Universit´e Cadi Ayyad El Jadida, BP20 Maroc E-mail address:[email protected] Primera versi´o rebuda el 28 de setembre de 2000, darrera versi´o rebuda el 23 de febrer de 2001.