Separable solutions of some quasilinear equations with source reaction
aa r X i v : . [ m a t h . A P ] A ug Separable solutions of some quasilinearequations with source reaction
Marie-Fran¸coise Bidaut-V´eron
Department of Mathematics, Universit´e Fran¸cois Rabelais, Tours, FRANCE
Mustapha Jazar ∗ Department of Mathematics, Universit´e Libanaise, Beyrouth, LIBAN
Laurent V´eron
Department of Mathematics, Universit´e Fran¸cois Rabelais, Tours, FRANCE
Abstract
We study the existence of singular solutions to the equation − div ( | Du | p − Du ) = | u | q − u under the form u ( r, θ ) = r − β ω ( θ ), r > , θ ∈ S N − . We prove the existence of an exponent q belowwhich no positive solutions can exist. If the dimension is 2 we use a dynamical system approach toconstruct solutions. . . Key words . p -Laplacian, Singularities, Phase-plane analysis, Poincar´e map, Painlev´e integral The study of isoslated singularities of solutions of quasilinear equations started with thecelebrated works of Serrin [20][21] dealing with expressions such asdiv A (( x, u, Du )) + B ( x, u, Du ) = 0 (1.1)where A and B are respectively vector valued and real valued Caratheodory functions sat-isfying the same power p -growth with p ≥
1. One of the main results of these works statedthat the type of singularities is dictated by the diffusion operator A . Later on the particularcases of superlinear semilinear elliptic equations was considered, either with an absorption − ∆ u + | u | q − u = 0 (1.2)[5], [24], or with a source reaction ∆ u + u q = 0 (1.3)[17], [10], [2], and in all cases q >
1. One of the main facts of these studies relied in theexistence of critical thresholds where the interaction of the diffusion and the reaction terms ∗ Supported by a grant from the Lebanese University − div (cid:16) | Du | p − Du (cid:17) + | u | q − u = 0 (1.4)[9], and div (cid:16) | Du | p − Du (cid:17) + u q = 0 (1.5)[22], in the range 0 < p − < q . In all these works, the radial explicit solutions, wheneverthey exist, played a key role.Similarly, the study of the boundary behaviour of solutions of quasilinear equations, hasa natural starting point in the description of their isolated singularities on the boundary.Besides the historical results of Fatou, Herglotz and Doob on the boundary trace of positive harmonic and super harmonic functions, equations of types (1.2 ), (1.3 ) and (1.4 )have alredy been considered ([4],[11],[6]). In the present article we consider equations oftype (1.5 ). The problem can be stated under the following form: Assume Ω is an opensubset of R N , a ∈ ∂ Ω and u ∈ C (Ω \ { a } ) ∩ C (Ω) is a solution of one of the above equationswhich vanishes on ∂ Ω \ { a } , what is the behaviour of u ( x ) when x → a . The simplestconfiguration corresponds to Ω = R N + , and a = 0 (or more generaly, if Ω is a cone and thesingular point a its vertex 0). For such geometry, the key-stone element for describing thebehaviour of u near 0 is played by separable solutions , whenever they exist. These solutions,which have the form u ( x ) = u ( r, σ ) = r − β ω ( σ ) r > , σ ∈ S N − , (1.6)have already proved their importance for (1.2 ), (1.3 ) and (1.4 ). It is expected that suchwill be the case for (1.5 ), even if the full theory will be much more difficult to developbecause of the absence of comparison principle and a priori estimates near x = 0. It isstraightforward that, if u is a separable solution of (1.5 ) in R N , β = pq + 1 − p := β q , (1.7)which is positive since q > p −
1. Furthermore ω is a solution of −∇ ′ . (cid:18)(cid:16) β q ω + |∇ ′ ω | (cid:17) p/ − ∇ ′ ω (cid:19) − | ω | q − ω = λ q,p (cid:16) β q ω + |∇ ′ ω | (cid:17) p/ − ω, (1.8)in S N − , where ∇ ′ is the covariant gradient on S N − , ∇ ′ . the divergence operator acting onvector fields on S N − and λ q,p = β q ( qβ q − N ) . When p = 2, β q = 2 / ( q −
1) and (1.8 ) becomes − ∆ ′ ω − | ω | q − ω = λ q, ω, (1.9)where ∆ ′ is the Laplace-Beltrami operator on S N − and λ q, = 2 q − (cid:18) qq − − N (cid:19) . S is a subdomain of S N − , equation (1.9 ), considered in S , is the Euler-Lagrange variationof the functional I ( ψ ) = Z S (cid:18) |∇ ψ | + λ q, ψ − q + 1 | ψ | q +1 (cid:19) dσ. (1.10)For any 1 < q < ( N + 1) / ( N −
3) (any q > N = 2 or 3) this functional satisfies thePalais-Smale condition. Furthermore, if λ q, < λ S, , ( λ S, is the first eigenvalue of − ∆ ′ in W , ( S )), Ambrosetti-Rabinowitz theorem [1] or Pohozaev fibration method [18], [19] applyand yield to the existence of non-trivial positive solutions to (1.9 ) in S vanishing on ∂S ;while if λ q, ≥ λ S, no such solution exists.When p = 2, equation (1.8 ) cannot be associated to any functional defined on S N − ,except if q = q c = ( N ( p −
1) + p ) / ( N − p ) (the critical Sobolev exponent for W ,p , when N > p ); therefore, finding functions satisfying it is not straightforward. Besides the constantsolutions which exist as soon as qβ q < N , it is not easy to prove the existence of non-constantsolutions. As in the case p = 2, it is remarkable to see that existence, or non existence, ofsolutions of (1.8 ) is associated to some spectral problem, although this problem is notstandard at all: if one looks for the existence of a positive p-harmonic function v in the cone C S = { ( r, σ ) : r > , σ ∈ S } vanishing on ∂S , under the form v ( r, σ ) = r − β φ ( σ ), one findsthat φ is a positive solution of the so-called spherical p -harmonic spectral equation on S ,namely −∇ ′ . (cid:18)(cid:16) β φ + |∇ ′ φ | (cid:17) p/ − ∇ ′ φ (cid:19) = λ (cid:16) β φ + |∇ ′ φ | (cid:17) p/ − φ in S φ = 0 in ∂ S , (1.11)and λ = β ( β ( p −
1) + p − N ). The difficulty of this problem is two-fold since β is unknownand (1.11 ) is not the Euler-Lagrange equation of any functional. However, given a smoothsubdomain S ⊂ S N − , it is proved in [25], following a shooting method due to Tolksdorff[23], that there exists a couple ( β, φ ) = ( β S , φ S ), where β S > φ S is definedup to an homothethy, such that (1.11 ) holds. Denoting λ S = β S ( β S ( p −
1) + p − N ) , the couple ( φ S , λ S ) is the natural generalization of the first eigenfunction and eigenvalue ofthe Laplace-Beltrami operator in W , ( S ) since λ S = λ S, when p = 2. Our first theorem isa non-existence which extends the one already mentioned in the case p = 2. Theorem 1.
Let S ⊂ S N − be a smooth subdomain. If β q ≥ β S there exists no positivesolution of (1.8 ) in S which vanishes on ∂S . Apart the case p = 2, the existence counterpart of this theorem is not known in arbitrarydimension, except if q = q c in which case (1.5 ) is the Euler-Lagrange equation of thefunctional J ( ψ ) = Z S (cid:18) p (cid:16) β q c ψ + |∇ ′ ψ | (cid:17) p/ − q c + 1 | ψ | q c +1 (cid:19) dσ, (1.12)3nd applications of the already mentioned variational methods lead to an existence result.However, when N = 2 the problem of finding solutions of (1.5 ) under the form (1.6 )can be completely solved using dynamical systems methods. In order to point out a richerclass of phenomena, we shall imbed this problem into a more general class of quasilinearequations with a potential, authorizing even the value p = 1. This equation is the following,div (cid:16) | Du | p − Du (cid:17) + | u | q − u − c | x | p | u | p − u = 0 (1.13)in R \ { } , with q > p − ≥ c ∈ R . If u is a solution under the form (1.6 ), β is beequal to β q , while ω is any 2 π -periodic solution of ddσ β q ω + (cid:18) dωdσ (cid:19) ! p − / dωdσ + λ q " β q ω + (cid:18) dωdσ (cid:19) ( p − / ω + | ω | q − ω − c | ω | p − ω = 0 , (1.14)where λ q = β q ( qβ q −
2) = β q ( p − p − β q ) . (1.15)If we set c q = β p − q λ q = p p − ( p − q + 2( p − q + 1 − p ) p , (1.16)then, if c ≤ c q , the only constant solution is the zero function, while if c > c q , there existtwo other constant solutions ± ( c − c q ) / ( q +1 − p ) . Let us denote by E + the set of positivesolutions of (1.14 ) on S , E the set of sign changing solutions and F = ±E + ∪ E the set ofall nonzero solutions. Our main result which gives the struture of the sets E and E + is thefollowing: Theorem 2.
Assume p > , q > p − . Then(i) E = ∞ [ k ∈ N ,k = k q (cid:8) ω k ( . + ψ ) : ψ ∈ S (cid:9) , (1.17) in which expression ω k is a function with least period π/k , and k q = 1 if c ≥ c q , or k q isthe smallest positive integer such that k q > M q , where M q = πβ − pq Z π/ p −
1) tan θβ pq ( p −
1) tan θ + c q − c cos p − θ dθ , (1.18) if c < c q .(ii) If c ≤ c q , E + is empty. If < c − c q ≤ β p − q /p, E + is reduced to the constant function ( c − c q ) / ( q +1 − p ) . If c − c q > β p − q /p, E + contains the constant function ( c − c q ) / ( q +1 − p ) nd the set E + ∗ = k + q [ k ∈ N ,k =1 (cid:8) ω + k ( . + ψ ) : ψ ∈ S (cid:9) , (1.19) where ω + k is a non-constant positive function with least period π/k, and k + q is the largestinteger smaller than ( pβ − pq ( c − c q )) / . Since separable solutions of (1.5 ) defined in a cone C S and vanishing on ∂C S are asso-ciated to elements of E , we can prove the existence counterpart of Theorem 1 in dimension 2. Corollary 1.
Let N = 2 and S be any angular sector of S . Then there exists a positivesolution of (1.8 ) vanishing at the two end points of S if and only if β q < β S . Further-more this solution is unique. In particular, existence holds for any sector if p < and q ≥ p − / (2 − p ) . The case p = 1 appears as a limiting case of the preceding one. In that case we observethat u is a positive solution of (1.13 ) if and only if v = u q is a solution of the same equationrelative to q = 1, div (cid:16) | Dv | − Dv (cid:17) + v − c | x | = 0 . (1.20)The initial case c = 0 is easily treated, but the case c = 0, that we shall analyse in fullgenerality, is much richer and delicate and shows a large variety of solutions depending onvarious parameters. Theorem 3.
Assume p = 1 and q > . Then(i) If c = 0 , or c = 0 and q > , E is empty. If c = 0 and q ≤ , E = (cid:8) ω ( . + ψ ) : ψ ∈ S (cid:9) , where σ ω ( σ ) := 2 /q | sin σ | (1 − q ) /q sin σ is a C solution of (1.14 ).(ii) If c ≤ − , E + is empty. If − < c < , E + is reduced to the constant function ( c + 1) /q .If c > , E + = n ( c + 1) /q o ∪ k [ k ∈ N ,k = k (cid:8) ω + k ( . + ψ ) : ψ ∈ S (cid:9) , in which expression ω + k is a positive function with least period π/k , k is the largest integerstrictly smaller than ( c +1) / and k is the smallest integer greater than π Z π/ q cos θ cos θ +2 c dθ .Finally, if c = 0 , E + = { } ∪ [ K ∈ (0 , (cid:8) ω + K ( . + ψ ) : ψ ∈ S (cid:9) ∪ ( ∅ if q ≥ (cid:8) ω +0 ( . + ψ ) : ψ ∈ S (cid:9) if q < where the functions ω + K and ω +0 are explicitely given by ω + K = (cid:16)p − K sin σ − K cos σ (cid:17) /q , and ω +0 = (2 | sin σ | ) /q ∀ σ ∈ S . c = 0.Our paper is organized as follows: 1- Introduction. 2- The N-dimensional case. 3- The 2-dimdynamical system. 4- The case p >
1. 5- The case p = 1. If p ≥ β > λ ∈ R we denote by T β,λ the operator defined on C ( S N − ) by ϕ T β,λ [ ϕ ] = −∇ ′ . (cid:16) ( β ϕ + |∇ ′ ϕ | ) ( p − / ∇ ′ ϕ (cid:17) − λ ( β ϕ + |∇ ′ ϕ | ) ( p − / ϕ. (2.1)Let q > p − > S be a smooth connected domain on S N − and C S the cone with vertex0 generated by S . If u is a positive solutions of − div( | Du | p − Du ) = u q , (2.2)in C S \ { (0) } vanishing on ∂C S \ { (0) } , under the form u ( r, σ ) = r − β ω ( σ ) , (2.3)then β = p/ ( q + 1 − p ) := β q and ω solves ( T β q ,λ q,p [ ω ] − ω q = 0 in Sω = 0 on ∂S, (2.4)where λ q,p = β q ( qβ q − N ) . We denote by β S the exponent corresponding to the first spherical singular p -harmonicfunction and by φ S the corresponding function. Thus β S > u ( r, σ ) = r − β S φ S ( σ ) is p -harmonic in C S \ { (0) } and vanishes on ∂C S \ { (0) } . Furthermore φ = φ S > ( T β S ,λ S [ φ ] = 0 in Sφ = 0 on ∂S, (2.5)where λ S = β S ( β S ( p −
1) + p − N ) . We recall that ( β S , φ S ) is unique up to an homothety upon φ . Furthermore φ S is positivein S , ∂φ S /∂ν < ∂S and S ′ ⊂ S, S ′ = S = ⇒ β S ′ > β S . .2 Non-existence Proof of Theorem 1.
We put θ = β q β S and η = φ θS . Then θ ≥ ∇ ′ η = θφ θ − S ∇ ′ φ S ,β q η + |∇ ′ η | = θ φ θ − S ( β S φ S + |∇ ′ φ S | ) , ( β q η + |∇ ′ η | ) ( p − / = θ p − φ ( p − θ − S ( β S φ S + |∇ ′ φ S | ) ( p − / , ∇ ′ . ( β q η + |∇ ′ η | ) ( p − / ∇ ′ η = θ p − φ ( p − θ − S ∇ ′ . ( β S φ S + |∇ ′ φ S | ) ( p − / ∇ ′ φ S + ( p − θ − θ p − φ ( p − θ − − S ( β S φ S + |∇ ′ φ S | ) ( p − / |∇ ′ φ S | Using (2.5 ) with φ = φ S , we derive T β q ,λ q,p [ η ] = − ( p − θ p − ( θ − φ ( p − θ − − S ( β S φ S + |∇ ′ φ S | ) p/ in S. (2.6)Because ω is a nonnegative nontrivial solution of (2.4 ), it is nonpositive in S . Furthermore ∂ω/∂ν < ∂S . Therefore we can choose φ S as the maximal positive solution of (2.5 )such that η ≤ ω . If θ > σ ∗ ∈ S such that ω ( σ ∗ ) = η ( σ ∗ ) > ω ( σ ) ≥ η ( σ ) ∀ σ ∈ ¯ S. (2.7)If θ = 1, the graphs of ω and η could be tangent only on ∂S . This means that either (2.7 )holds, or there exists ¯ σ ∈ ∂S such that ∂ω (¯ σ ) /∂ν = ∂η (¯ σ ) /∂ν < ω ( σ ) < η ( σ ) ∀ σ ∈ S. (2.8)Let ψ = ω − η and we first consider the case where (2.7 ) holds. Let g = ( g ij ) be the metrictensor on S N − . We recall the following expressions in local coordinates σ j around σ ∗ , |∇ ′ ϕ | = X j,k g jk ∂ϕ∂σ j ∂ϕ∂σ k , for any ϕ ∈ C ( S ), and ∇ ′ .X = 1 p | g | X ℓ ∂∂σ ℓ (cid:16)p | g | X ℓ (cid:17) = 1 p | g | X ℓ,i ∂∂σ ℓ (cid:16)p | g | g ℓi X i (cid:17) , for any vector field X ∈ C ( T S N − ), if we lower the indices by setting X ℓ = X i g ℓi X i . Wederive from the mean value theorem( β q ω + |∇ ′ ω | ) ( p − / ∂ω∂σ i − ( β q η + |∇ ′ η | ) ( p − / ∂η∂σ i = X j α ij ∂ ( ω − η ) ∂σ j + b i ( ω − η ) , b i = ( p − (cid:16) β q ( η + t ( ω − η )) + |∇ ′ ( η + t ( ω − η )) | (cid:17) ( p − / × ( η + t ( ω − η )) ∂ ( η + t ( ω − η )) ∂σ i , and α ij = ( p − (cid:16) β q ( η + t ( ω − η )) + |∇ ′ ( η + t ( ω − η )) | (cid:17) ( p − / × ∂ ( η + t ( ω − η )) ∂σ i X k g jk ∂ ( η + t ( ω − η )) ∂σ k + δ ji (cid:16) β q ( η + t ( ω − η )) + |∇ ′ ( η + t ( ω − η )) | (cid:17) ( p − / . Since the graph of η and ω are tangent at σ ∗ , η ( σ ∗ ) = ω ( σ ∗ ) = P > ∇ ′ η ( σ ∗ ) = ∇ ′ ω ( σ ∗ ) = Q. Thus b i ( σ ∗ ) = ( p − (cid:16) β q P + | Q | (cid:17) ( p − / P Q i , and α ij ( σ ∗ ) = (cid:16) β q P + | Q | (cid:17) ( p − / δ ji ( β q P + | Q | ) + ( p − Q i X k g jk Q k ! . Now T β q ,λ q,p [ ω ] − T β q ,λ q,p [ η ] = ω q + ( p − θ p − ( θ − φ ( p − θ − − S ( β S φ S + |∇ ′ φ S | ) p/ = − p | g | X ℓ,i ∂∂σ ℓ (cid:20)p | g | g ℓi (cid:18) ( β q ω + |∇ ′ ω | ) p − ∂ω∂σ i − ( β q η + |∇ ′ η | ) p − ∂η∂σ i (cid:19)(cid:21) − λ q,p (cid:16) ( β q ω + |∇ ′ ω | ) p − ω − ( β q η + |∇ ′ η | ) p − η (cid:17) , = − p | g | X ℓ,i ∂∂σ ℓ p | g | g ℓi X j α ij ∂ ( ω − η ) ∂σ j + b i ( ω − η ) + X i C i ∂ ( ω − η ) ∂σ i = − p | g | X ℓ,j ∂∂σ ℓ (cid:20) a ℓj ∂ ( ω − η ) ∂σ j (cid:21) + X i C i ∂ ( ω − η ) ∂σ i , where the C i are continuous functions and a ℓj = p | g | X i g ℓi α ij . (cid:0) α ij ( σ ) (cid:1) is symmetric, definite and positive since it is the Hessian of the strictlyconvex function X = ( X , . . . , X n − ) p (cid:16) P + | X | (cid:17) p/ = 1 p P + X j,k g jk X j X k p/ . Therefore (cid:0) α ij (cid:1) has the same property in some neighborhood of σ ∗ , and the same holds truewith (cid:0) a ℓj (cid:1) . Finally the function ψ = ω − η is nonnegative, vanishes at σ ∗ and satifies − p | g | X ℓ,j ∂∂σ ℓ (cid:20) a ℓj ∂ψ∂σ j (cid:21) + X i C i ∂ψ∂σ i ≥ . (2.9)Then ψ = 0 in a neighborhood of S . Since S is connected, ψ is identically 0 which acontradiction.If (2.8 ) holds, then θ = 1 and the graphs of η and ω are tangent at ¯ σ . Proceedingas above and using the fact that ∂η/∂ν exists and never vanishes on the boundary, we seethat ψ = η − ω satisfies (2.9 ) with a strongly elliptic operator in a neighborhood N of ¯ σ .Moreover ψ > N , ψ (¯ σ ) = 0 and ∂ψ/∂ν (¯ σ ) = 0. This is a contradiction, which ends theproof. (cid:3) Remark. If p = 2, the proof of non-existence is straightforward by multiplying the equationin ω by the first eigenfunction φ S and get Z S (( λ S − λ q, ) ω − ω q ) φ S dσ = 0 , a contradiction since λ S ≤ λ q, . Let us consider the case q = q c = ( N ( p −
1) + p ) / ( N − p ) ( N > p > S be anysmooth subdomain of S N − . Since in that case λ q,p = − β q c , the research of solutions of(1.5 ) under the form (1.6 ) vanishing on ∂C S leads to ( T β qc , − β qc [ ω ] − | ω | q c − ω = 0 in Sω = 0 in ∂S, (2.10)where β q c = N/p −
1. This equation is the Euler-Lagrange variation of the functional J defined on W ,p ( S ) by J ( ψ ) = Z S (cid:18) p (cid:16) β q c ψ + |∇ ′ ψ | (cid:17) p/ − q c + 1 | ψ | q c +1 (cid:19) dσ. (2.11) Theorem 2.1
Problem (2.10 ) admits a positive solution. roof. Clearly the functional is well defined on W ,p ( S ) since q c is smaller than the Sobolevexponent p ∗ N − for W ,p in dimension N-1. For any ψ ∈ W ,p ( S ), lim t →∞ J ( tψ ) = −∞ .Furthermore there exist δ > ǫ > J ( ψ ) ≥ ǫ for any ψ ∈ W ,p ( S ) such that k ψ k W ,p = δ . Assume now that { ψ n } is a sequence of W ,p ( S ) such that J ( ψ n ) → α and k DJ ( ψ n ) k W − ,p ′ → n → ∞ . Then T β qc , − β qc [ ψ n ] − | ψ n | q c − ψ n = ǫ n → . Then Z S (cid:18)(cid:16) β q c ψ n + |∇ ′ ψ n | (cid:17) p/ − | ψ n | q c +1 (cid:19) dσ = h ǫ n , ψ n i . Since J ( ψ n ) → α it follows Z S (cid:16) β q c ψ n + |∇ ′ ψ n | (cid:17) p/ dσ → p ( q c + 1) α/ ( q c + 1 − p ) . Therefore { ψ n } remains bounded in L q c +1 ( S ), and relatively compact in L r ( S ), for any 1
0. In fact we can easily reduce the problem to a simpler form, andparticularly in the case p = 1 , where the equation has a remarkable homogeneity property.The next statement is a straightforward computation which transforms the equation satisfiedby ω into two more canonic forms. 10 emma 3.1 Let ω be a solution of (3.1 ).(i) Assume p > . If we set τ = βσ , ω ( σ ) = β p/ ( q +1 − p ) w ( τ ) and w ′ = dwdτ , (3.3) then w satisfies ddτ (cid:16)(cid:0) w + w ′ (cid:1) p/ − w ′ (cid:17) − b (cid:0) w + w ′ (cid:1) p/ − w + f ( w ) − d | w | p − w = 0 , (3.4) where b = − λβ , d = cβ p , f ( s ) = β − pq/ ( q +1 − p ) g ( β p/ ( q +1 − p ) s ) . (3.5) In particular f satisfies the same assumptions (3.2 ) as g. (ii) Assume p > . If on any open interval I ⊂ (0 , π ) where ω ( σ ) = 0 , we set τ = βq σ, and ω ( σ ) = ( βq ) /q | w ( τ ) | /q − w ( τ ) , (3.6) then w satisfies (3.4 ) on I , with b = − λ/β q, d = c/βq, f ( s ) = β − q g (( βqs ) /q ) . (3.7) Furthermore f satisfies the assumptions (3.2 ) with q = 1 , i.e. lim s → f ( s ) /s = 1 , lim s →∞ f ( s ) = ∞ , f ′ ( s ) > on (0 , ∞ ) . (3.8)Due to this result, the changes of variables (3.3 ) and (3.6 ) reduce the problem to thestudy both of existence of periodic solutions of equation (3.4 ), and to characterizing theperiod function of these solutions, in the range q > p − p > , and q > p = 1 . We re-write (3.4 ) as the system, w ′ = F ( w, y ) = yy ′ = G ( w, y ) = bw + ( b + 2 − p ) w y − ( f ( w ) − d | w | p − w )( w + y ) − p/ w + ( p − y , (3.9)and we denote by h the odd function defined on R by h ( s ) = ( f ( s ) / | s | p − s if s = 00 if s = 0 . (3.10)11f b + d ≤ , (3.9 ) has no non-trivial stationary point, while if b + d > , it admits the twostationary points ± P , with P = ( a,
0) and a = h − ( b + d ) . Furthermore P is a center sincethe linearized system at P is given by the matrix (cid:18) − ah ′ ( a ) 0 (cid:19) . System (3.9 ) is clearly singular at (0 , . Furthermore it could singular be along the line w = 0 if p = 1, if q <
1, and if p < d = 0. Actually, for p > , σ ) with σ = 0. This can be checked as follows: consider the Cauchy problem (cid:26) w ′′ = G ( w, w ′ ) , t ∈ ( − δ, δ ) w (0) = 0 , w ′ (0) = σ, (3.11)and let w be any local solution; since near (0 , σ ) , G is continuous with respect to w and C with respect to y, w is C ; because σ = 0, t can be expressed locally in terms of w . Defining w ′ ( t ) = p ( w ) , then p is C near 0 , p (0) = 1 and satisfies dpdw = G ( w, p ) p , with J ( w, p ) = G ( w, p ) /p . Clearly is C with respect to p and continuous with respect to w, thus one gets local uniqueness of p. and then the local uniqueness of problem w ′ ( t ) = p ( w ( t )) ,w (0) = 1, since p is of class C . The phase plane of the system (3.9 ) is equivariant under symmetries with respect to thetwo axes of coordinates, because F is even with respect to w and odd with respect to y, and G is odd with respect to w and even with respect to y. Thus from now we can restrict thestudy to the first quadrant
Q\ { (0 , } , where Q = (0 , ∞ ) × (0 , ∞ ) , where, in particular, w ≥ . Due to the symmetries, in the case p > , , any trajectory whichmeets the two axes in finite times τ, τ + T is a closed orbit of period 4 T. Remark.
It is useful to introduce the slope ξ = w ′ /w, (or a function of the slope) as a newvariable. This was first used for p > ddτ (cid:16)(cid:0) w + w ′ (cid:1) p/ − w ′ (cid:17) − b (cid:0) w + w ′ (cid:1) p/ − w = 0 . In that case the function ξ satisfies ddτ (cid:16)(cid:0) ξ (cid:1) p/ − ξ (cid:17) = − (( p − ξ − b ) (cid:0) ξ (cid:1) p/ − , for w >
0, and this equation is completely integrable in terms of u = (cid:0) ξ (cid:1) p/ − ξ.
12y using polar coordinates in Q ( w, y ) = ( ρ cos θ, ρ sin θ ) , ρ > , θ ∈ (0 , π/ , we transform (3.9 ) into θ ′ = b − ( p −
1) tan θ + ( d − h ( ρ cos θ )) cos p − θ p −
1) tan θρ ′ = ρ (1 + θ ′ ) tan θ. (3.12)Equivalently, if we introduce the slope ξ = tan θ ∈ (0 , ∞ ), and set u = φ ( ξ ) = cos − p θ sin θ, φ ( ξ ) = (1 + ξ ) ( p − / ξ, (3.13)then φ ′ ( ξ ) = (1 + ξ ) ( p − / (1 + ( p − ξ ); thus φ is strictly increasing: from (0 , ∞ ) into(0 , ∞ ) when p > , and from (0 , ∞ ) into (0 ,
1) when p = 1 . Defining ϕ = φ − , and E ( ξ ) = (cid:0) ( p − ξ − b (cid:1) (1 + ξ ) p/ − , (3.14)we obtain ( w ′ = wϕ ( u ) ,u ′ = − E ( ϕ ( u )) − h ( w ) + d. (3.15)This system is still singular on the line w = 0 if h C ([0 , ∞ )) near 0 . In the sequel we setΨ( u ) = u Z ϕ ( s ) ds. (3.16)Noticing that E ′ ( ξ ) = (cid:0) p ( p − ξ + 2( p − − ( p − b (cid:1) (1 + ξ ) ( p − / ξ, (3.17)we derive that E is increasing on (0 , ∞ ) when ( p − b ≤ p − p − b > p − E is decreasing on (0 , η ) and then increasing, where η is defined by p ( p − η = ( p − b − p − , (3.18)and min E = E ( η ) = − p − (cid:18) ( p − b + p − p ( p − (cid:19) p/ . (3.19)In the case of initial problem (1.14 ), E is increasing. Remark. If p >
1, system (3.9 ) is singular at (0 , . If we replace the assumption lim s → f ( s ) /s q =1 , by the stronger one lim s → f ′ ( s ) /s q − = q, (3.20)13e can transform system (3.15 ) in (0 , ∞ ) × R in a system of the same type, but withoutsingularity: this is obtained by performing the substitution v = w q +1 − p . Then (cid:26) v ′ = ( q + 1 − p ) vϕ ( u ) u ′ = − E ( ϕ ( u )) − ˜ h ( v ) + d, (3.21)where v ˜ h ( v ) = h ( v / ( q +1 − p ) ∈ C ([0 , . In particular, if f ( w ) = | w | q − w , we find (cid:26) v ′ = ( q + 1 − p ) vϕ ( u ) u ′ = − E ( ϕ ( u )) − v + d. (3.22) Remark.
In the case f ( w ) = | w | q − w, we can differentiate the equation relative to u ′ andobtain that u satisfies the following equation u ′′ = B ( ϕ ( u )) u ′ + ( q + 1 − p )( E ( ϕ ( u )) − d ) ϕ ( u ) , (3.23)where E is given above, and B ( ξ ) = ( p − b + q − p −
1) + ( q + 1 − p )( p − ξ p − ξ ξ. (3.24)Notice that equation (3.23 ) has no singularity for p > . p > A natural question is to see if equation (3.4 ) admits a variational structure. When p = 2,it is the case, for any b and d . Since (3.4 ) takes the form w ′′ − ( b + d ) w + f ( w ) = 0 , it is the Euler equation of the functional H ( w, w ′ ) = w ′ b + d ) w − F ( w )where F ( w ) = Z w f ( s ) ds. Thus the function w ′ = ( b + d ) w − F ( w ) is constant along thetrajectory. When p = 2 , p >
1, we find that a first integral exists only in the case b = 1 . Insuch a case (3.4 ) is the Euler equation of the functional H ( w, w ′ ) = (cid:0) w + w ′ (cid:1) p/ p + d | w | p p − F ( w ) . Therefore, the associated Painlev´e integral P ( w, w ′ ) = 1 p (cid:0) w + w ′ (cid:1) p/ − (cid:0) ( p − w ′ − w (cid:1) − d | w | p p + F ( w ) (4.1)14s constant along the trajectories. Using the function E introduced at (3.14 ), then (4.1 ) isequivalent to E (cid:18) w ′ w (cid:19) = E ( ϕ ( u )) = d − p K + F ( w ) w p (4.2)for w >
0. Hence E is increasing on (0 , ∞ ) from − b = − ∞ .In the general case, we cannot use a first integral for studying the periodicity propertiesof the solutions, while it was the main tool in [3] for p = 2. This is the reason for whichwe are lead to use phase plane techniques. Notice that, for the initial problem (1.14 ), thevalue b = 1 corresponds to the case p < q = (3 p − / (2 − p )) . In this section we describe in full details the trajectories of system (3.9 ) in the phase plane( w, y ). Notice that the system can be singular on the axis w = 0 . Proposition 4.1
Assume p > . Then all the orbits of system (3.9 ) are bounded. Anytrajectory T [ P ] issued from a point P in Q is(i) either a closed orbit surrounding (0 , ,(ii) or, if b + d > , a closed orbit surrounding P but not (0 , ,(iii) or an homoclinic orbit defined on R , starting from (0 , with initial slope lim t →−∞ w ′ ( t ) w ( t ) = m where m is defined E ( m ) = d , and ending at (0 , with lim t →∞ w ′ ( t ) w ( t ) = − m. Proof.
We recall that E and u are defined by (3.13 ) and (3.14 ), by using polar coordinates( ρ, θ ) in the ( w, y )-plane.First look at the vector field on the boundary of Q . At any point (0 , σ ) with σ > , it isgiven by ( σ, , thus it is transverse and inward. At any point ( ¯ w,
0) with ¯ w > , it is givenby (0 , ¯ w ( b + d − h ( ¯ w )) . Thus it is transverse and outward whenever b + d ≤ b + d > w > a, and inward whenever b + d > w < a. Consider any solution ( w, y ) of the system, such that P = ( w (0) , y (0)) ∈ Q , and let( τ , τ ) be its maximal interval existence in Q . At any point τ where u ′ ( τ ) = 0 and u ( τ ) > u ′′ ( τ ) = − h ′ ( w ) wϕ ( u ) < τ exists, it is unique, and it isa maximum for u .Since w ′ = y > w has the limits ℓ ∈ (0 , ∞ ] as τ ↑ τ and ℓ ∈ [0 , ∞ ) as τ ↓ τ .Therefore u is strictly monotonous near τ , and τ thus it has limits u , u ∈ [0 , ∞ ] , in otherwords θ has limits θ , θ ∈ [0 , π/ u is increasing, one has E ( ϕ ( u )) ≤ d, thus u is bounded and, consequently, u is finite. If ℓ = ∞ , then θ ′ ( τ ) → −∞ , as τ ↑ τ ;by (3.13 ), ρ is decreasing, thus it is bounded, which is contradictory; thus ℓ is finite. If15 > ℓ , ℓ ϕ ( u )) is stationary, which is impossible. Thus u is decreasing to 0 , andthe trajectory converges to ( ℓ , . If b + d > ℓ = a , u ′ tends to 0 from (3.15 ), and u ′′ = − ( E ◦ ϕ ) ′ ( u ) u ′ − h ′ ( w ) wϕ ( u ) = − h ′ ( a ) aϕ ( u )(1 + o (1));therefore u ′′ < τ , which is impossible. Finally, either b + d ≤ , or b + d > w > a, and τ is finite, the trajectory leaves Q transversally at τ . (ii) Next let us go backward in time. • Suppose u = 0. Clearly the trajectory converges to ( ℓ , b + d > ℓ ≤ a, thus ℓ < a as above. The trajectory enters Q transversally at τ , and from thesymmetries it is a closed orbit surrounding only the stationnary point P . • Next, suppose u = ∞ . It means that θ tends to π/ . Then from (3.12 ), θ ′ tends to1 , thus τ is finite, π/ − θ = ( τ − τ )(1 + o (1)) , tan θ = ( τ − τ ) − (1 + o (1)) , and ( p − τ − τ ) − ρ ′ ρ = ( b + 1 + ( d − h ( ρ cos θ )) cos p − θ )(1 + o (1) . If p ≥ , then ρ ′ /ρ = O (( τ − τ )); if p < ρ ′ /ρ = O (( τ − τ ) p − ) . In any case, ln ρ caseis integrable, thus ρ has a finite limit ¯ y > . Then the trajectory enters Q transversally at τ and from the symmeries it is a closed orbit surrounding (0 , . From the considerationsin § y > Q the slope w ′ /w = ξ = ϕ ( u ) is decreasing from ∞ to 0; indeed it decreases near τ and τ and can onlyhave a maximal point. • At end, suppose 0 < u < ∞ . If ℓ > , then ( ℓ , ℓ ϕ ( u )) is stationary, which isimpossible. Thus ( y, w ) converges to (0 , . And w ′ /w tends to ϕ ( u ) , thus τ = −∞ . And u ′ converges to d − E ( ϕ ( u )) , thus tan θ = ϕ ( u ) has a limit m ≥ E ( m ) = d. From the symmetries the trajectory is homoclinic and the solution w is defined on R . (cid:3) The next theorem studies the precise behaviour of solutions according to the sign of b + d. Theorem 4.2
Assume p > and consider system (3.9 ) in the ( w, y ) -plane.(i) Assume b + d > . Then there exists a unique homoclinic trajectory H starting from (0 , in Q with initial slope m d = E − ( d ) ( m = p b/ ( p − if d = 0) , ending at (0 , withthe slope − m d , and surrounding P . Up to the stationary points, the other orbits are closed,and either they surround only one of the points P or − P , in the domain delimitated by H , corresponding to solutions w of constant sign, or they are exterior to ±H and surround (0 , and ± P , corresponding to sign changing solutions w .(ii) Assume b + d ≤ . Then • if ( p − b ≤ p − , or [( p − b > p − and d < E ( η )] , there is no homoclinictrajectory. • if [( p − b > p − and E ( η ) < d ≤ − b ] ; then denoting by m ,d < m ,d the twopositive roots of equation E ( m ) = d , there exist infinitely many homoclinic trajectories H starting from (0 , in Q with the initial slope m ,d and ending at (0 , with the final slope − m ,d , and a unique homoclinic trajectory H starting from (0 , in Q with initial slope m ,d and ending at (0 , with final slope − m ,d . roof. (i) Case b + d > . Then the equation E ( m ) = d has a unique positive solution m = E − ( d ); and w ′ /w tends to m ; thus the trajectory starts from (0 ,
0) with a slope m. Then for any P ∈ Q , the trajectory T [ P ] passing through P meets the axis y = 0 after P atsome point ( µ,
0) with µ > a.
Denote U = (cid:8) P ∈ Q : T [ P ] ∩ { (0 , σ ) : σ > } 6 = ∅ (cid:9) , V = (cid:8) P ∈ Q : T [ P ] ∩ { ( µ,
0) : 0 < µ < a } 6 = ∅ (cid:9) . (4.3)Then either P ∈ U and the trajectory is a closed orbit surrounding (0 ,
0) and ± P , and in Q . Or P ∈ V and the trajectory is a closed orbit surrounding only P . Or T [ P ] is an homoclinicorbit H starting from (0,0) with the slope m , where m is the unique solution of equation E ( m ) = d (such that m > η if E is not monotone, see (3.17 )). Next U and V are open,since the vector field is transverse on the axes, thus U ∪ V 6 = Q . This shows the existence ofsuch an orbit H . (ii) Case b + d ≤ . • Either b + d < E is increasing, or E has a minimum at η and d < E ( η ). Insuch a case equation E ( m ) = d has no solution, and there is no homoclinic orbit . Or E isincreasing and b + d = 0; then E ( ϕ ( u )) > − b = d, thus u ′ < , thus u cannot tend to 0, andthe same conclusion holds. • E has a minimum at η and E ( η ) < d ≤ − b . In that case the equation E ( m ) = d hastwo roots m , m such that 0 ≤ m < η < m ≤ m b , where m ( b ) is defined by E ( m b ) = − b .Any trajectory T [ P ] such that P ∈ U (see (4.3 ) for the definition) satisfies u ′ < , itmeans h ( w ) > d − E ( ϕ ( u )) and the range of u is (0 , ∞ ) , therefore there exists τ such that ϕ ( u )( τ ) = η, hence h ( w ( τ )) > d − E ( η ) and y ( τ ) = ηw ( τ ) . Next consider any trajectory T [ ˜ P ] starting from ˜ P = ( ˜ w, η ˜ w ) such that h ( ˜ w ) ≤ d − E ( η ). It cannot be a trajectory of thepreceding type , thus ( y, w ) → (0 ,
0) as τ → τ , and θ tends to θ , with tan θ = m or m ;moreover u ′ (0) ≥
0, and u ′ < τ , thus there exists a unique τ ≥ u ′ ( τ ) = 0;then u ′ > τ , τ ), therefore tan θ < η, and finally tan θ = m . Consequently there existinfinitely many such trajectories H , with initial slope m . Next fix one trajectory T [ ˜ P ] suchthat h ( ˜ w ) ≤ d − E ( η ). Let R be the subdomain of Q delimitated by T [ ˜ P ] and T [(0 , and V = (cid:8) P ∈ R : T [ P ] ∩ { ( w, ηw ) : 0 < w < ˜ w } 6 = ∅ (cid:9) . The set V is open because the intersection with the line y = ηw for w < ˜ w is transversesince at the intersection point, h ( ˜ w ) < d − E ( η ) , thus u ′ > , and y/w = ϕ ( u ) = η, and y ′ y = ϕ ( u ) + ϕ ′ ( u ) ϕ ( u ) u ′ > η = w ′ w . Then (
U ∩ R ) ∪ V 6 = R . Then there exists at least a trajectory H , ∗ starting from (0 ,
0) withinitial slope m . (iii) Uniqueness of H and H . Let m = m or m ,d . Suppose that system (3.9 ) hastwo solutions ( w , y ) , ( w , y ) defined near −∞ , such that w i > w i ( τ ) tends to 0 and y i ( τ ) /w i ( τ ) tend to m as τ ↓ −∞ . Then the system (3.15 ) has two local solutions ( w , u ) , w , u ) such that ϕ ( u i ) tends to m at −∞ . Then w ′ i > u i asa function of w i . Then at the same point w,w d (Ψ( u i ) dw = wϕ ( u i ) du i dw = − E ( ϕ ( u i )) − h ( w ) + d,w d (Ψ( u ) − Ψ( u )) dw = − ( E ( ϕ ( u )) − E ( ϕ ( u )) = E ′ ( ϕ ( u ∗ )) ϕ ′ ( u ∗ )( u − u )for some u ∗ between u and u , and E ′ ( ϕ ( u ∗ )) = E ′ ( m )(1 + o (1)); and E ′ ( m ) > . Then forsmall w d (Ψ( u ) − Ψ( u )) dw (Ψ( u ) − Ψ( u )) < , which implies that (Ψ( u ) − Ψ( u )) is decreasing, with limit 0 at 0. Therefore Ψ( u ) =Ψ( u ) , thus u ≡ u near −∞ ; but from (3.15 ), h ( w ) = h ( w ) , and since h is one to one, itfollows w ≡ w near −∞ . The global uniqueness follows, since the system is regular exceptat (0 , . All the trajectories are described. (cid:3)
Remark.
Under the assumption (3.20 ), existence and uniqueness of H and H can be ob-tained in a more direct way whenever d = E ( η ) . Indeed the system (3.21 ) relative to( v, u ) is regular, with stationary points (0 , , (0 , ± ϕ − ( m )) , where m = m , m or m and also ( ± a,
0) if b + d > . The linearized system at (0 , ϕ − ( m )) is given by the matrix (cid:18) m ( q + 1 − p ) 00 K ( m ) (cid:19) , with K ( m ) = p ( p − η − m ) / (1 + ( p − m ) . If m = m ,d , then it is a source, and we find again the existence of an infinity of solutions. If m = m d or m = m ,d , then K ( m ) < , thus this point is a saddle point. Then in the phase plane ( v, u ) , there exists precisely one trajectory defined near −∞ , such that v > , m ) at −∞ , and u/v converges to 0 . (cid:3) Remark.
Suppose f ( w ) = | w | q − w, then we can study the critical case ( p − b > p − E ( η ) = d : there exist infinitely many homoclinic trajectories H starting from (0 , Q with an infinite initial slope and ending at (0 ,
0) with an infinite slope, and a uniquehomoclinic trajectory H starting from (0 ,
0) in Q with the initial slope η and ending at(0 ,
0) with the slope − η. Indeed using system (3.22 ) and setting u = ϕ − ( η ) + z, and ζ = ( q + 1 − p ) ηz + v, it can be written under the form ζ ′ = P ( ζ, v ) , v ′ = ( q + 1 − p ) ηv + Q ( ζ, v ) , where P and Q both start with quadratic terms. Moreover the quadratic part of P ( ζ, v ) isgiven by p , ζ + p , ζv + p , v , where by computation, p , = − p ( p − q + 1 − p ηϕ ′ ( ϕ ( η ))(1 + η ) ( p − / < . The results follow from the description of sadle-node behaviour given in [13, Theorem 9.1.7].
Remark.
In the case b = 1 > − d, we have a representation of the homoclinic trajectory : itcorresponds to K = 0 in (4.2 ) . In the case f ( w ) = | w | q − w , in terms of u we obtain u ′ = q + 1 − pp ( E ( ϕ ( u )) − d ) , u by a quadrature. First we consider the sign changing solutions
Theorem 4.3
Assume p > . For any ν > let T [(0 ,ν )] be the trajectory which starts from (0 , ν ) , and let T ( ν ) be its least period. Then ν T ( ν ) is decreasing on (0 , ∞ ) . Furthermorethe range of T ( . ) can be computed in the following way.(i) If b + d ≤ and m E ( m ) is increasing, or if d < min E, then T ( . ) decreases from T d to , where T d = 4 Z ∞ duE ( ϕ ( u )) − d = 4 Z π/ p −
1) tan θ ( p −
1) tan θ − b − d cos p − θ dθ, (4.4) and T d is finite if and only if b + d < . If b < d, then T = 2 π ( p − γ + 1( p − γ ( γ + 1) with γ = p | b | / ( p − . (4.5) (ii) If b + d > or b + d ≤ and d ≥ min E, then T ( . ) decreases from ∞ to .Proof. Step 1. Monotonicity of T . Consider the part of the trajectories T [(0 ,ν )] located in Q , given by ( w ν , y ν ) . We have already shown that u is decreasing with respect to τ from ∞ to 0 , then E ( ϕ ( u )) + h ( w ν ( u )) − d > w ν can be expressed in terms of u, and T ( ν ) = 4 Z ∞ duE ( ϕ ( u )) + h ( w ν ( u )) − d . (4.6)Let λ > . Since the trajectories T [(0 ,ν )] and T [(0 ,λν )] have no intersection point, w λν ( u ) >w ν ( u ) for any u > , and h is nondecreasing, thus T ( λν ) < T ( ν ) , and T is decreasing. Step 2. Behaviour near ∞ . Let ν n ≥ , such that lim ν n = ∞ . Observe that for fixed u, forany integer n ≥ , there exists a unique ˜ ν n > u ) , such that w ˜ ν n ( u ) = n ;let ˆ ν n = max(˜ ν n , n ) . Then h ( w ˆ ν n ( u )) ≥ h ( n ), thus h ( w ˆ ν n ( u )) converges to ∞ ; since ν h ( w ν ( u )) is nondecreasing then h ( w ν n ( u )) converges to ∞ , and T ( ν n ) converges to 0, usingthe Beppo-Levi theorem. Step 3. Behaviour near . • First assume b + d ≤ , and E is increasing, or d < E ( η ) . Then all the orbits are of thetype T [(0 ,ν )] . Let ν n ∈ (0 , , such that lim ν n = 0 . For fixed u and any integer n ≥ , thereexists a unique ¯ ν n > u ) , such that w ¯ ν n ( u ) = 1 /n ; let ˇ ν n = min(¯ ν n , /n ) . Then h ( w ˇ ν n ( u ) ≤ h (1 /n ), thus h ( w ˇ ν n ( u )) converges to 0 , and again h ( w ν n ( u )) converges to0 . Then T ( ν n ) converges to T d given by (4.4 ), using the Beppo-Levi theorem. If b + d < , then T d is finite: indeed near ∞ , E ( ϕ ( u )) = ( p − u p/ ( p − (1 + o (1)); if E is increasing,then E ( ϕ ( u )) − d > − ( b + d ) >
0; if d < E ( η ) , then E ( ϕ ( u )) − d ≥ E ( η ) − d > b + d = 0 and E is increasing, then T d = ∞ : indeed near 0 ,E ( ϕ ( u )) − d = u ( E ′′ (0) / o (1))19nd E ′′ (0) = 2( p − − ( p − b if ( p − b = 2( p − . Therefore E ( ϕ ( u )) − d = ( p ( p − / u (1 + o (1)) . In all the cases the integral (4.6 ) giving T is divergent.When b < d, one can compute T : T Z ∞ duE ( ϕ ( u )) = Z ∞ φ ′ ( ξ ) dξE ( ξ ) = Z ∞ p − ξ ( | b | + ( p − ξ )(1 + ξ ) dξ = π p − − γ ) Z ∞ ds ( γ + s )(1 + s ) = π − ( p − γ ( p − γ ( γ + 1) ) . Hence (4.5 ) holds. • Next assume d > E ( η ) . Considering ν n as above, for any fixed u such that ϕ ( u ) > m , there exists a unique ¯ ν n > u ) , such that w ¯ ν n ( u ) = 1 /n. As above, Z ∞ ϕ − ( m ) duE ( ϕ ( u )) + h ( w ¯ ν n ) − d → Z ∞ ϕ − ( m ) duE ( ϕ ( u )) − d = ∞ , since E ′ ( m ) is finite. As a consequence, T ( ν n ) tends to ∞ . If d = E ( η ) , the same proofstill works with m replaced by η : the integral is still divergent because the denominator isof order 2 in u − ϕ − ( η ), as, near 0 , there holds E ( ϕ ( u )) − d = 12 E ′′ ( η ) ( ϕ ( u ) − η ) (1 + o (1)) = 12 E ′′ ( η ) ( ϕ ( u ) − η ) (1 + o (1) , and E ′′ ( η ) = 2 p ( p − η (1 + η ) ( p − / > . At last suppose b + d >
0; the same proof with m replaced by m shows that T ( ν ) convergesto ∞ as ν tends to 0 , since E ′ ( m ) at m = E − ( d ) is finite. (cid:3) The monotonicity of the period function is a more general property, since we have thefollowing result.
Proposition 4.4
Let
F, G ∈ C ( R \ (0 , are such that F (resp. G ) is odd with respect to y (resp. x ) and even with respect to x (resp. y ) , with F ( w, y ) > in Q . Assume that forany ( w, y ) ∈ Q , and any λ > ,∂∂λ (cid:18) F ( λw, λy ) λ (cid:19) ≥ resp. ≤ and ∂∂λ (cid:18) G ( λw, λy ) λ (cid:19) < resp. > . (4.7) Assume also that for any σ in some interval ( σ , σ ) (where < σ < σ ), the trajectory T [(0 ,σ )] of solution of the system (cid:26) w ′ = F ( w, y ) y ′ = G ( w, y ) (4.8) passing through (0 , σ ) (necessarily entering Q since F (0 , σ ) > leaves Q transversally in afinite time T ( σ ) / at some point ( c ( σ ) , (thus G ( c ( σ ) , < . Then (from the symmetries), T [(0 ,σ )] is a closed orbit surrounding (0 , , with period T ( σ ) , and σ T ( σ ) is decreasing(resp. increasing) on ( σ , σ ) . emark. We can notice the condition on F is equivalent to F ( λw, λy ) ≥ λF ( w, y ) for any λ >
1. The second condition implies that for any λ > ,G ( λw, λy ) < λG ( w, y ) (resp. G ( λw, λy ) > λG ( w, y )) . Proof of Proposition 4.4 . In polar coordinates ( ρ, θ ) in Q , we get ρ ′ = F cos θ + G sin θ, θ ′ = 1 ρ ( G cos θ − F sin θ ) . At each point τ where θ ′ ( τ ) = 0 , there holds ρθ ′′ ( τ ) = (cid:18) ∂G∂ρ cos θ − ∂F∂ρ sin θ (cid:19) , ρ ′ ( τ ) = F cos θ (cid:18) ∂G∂ρ cos θ − ∂F∂ρ sin θ (cid:19) . But (4.7 ) is equivalent to ∂F/∂ρ ≥ F/ρ and ∂G/∂ρ < G/ρ (resp > ), thus ρθ ′′ ( τ ) < Fρ cos θ ( G cos θ − F sin θ ) = 0 (resp. > )In both case θ ′′ has a constant sign. But θ ′ (0) = − F (0 , σ ) < θ ′ ( σ ) = G ( c ( σ ) , < θ ′ ( τ ) = 0 , which satisfies θ ′′ ( τ ) ≥ ≤ . Thus θ is decreasing from π/ . Then the curvescan be represented in function of θ by ( ρ ( σ, θ ) , θ ( σ )) , and T ( σ ) = 4 Z π/ dθH ( ρ ( σ, θ ) , θ )with H ( ρ, θ ) = 1 ρ ( F ( ρ cos θ, ρ sin θ ) sin θ − G ( ρ cos θ, ρ sin θ ) cos θ )Let λ > . Since the trajectories T [(0 ,σ )] and T [(0 ,λσ )] . have no intersection point, then ρ ( λσ, θ ) > ρ ( σ, θ ) for any θ ∈ (0 , π/
2) ; by assumption, for fixed θ, the function ρ F ( ρ cos θ, ρ sin θ ) /ρ is nondecreasing (resp. nonincreasing) and ρ G ( ρ cos θ, ρ sin θ ) /ρ isdecreasing (resp. increasing), thus H ( ρ ( λσ, θ ) , θ ) > H ( ρ ( σ, θ ) , θ ), which yields to T ( λσ )
Assume p > and b + d > . Consider the trajectories T [( µ, in the phaseplane ( w, y ) which goes through ( µ, , for some µ ∈ (0 , a ) . Let T + ( µ ) be their least period.Then lim µ → T + ( µ ) = ∞ , lim µ → a T + ( µ ) = 2 π p ah ′ ( a ) . In particular if f ( w ) = | w | q − w , then lim µ → a T + ( µ ) = 2 π/ ( q + 1 − p )( b + d ) . roof. We notice that the trajectory T [( µ, intersects the line y = 0 at ( µ,
0) and anotherpoint ( g ( µ ) , , with µ < a < g ( µ ) , and g is decreasing. Step1. Behaviour near a . When µ tends to a, then also g ( µ ) tends to a. Indeed for anysmall ε > , then g ( µ ) − a < ε as soon as µ − a < min( ε, a − g − ( a + ε )) . Since, along sucha trajectory in Q , ξ = ϕ ( u ) varies from 0 to 0, it has a maximal ξ ∗ , where u ′ = 0 , thus E ( ξ ∗ ) = h ( w ∗ ) . When µ tends to a, then h ( w ∗ ) tends to b, thus ξ ∗ tends to E − ( b ) = 0 , thus also max y ∈T [( µ, | y | tends to 0. Using the linearized form of the system at P , andpolar coordinates with center ( a, , w = a + r cos η, y = p ah ′ ( a ) r sin η, then r tends to 0as µ tends to a, and one finds η ′ = − p ah ′ ( a ) + R/r, where R involves the derivatives of G of order 2, which are bounded near the point ( a, , thus R/r is bounded. Therefore η ′ tends to − p ah ′ ( a ) , and finally T + ( µ ) tends to 2 π/ p ah ′ ( a ) . Step2. Behaviour near
0. On the trajectory T [( µ, , the function u is increasing up to amaximal value u ∗ ( µ ) , and then decreasing; moreover u ∗ is a nonincreasing function of µ, because two different trajectories have no intersection. Let µ n ∈ (0 , a ) , such that lim µ n = 0 . For any n there exists ˜ µ n ∈ (0 , a ) such that the orbit T [(˜ µ n , , contains a point above theline y = ϕ − ( m )(1 − /n ) w, let ˆ µ n = min( µ n , /n ) . Then u ∗ (ˆ µ n ) ≥ ϕ − ( m )(1 − /n ) , thus u ∗ ( µ n ) tends to m ; then from the Beppo-Levi theoremlim inf T + ( µ ) ≥ lim Z ∞ u ∗ ( µ ) duE ( ϕ ( u )) − d + h ( w ( µ, u )) = Z ∞ m duE ( ϕ ( u )) − d + h ( w ( u ))where w is the solution defining H , and this integral is infinite. (cid:3) Remark.
Here the question of the monotonicity of the period is difficult to answer, even for p = 2, where it is solved by using the first integral, see [3]. It is open in the general case.More generally, if a dynamical system a center, the description of the period function is stilla chalenging problem. For example, one can contruct a quadratic dynamical system with acenter, the associated period function of which is not monotone, and even with at least twocritical points, see [7] and [8]. Remark.
In the case b = 1 , we can compute theoretically the period T + by using the firstintegral (4.1 ). The stationary point P = ( h − (1) ,
0) is obtained for K a = a p /p − F ( a ) > K a = ( q + 1 − p ) /p ( q + 1)) . The positive solutions correspond totrajectories T K with K ∈ (0 , K a ) , intersecting the axis y = 0 at points ( w , w ,
0) with w < a < w defined by w pi /p − F ( w i ) = K, and the period is given by T + = 2 Z w w dwwE − ( − p K + F ( w ) w p ) . Unfortunately, this formula does not allow us to prove the monotonicity of the period func-tion for p = 2 . It is remarkable that, in the case f ( w ) = | w | q − w, one can solve completely the problemin the particular case where b = 1 and q = 2 p − , using the equation (3.23 ) satisfied by u . Proposition 4.6
Suppose that f ( w ) = | w | q − w, and b = 1 and q = 2 p − , p > and d + 1 > . If p > or d + 1 < / (2 − p ) , then T + is decreasing on (0 , a ) . roof. Since B ( ξ ) = 0 by (3.24 ), equation (3.23 ) turns to u ′′ = ( q + 1 − p )( E ( ϕ ( u ) − d ) ϕ ( u ) = ( q + 1 − p )( − (1 + d ) + p Ψ . ( u )) ϕ ( u )Henceforth 1 q + 1 − p u ′′ u ′ = − (1 + d )Ψ ′ ( u ) u ′ + p Ψ( u )Ψ ′ ( u ) u ′ , from which expression we derive the first integral,1 q + 1 − p u ′ = C − U ( u )) , U = M ◦ Ψ , M ( t ) = 2(1 + d ) t − pt . (4.9)From (4.9 ) the integral curves S in the ( u, u ′ )-plane are symmetric with respect to the axis u ′ = 0 . The times for going from u = 0 to u = u ∗ and from u ∗ to 0 are equal, and u ∗ isgiven by C = M (Ψ( u ∗ )) . The computation of the period is reduced to the part relative tothe first quadrant. Here we follow the method of [3]: we get T + ( u ∗ ) = 4 Z u ∗ dη p U ( u ∗ ) − U ( η ) = 4 Z u ∗ ds p U ( u ∗ ) − U ( su ∗ ) . Then dT + ( u ∗ ) du ∗ = 4 Z (Θ( u ∗ ) − Θ( su ∗ )) ds ( U ( u ∗ ) − U ( su ∗ )) / , with Θ( u ∗ ) = U ( u ∗ ) − u ∗ U ′ ( u ∗ ) / , and 2 d Θ( u ∗ ) du ∗ = 2Θ ′ ( u ∗ ) = U ′ ( u ∗ ) − u ∗ U ′′ ( u ∗ ) . In the interval of study, ϕ ( u ∗ ) < E − ( d ), ( E ◦ ϕ )( u ) < d from (3.14 ), thus Ψ( u ) < (1 + d ) /p, and M is increasing for 0 < t < (1 + d ) /p, thus U ′ > . Then at any point u , Θ ′ ( u ) > ⇐⇒ ( U ′ /u ) ′ < . Now U ′ ( u )2 pu = ( − E ( ϕ ( u )) + d ) ϕ ( u ) u = 1 − ( p − ϕ ( u ) + d (1 + ϕ ( u )) (2 − p ) / , hence ( U ′ /u ) ′ = 2 X ( u ) ϕ ( u ) ϕ ′ ( u ) , with X ( u ) = − ( p −
1) + (2 − p ) d (1 + ϕ ( u )) − p/ , and d > E ( ϕ ( u )); it implies X ( u ) < p > p < d < ( p − / (2 − p ) . HenceforthΘ is increasing, and the same holds for P as a function of u ∗ . Finally u ∗ is decreasing withrespect to µ, and consequently P is decreasing with respect to µ. (cid:3) Remark.
When p = 2 , and q = 2 p − , equation (3.23 ) reduces to u ′′ = − u + 2 u , which, surprisingly, is an equation correponding to the problem with absorption, and (3.4 )reduces to w ′′ − w + w = 0. In this case, all the solutions can be expressed in terms ofelliptic integrals, see [3]. 23 .4 Returning to the initial problem Proof of Theorem 2.
Here β = β q = p/ ( q +1 − p ), λ = λ q is given by (1.15 ) and c q = β p − q λ q by (1.16 ). Moreover ω ( σ ) = β β q q w ( β q σ ) from (3.3 ), b = − λ q /β q = − c q /β pq and d = c/β pq from (3.5 ). At end f ( s ) = g ( s ) = | s | q − s and h ( s ) = | s | q − p s. Thus c > c q is equivalent to b + d > , and then the constant solutions w ≡ ± ( b + d ) / ( q − p +1) of (3.4 ) correspond to theconstant solutions ω ≡ ± ( c − c q ) / ( q +1 − p ) of equation (1.14 ). For any integer k ≥ , welook for periodic solutions ω of smallest period 2 π/k, or equivalently solutions w of period T k = 2 πβ q /k. From (3.17 ), the function E is increasing. First consider the sign changingsolutions: if c ≥ c q , then from Theorem 4.3, the period function T of w is decreasing from ∞ to 0, hence for any k ≥ T k . If c < c q , then T decreasesfrom T d given by (4.4 ) to 0 , thus it takes once the value T k for any k > M q = T d / πβ q givenat (1.18 ). Next consider the positive solutions: from Proposition 4.5, the period functionof w takes any value between ∞ and 2 π/ p ( q + 1 − p )( b + d ) , thus it takes the value T k forany k < ( pβ − pq ( c − c q )) / , which ends the proof. (cid:3) In the case of equation (1.14 ) (i.e. c = 0), we obtain the following description of thesets E and E + : Corollary 4.7
Assume p > , q > p − , and c = 0 . (i) Then the set E of changing sign solutions of (1.14 ) is given by (1.17 ), where k q = 1 if p < and q ≥ p − / (2 − p ) , and k q > M q if p ≥ or ( p < and q < p − / (2 − p )) , where M q = 2 / ( q − , (4.10) if p = 2 , and M q = ( p − m q (( p − m q + 1)( m q − , with m q = s (2( p −
1) + ( p − qp ( p − , (4.11) if p = 2 .(ii) If p ≥ or ( p < and q < p − / (2 − p )) , then E + = ∅ . If p < and q ≥ p − / (2 − p ) , then E + = (cid:8) ( − c q ) / ( q +1 − p ) (cid:9) . Proof.
Here c q < p < q > p − / (2 − p ) . Furthermore M q = T / πβ q can be computed from (4.5 ), which gives (4.10 ), (4.11 ). Moreover in any case c q + β p − q /p = β pq ( p − q + 1) /p > (cid:3) Proof of Corollary 1.
Let S be a sector on S with opening angle θ ∈ (0 , π ). From [14,Th 3.3], β S is the positive solution of equation φ ( β S ) = (cid:18) k (cid:19) (cid:18) β S + p − p − β S − ( β S + 1) (cid:19) = 0 , where k = π/θ ≥
1. Using Corollary 4.7 (applied without assuming that k is an integer) wedistinguish two cases: 24i) p < q ≥ p − / (2 − p ). Then there always exists a solution to the Dirichletproblem in S . Notice that 0 < β q ≤ (2 − p ) / ( p − φ ( β S ) < β q < β S .(ii) p > p < q < p − / (2 − p ). The existence is equivalent to k > M q (see(4.11 )). It means (cid:18) k (cid:19) < ( p − m q − m q ( p − ! = ( β q + 1) β q m q = ( β q + 1) β q ( β q + ( p − β q / ( p − . Thus φ ( β q ) <
0. Equivalently, β q < β S . (cid:3) p = 1 As shown in Lemma 3.1, we can reduce the study to ddτ (cid:18) w ′ √ w + w ′ (cid:19) − b w √ w + w ′ + f ( w ) − d | w | − w = 0 , (5.12)where f satisfies (3.8 ); in particular we are interessed by the case f ( s ) = s. Here the problem is variational: if S ( w ) is any primitive of w
7→ | w | b − f ( w ) and R ( w ) = | w | b /b if b = 0 , R ( w ) = ln | w | if b = 0 , then (5.12 ) is the Euler equation of thefunctional H ( w, w ′ ) = | w | b − p w + w ′ − S ( w ) + dR ( w ) . Thus the following Painlev´e first integral is constant along the trajectories P ( w, w ′ ) = | w | b +1 √ w + w ′ − S ( w ) + dR ( w ) . (5.13)The system (3.9 ) reads as w ′ = yy ′ = G ( w, y ) = bw + ( b + 1) w y − ( f ( w ) − d | w | − w )( w + y ) / w , and it is singular on the line w = 0 . For w > ( w ′ = wϕ ( u ) = w u √ − u u ′ = b √ − u − f ( w ) + d. (5.14)In the case f ( w ) = w , the equation satisfied by u is u ′′ = (1 − b ) u √ − u u ′ − bu − d u √ − u . (5.15)25 .2 Existence of periodic solutions From the Painlev´e integral (5.13 ), we can describe the solutions, in the phase plane ( w, y ) . Since a complete description is rather long, we reduce it to the research of periodic solutions.
Proposition 5.1
Let p = 1 , and consider equation (5.12 ).(i) If d = 0 , there is no periodic sign changing solution. If d = 0 there exists such a solutionif and only if b > − , and then it is unique (up to a translation).(ii) There exists periodic positive solutions if and only if b + d > . (iii) Suppose moreover that f ( w ) = w. Then the sign changing solution is given by w ( τ ) = ( b + 1) cos( τ − τ ); it has period π . The orbits T [( µ, of the periodic solutions intersect the axis y = 0 at afirst point ( µ, such that µ < a = b + d, and µ describes µ ∈ (¯ µ, a ) with ¯ µ = 0 if d ≤ , and ¯ µ > if d > it is given by (5.20 ),(5.18 ),(5.19 ).Proof. By symmetry we reduce the study to the case w ≥ w b p − u − S ( w ) + dR ( w ) = C, (5.16)where we denote S ( w ) = Z w s b − f ( s ) ds if b > − ,S ( w ) = Z w s b − f ( s ) ds + 1 b + 1 if b < − , and S ( w ) = Z w s − f ( s ) ds if b = − . Step 1. Periodic sign changing solutions.
The curves in the phase plane ( w, y ) are given,for w >
0, by y = (cid:18) w b +1 C − dR ( w ) + S ( w ) (cid:19) − w = w (cid:0) w b + dR ( w ) − S ( w ) − C (cid:1) ( w b − dR ( w ) + S ( w ) + C )( S ( w ) + C − dR ( w )) , which defines ± y in function of w. If there exists a sign changing periodic solution, thetrajectory intersects the axis w = 0 at some point (0 , ℓ ) with ℓ ≥ , thus y needs to ℓ as w tends to 0 . From (5.16 ), it is impossible if b ≤ − . Assume d = 0; if − < b, then near w = 0 , in any case y ≤ ( b /d + 1) w , thus ℓ = 0 and w ′ /w is bounded, thus the maximalinterval of existence is infinite, and we reach a contradiction. If d = 0 , and C = 0 , then y = − w (1 + o (1)) , which is impossible. If d = C = 0 , then y = w ( w b /S ( w ) −
1) ;observing that the function w χ ( w ) = w − b S ( w ) is increasing from 0 to ∞ , the curveintersects the two axis at (0 , b + 1) and ( χ − (1) ,
0) and this corresponds to a closed orbit.26 tep 2. Existence of periodic positive solutions.
If we look at the intersection points of anytrajectory in the phase plane with the axis y = 0, we find that they are given by H ( w ) = C, where H ( w ) = w b + dR ( w ) − S ( w ) . Then H ′ ( w ) = w b − ( b + d − f ( w )) . If b + d ≤ , then H is decreasing, thus there ex-ist no positive periodic solutions. If b + d > , the function H is increasing on (0 , a )where it reaches a maximum M, and decreasing on ( a, ∞ ). The stationary point ( a,
0) with a = f − ( b + d ) corresponds to C = M. If b > , then lim w → H = 0 , while, if b ≤ , then lim w → H = −∞ . Equation H ( w ) = C has two roots 0 < w < w , if and only if C ∈ (max { lim w → H, lim w →∞ H } , M ) . Moreover, if there exists a trajectory going through( w ,
0) and ( w ,
0) and if one denotes K ( w ) = dR ( w ) − S ( w ) = H ( w ) − w b , one has K ( w ) < C on ( w , w ), thus C > M ′ = max K = K ( f − ( d )) . Conversely, ifmax n lim w → H, lim w →∞ H, M ′ o < C < M, M = H ( f − ( b + d )) , M ′ = K ( f − ( d )) , (5.17)then there exists a closed orbit going through ( w ,
0) and ( w , Step 3. End of the proof.
The sign changing solution is given by w + w ′ = ( b + 1) andits trajectory is a circle with center 0 and radius b + 1; for w > , w = ( b + 1) √ − u = b √ − u − u ′ , thus u ′ = −√ − u , and θ ′ = − , then w ( τ ) = ( b + 1) cos( θ − τ ) , periodicsolution with period 2 π. Now consider the positive periodic solutions. Here a = b + d, and H ( w ) = (1 + d/b ) w b − w b +1 / ( b + 1), if b = 0 , − , d ln w − w, if b = 0 , (1 − d ) w − − ln w, if b = − , (5.18) M = a b +1 /b ( b + 1) , M ′ = ( d + ) b +1 /b ( b + 1)) , if b = 0 , − ,M = 1 + d ln d − d, M ′ = d ln d − d, if b = 0 .M = − − ln ( d − , M ′ = − − ln d )) , if b = − , (5.19)If d ≤ , thus b >
0, then any C ∈ (0 , M ) corresponds to a closed orbit, thus for any µ ∈ (0 , a ) , one has a closed orbit passing through ( µ, , of period still denoted by T + ( µ ) . If d > , in any case, any C ∈ ( M ′ , M ) corresponds to a closed orbit. If − < b < H is increasing on (0 , a ) from −∞ to M < , and then decreasing on ( a, ∞ ) from M to −∞ . If b < − , then M > M ′ > . If b < − , then d > , and lim w → H = −∞ , lim w →∞ H = 0 , < M ′ < M . If b = 0 , thus d > , then lim w → H = −∞ , then any C ∈ ( M − , M ) corresponds to a closed orbit. Then H is increasing on (0 , d ) from −∞ to M = 1 + d ln d − d ≥ , (notice that M = 0 ⇔ d = 1) and then decreasing on ( a , ∞ ) from M to −∞ ; let ¯ µ ∈ (0 , b + d ) be defined by H (¯ µ ) = M ′ , (5.20)27hus for any µ ∈ (¯ µ, a ) , one has a closed orbit passing through ( µ, , with a periodstill denoted by T + ( µ ) . If − ≤ b < , thus d > − b > , then lim w → H = −∞ =lim w →∞ H, M <
0, and any C ∈ ( M ′ , M ) corresponds to a closed orbit (if b = − , then H ( w ) = (1 − d ) w − − ln w, M = − − ln ( d − , M ′ = − − ln d ) . Returning to equation(1.14 ), the conclusion follows with ¯ µ q = ¯ µ q . (cid:3) Let p = 1 , b + d > . Consider the equation (5.12 ). Let T + ( µ ) be the least period of theperiodic positive solutions corresponding to the orbit T [( µ, . As in the case p > , we havea general result: lim µ → a T + ( µ ) = 2 π p af ′ ( a ) . (5.21)Next we study the variations of the period in the case of a power f ( w ) = w . Theorem 5.2
Assume p = 1 , b + d > and f ( w ) = w. Then lim µ → a T + ( µ ) = 2 π/ √ b + d. If d < , then lim µ → T + ( µ ) = ∞ . If d ≥ , then lim µ → ¯ µ T + ( µ ) = ¯ T + is finite, and givenby (5.23 ) if b / ∈ { , − } , by (5.24 ) if b = 0 , and by (5.25 ) if b = − . If d = 0 , then ¯ T + = π (1 + 1 /b ) . Proof. Step 1. Assume b / ∈ { , − } . From (5.16 ), the solutions of (5.12 ) satisfy w b p − u − w b +1 b + 1 + db w b = C, thus u ′ = b p − u − w + d = − p − u − db + C (1 + b ) w b . Eliminating w between the two relations, we find that Cb ( b + 1) > (cid:16) d + b p − u − u ′ (cid:17) b/ ( b +1) (cid:16) d + b p − u + bu ′ (cid:17) / ( b +1) = ( Cb (1 + b )) / ( b +1) := A. When a solution goes through the half-part of its trajectory T located in Q , the associatedfunction u increases from 0 to some u ∗ ∈ (0 ,
1) where the derivative u ′ vanishes and d + b √ − u ∗ >
0; next d + b √ − u is monotone and positive at 0 and u ∗ , thus d + b √ − u > A = d + b √ − u ∗ = w ∗ (the value of w when u = u ∗ ). Let z = u ′ d + b √ − u and G ( s ) = (1 − s ) b/ ( b +1) (1 + bs ) / ( b +1) . If b > , then z ∈ ( − /b,
1) ; if b < − z ∈ ( −∞ ,
1) ; if − < b < z ∈ ( −∞ , / | b | ) , and G ( z ) = Ad + b √ − u . Since G ′ ( s ) = − bs (1 − s ) − / ( b +1) (1 + bs ) − b/ ( b +1) , G ′′ ( s ) = − b (1 − s ) − ( b +2) / ( b +1) (1 + bs ) − (2 b +1) / ( b +1) , it follows G (0) = 1, and 0 is a maximum if b > b <
0: if b > G increases on ( − /b,
0) from 0 to 1 and decreases on (0 ,
1) from 1 to 0; if b < , G decreaseson ( −∞ ,
0) from ∞ to 1 and increases on (0 , min(1 , / | b | )) from 1 to ∞ . Thus it has twoinverse functions − L and L : for b > , L maps (0 ,
1) into (0 , /b ) and L maps (0 , ,
1) ; for b < , L maps (1 , ∞ ) into (0 , ∞ ) and L maps (1 , ∞ ) into (0 , min(1 , / | b | )) . Then T + = T +1 + T +2 , T + i = Z ψ i,u ∗ ( λ ) dλ, (5.22)where ψ i,u ∗ ( λ ) = 2 u ∗ ( d + b √ − λ u ∗ ) L i (( d + b √ − u ∗ ) / ( d + b √ − λ u ∗ )) . • First suppose d < b > C →
0, thus √ − u ∗ → − d/b, thus u ∗ → ¯ u = p − d /b . Near ¯ u,ψ i,u ∗ ( λ ) ≥ u ∗ b ( √ − λ u ∗ − √ − u ∗ ) L i (0) ≥ − dbL i (0)(1 − λ ) , therefore T + i tends to ∞ . • Suppose d ≥ , b > . C → M ′ , thus u ∗ → . Thereexists a constant m > ≤ − G ( s ) = G (0) − G ( s ) ≤ m s on [ − /b, . Indeed G ′ (0) = 0 and G ′′ is bounded on [ − / b, / , and on [ − /b, − / b ] ∪ [1 / ,
1] thequotient ( G (0) − G ( s )) /s is bounded. Thus 1 /L i ( η ) ≤ m/ √ − η on [0 , , hence taking η = ( d + b √ − u ∗ ) / ( d + b √ − λ u ∗ ) , and computing + − η = b (1 − λ ) u ∗ ( d + b √ − λ u ∗ ) (cid:0) √ − u ∗ + √ − λ u ∗ (cid:1) , one finds ψ i,u ∗ ( λ ) ≤ m/ p b (1 − λ ) . From the Lebesgue theorem, as u ∗ → , T + tends tothe finite limit¯ T + = ¯ T +1 + ¯ T +2 , ¯ T + i = 2 Z dλ ( d + b √ − λ ) L i ( d/ ( d + b √ − λ )) (5.23)in particular if d = 0 , then L (0) = 1 /b, L (0) = 1 , thus ¯ T +1 , = π and ¯ T +1 , = π/b. • Suppose b < , thus d > − b > . Then again C → M ′ , consequently u ∗ →
1. Thefunction u ∗ → Q ( u ∗ , λ ) = η = d + b √ − u ∗ d + b √ − λ u ∗ = 1 − b (1 − λ ) u ∗ (cid:0) d + b √ − λ u ∗ (cid:1) ( √ − u ∗ + √ − λ u ∗ )is increasing on (0 ,
1) from 1 to d/ ( d + b √ − λ ) and d/ ( d + b √ − λ ) ≤ d/ ( d + b ) = α. Thereexists m > ≤ G ( s ) − ≤ m s on [ − L ( α ) , L ( α )] , thus 1 /L i ( η ) ≤ m/ √ η − /d/ ( d + b )] . Thus as above, ψ i,u ∗ ( λ ) ≤ m/ p | b | (1 − λ ) , and T + tends to ¯ T + definedat (5.23 ). 29 tep 2. Assume b = 0 . There exist periodic solutions for any C ∈ ( M − , M ) . The solutionsare given by p − u + H ( w ) = p − u + d ln w − w = C and u ′ = − w + d, thus u is maximal (= u ∗ ) for w = d : therefore √ − u ∗ + H ( d ) = C, then H ( d − u ′ ) = H ( d ) + p − u ∗ − p − u and H has two inverse functions H i from ( −∞ , H ( d )) into (0 , d ) and ( d, ∞ ) , thus (5.22 )holds with ψ i,u ∗ ( λ ) = 2 u ∗ dλ ( d − H i ( H ( d ) + √ − u ∗ − √ − λ u ∗ )and ξ = H ( d ) + √ − u ∗ − √ − λ u ∗ = H ( d ) − k = H ( d + h ) stays in ( M − , M ) =( H ( d ) − , H ( d )) , and H ( d + h ) − H ( d ) ≥ − m h for H ( d + h ) ∈ ( M − , M ) , thus H ( d ) − ξ = k ≤ m ( d − H i ( ξ )) , thus ψ i,u ∗ ( λ ) ≤ m √ k = 2 m (cid:0) √ − u ∗ + √ − λ u ∗ (cid:1) √ − λ ≤ m √ − λ . Therefore, as u ∗ → , T + tends to the finite limit¯ T + = ¯ T +1 + ¯ T +2 , ¯ T + i = 2 Z dλ ( d − H i ( H ( d ) − √ − λ ) . (5.24) Step 3. Assume b = − . In that case d >
1; let B = − ( C + 1) ∈ (ln( d − , ln d ) then B → ln d and u ′ + w = d − p − u = ( B + 1) w − w ln w = H B ( w )where H B is increasing on (cid:0) , e B (cid:1) from 0 to e B and decreasing on (cid:0) e B , ∞ (cid:1) from e B to −∞ ; it has two inverse functions L B,i from (cid:0) −∞ , e B (cid:1) into (0 , e B ) and ( e B − , ∞ ); and w ∗ = d − √ − u ∗ = e B ; then (5.22 ) holds with ψ i,u ∗ ( λ ) = 2 u ∗ d − √ − λ u ∗ − L B,i ( d − √ − λ u ∗ = 2 u ∗ (cid:12)(cid:12) H B − ( L B,i ( d − √ − λ u ∗ )) (cid:12)(cid:12) . Because H B − ( e B ) = 0 , H B − ( x ) − H B − ( e B ) = H ′ B − ( ξ )( x − e B ) and x ranges onto( H B, ( d − , H B, ( d − x ,B , x ,B ), when B → ln d, ( x ,B , x ,B ) → ( x , ln d , x , ln d ),it follows (cid:12)(cid:12) H ′ B − ( ξ ) (cid:12)(cid:12) ≥ /µ > B. Moreover H B ( x ) − H B ( e B ) = (1 / H ′′ B ( ξ )( x − e B ) = − (1 / ξ )( x − e B ) . Thus there exists m > H B ( x ) − H B ( e B ) ≤ m ( x − e B ) ≤ m µ H B − ( x ) . Therefore, near ln d, taking x = L B,i ( d − √ − λ u ∗ ) , one derive ψ i,u ∗ ( λ ) ≤ mµ p d − √ − λ u ∗ − e B = 2 mµ p √ − u ∗ − √ − λ u ∗ ≤ mµ √ − λ . u ∗ → , T +1 ,i tends to the finite limit¯ T + = ¯ T +1 + ¯ T +2 , ¯ T +1 ,i = 2 Z dλ (cid:12)(cid:12) H ln d − ( L ln d,i ( d − √ − λ )) (cid:12)(cid:12) . (5.25) (cid:3) Remark.
In the case d = 0 , b = 1 , notice that T +1 and T +2 converges to π/ √ b as µ tendsto b (one can verify it by linearizing the equation in u ) and respectively to π and π/b as µ tends to Thus if those functions are monotonous, they vary in opposite senses and it is noteasy to get the sense of variations of their sum T +1 . Moreover in the phase plane ( w, y ) , as µ tends to 0 , on can observe that the trajectory tends to a limit curve constituted of asegment [(0 , , (0 , b )] and half of the unique closed orbit surrounding (0 , , circle of center0 and radius b + 1, which is covered in a time π The case b = 1 is the most interesting for (5.12 ), since it corresponds to the initialproblem (1.14 ). In that case we improve the results by showing the monotonicity of theperiod function: Theorem 5.3
Assume b = 1 , d > − . When d = 0 the period function T + ( µ ) is constant,with value π, thus there exists an infinity of positive solutions w of (5.12 ), which are all π -periodic; they are explicitely given by w = p − K ∗ sin τ − K ∗ cos τ, τ ∈ [ − π, π ] , K ∗ ∈ (0 , . (5.26) When d = 0 , then T + ( µ ) is strictly monotone; if d < it decreases from ∞ to π/ √ d ; if d > it increases from ¯ T + = 4 Z du q ( d + √ − u ) − d = 4 Z π/ r cos θ cos θ + 2 d dθ (5.27) to π/ √ d. Proof. • If d = 0, then u ′′ = − u, from (5.15 ), and u = sin θ ∈ [0 , , thus the positivesolutions w are given in Q by u = K ∗ sin τ, K ∗ ∈ [0 , , τ ∈ [0 , π ] , and the period T + is constant, equal to 2 π . We obtain an infinity of positive solutions w ,given explicitely by w = p − u − u ′ = p − K ∗ sin τ − K ∗ cos τ, K ∗ ∈ (0 , y = 0 at points w i = (1 ∓ K ∗ ) . • In the general case d > − , we find (cid:16) d + p − u − u ′ (cid:17) (cid:16)p − u + u ′ + d (cid:17) = A G is symmetric: G ( s ) = √ − s , thus u ′ = ( d + p − u ) − A √ − u ∗ = A − d = √ C − d ; thus here T +1 = T +2 , and T + = 4 Z dλ p Ψ( u ∗ , λ ) , where Ψ( s, λ ) = ( d + √ − λ s ) − ( d + √ − s ) s . We show that the period function is strictly monotone with respect to u ∗ . Because s ∂ Ψ ∂s ( s, λ ) = d ( d + 1) (cid:16) / √ − s − / p − λ s (cid:17) > , we see that T + is increasing if d < d > d = 0). Also µ can be expressed explicitely in terms of u ∗ by µ = d + 1 − q ( d + 1) − ( d + p − u ∗ ) . Therefore µ is decreasing with respect to u ∗ , hence T + is decreasing with respect to µ if d < d > . (cid:3) Proof of Theorem 3.
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