characteristic numbers of non-autonomous emden-fowler type ...

Mathematical Modelling and Analysis 2005. Pages 403–408Proceedings of the 10 th International Conference MMA2005&CMAM2, Trakaic○ 2005 Technika ISBN 9986-05-924-0CHARACTERISTIC NUMBERS OFNON-AUTONOMOUS EMDEN-FOWLERTYPE EQUATIONSA. GRITSANS 1 and F. SADYRBAEV 21 Daugavpils UniversityDaugavpils, Parades str. 1E-mail: arminge@dau.lv2 Institute of Mathematics and Computer Science, University of LatviaRiga, Rainis blvd 29E-mail: felix@cclu.lvAbstract. Consider the Emden – Fowler equation x ′′ = −q(t)|x| 2ε x, ε > 0, inthe interval [a, b]. The coefficient q(t) is a positive valued continuous function. TheNehari’s characteristic number λ n associated withRthe Emden – Fowler equationε bcoincides with a minimal value of the functional1+ε a x′2 (t) dt over all solutions ofthe boundary value problemx ′′ = −q(t)|x| 2ε x, x(a) = x(b) = 0, x(t) has exactly n − 1 zeros in (a, b).The respective solution is called by Nehari’s solution. We construct an examplewhich shows that the Nehari’s extremal problem may have more than a uniquesolution.Key words: Characteristic numbers, Emden - Fowler equation, Nehari’s solutions1. Nehari’s SolutionsBehavior of solutions to the Emden – Fowler type equationx ′′ = −q(t)|x| 2ε x, ε > 0, (1.1)where q(t) is a positive valued continuous function, may be complicated if q(t)is a non-monotone function.Some regularity to the theory of the Emden – Fowler type equations ofthe form (1.1) is brought by the so called Nehari’s solutions.The Nehari’s theory applies to equations of the type (1.1).

404 A. Gritsans, F. SadyrbaevThe general theorem by Nehari ([2, Theorem 3.2]) when adapted to thecase under consideration states that the extremal problem below has a solution.Problem:H(x) =∫ ba[x ′2 − (1 + ε) −1 q(t)x 2+2ε] dt → inf, x ∈ Γ n , (1.2)where Γ n consists of all functions x(t), which are continuous and piece-wisecontinuously differentiable in [a, b]; there exist numbers a ν such thata = a 0 < a 1 < . . . < a n = b;x(a 0 ) = 0 and for ν = 1, . . . , n, x(a ν ) = 0 but x ≢ 0 in any [a ν−1 , a ν ], and∫ aνa ν−1x ′2 (t) dt =∫ aνa ν−1q(t)x 2 |x| 2ε dt. (1.3)The respective extremal functions x n (t) are those solutions of equation(1.1), which vanish at the points t = a and t = b, have exactly n − 1 zeros in(a, b) and satisfy the condition∫ bax ′2 dt =∫ bBy combining (1.3) with (1.4) one getsλ n (a, b) := min H(x) = H(x n ) =εx∈Γ n 1 + εaq(t)x 2 |x| 2ε dt. (1.4)∫ baq(t)x 2+2ε dt =ε1 + ε∫ bax ′2 (t) dt.Thus the characteristic number λ n (a, b) is (up to a multiplicative constant)a minimal value of the functional ∫ ba x′2 (t) dt over the set of all solutions ofthe boundary value problemx ′′ = −q(t)|x| 2ε x, x(a) = x(b) = 0, x(t) has n − 1 zeros in (a, b).We will call the characteristic numbers λ n by the Nehari’s numbers andthe respective solutions of the differential equation by the Nehari’s solutions.Remark 1. Nehari’s numbers λ n (a, b) are uniquely defined by the interval(a, b). In the work [2] Nehari mentioned that the theory could be developedmush easier if the associated Nehari’s solution be unique. It was shown theoreticallyin [3] that this is not the case. There exist equations of the type(1.1), which have more than one Nehari’s solution for certain a, b and n.2. Example: Nonuniqueness of the Nehari’s SolutionsWe construct the Emden - Fowler equation which possesses two Nehari’s solutions.

Characteristic Numbers of Emden – Fowler Equations 405In our considerations we use systematically the lemniscatic functions sl tand cl t which can be defined as solutions of the equation x ′′ = −2x 3 , subjectto the initial conditions x(0) = 0, x ′ (0) = 1 and x(0) = 1, x ′ (0) = 0respectively. Both functions are periodic with a minimal period of 4A, whereA =∫ 10ds√ . One may consult the paper [1] for more properties of these1 − s4functions. In many respects they behave like usual trigonometric functions.Equation. Consider equationtogether with the boundary conditionsx ′′ = −q(t) x 3 , t ∈ (−1, 1), (2.1)x(−1) = 0, x(1) = 0, x(t) > 0, t ∈ (−1, 1). (2.2)The coefficient q(t) is constructed as follows. Letq(t) =2(ξ(t)) 6,whereandξ(t) ={ξ1 (t), −1 ≤ t ≤ 0,ξ 2 (t), 0 ≤ t ≤ 1ξ 1 (t) = ht + η, −1 ≤ t ≤ 0,ξ 2 (t) = −ht + η, 0 ≤ t ≤ 1.Thus ξ(t) is a “Λ-shaped” piece-wise linear function, which depends on apositive valued parameter h, η depends on h as η = h + 1.Solutions. Our goal: we are looking for a solution (solutions) of the problem(2.1), (2.2).Consider two problemsk1 = −(ht + η) 6 x3 1, x 1 (−1) = 0, x 1 (0) = τ, x 1 (t) > 0, t ∈ (−1, 0);x ′′x ′′2 = − k(−ht + η) 6 x3 2 , x 2(0) = τ, x 2 (1) = 0, x 2 (t) > 0, t ∈ (0, 1),where τ > 0. Letx 1 (t) be a solution of the first equation of (2.3) in [−1; 0];x 2 (t) be a solution of the second equation of (2.3) in [0; 1].(2.3)

406 A. Gritsans, F. SadyrbaevThen the functionx(t) ={x1 (t), if −1 ≤ t ≤ 0,x 2 (t), if 0 ≤ t ≤ 1is a C 2 -solution of the problem (2.1), (2.2) if additionally the smoothnessconditionx ′ 1(0) = x ′ 2(0)is satisfied. The problem (2.3) can be explicitly resolved as( )x 1 (t, β 1 ) = β 1 21 (ht + η) · sl β 1 t + 121 ,ht + ηwhereandβ 1 = x ′ 1 (−1) > 0x 1 (0; β 1 ) = τ.The derivative is given by( )( )x ′ 1(t; β 1 ) = β 1 21 h · sl β 1 t + 1 −h + η21 + β 1ht + η ht + η · sl′ β 1 t + 121 .ht + ηSimilar formulas are valid for x 2 (t). Notice that x ′ 2 (1) = −β 2 < 0. In orderto get an explicit formula for a solution of the BVP (2.1), (2.2) one have tosolve a system of two equations with respect to (β 1 , β 2 )x 1 (0; β 1 ) = x 2 (0; β 2 ),x ′ 1(0; β 1 ) = x ′ 2(0; β 2 ).This system after replacements and simplifications looks as⎧ ( ) ( )⎪⎨ β 1 β 1 221 · sl 1= β 1 β 1 22η 2 · sl 2,η( ))( )⎪⎩ β 1 β 1 221 h · sl 1+ β 1(β 1 2η η · 1 sl′ = −β 1 β 1 22η2 h · sl 2− β 2(β 1 2η η · 2 sl′ η),where 0 < β 1 2 1η, β 2 1 2η< 2A. In new variables u := β 2 1 1η, v := β 1 2 2ηthe systemtakes the form{ u sl u = v sl v, 0 < u, v < 2A,hu sl u + u 2 sl ′ u = −hv sl v − v 2 sl ′ v, h > 0.(2.4)Notice that if a solution (ū, ¯v) of the system (2.4) exists, then a solution x(t)of the BVP (2.1), (2.2) can be constructed such thatx ′ (−1) = β 1 = ū 2 (h + 1) 2 , x ′ (1) = −β 2 = −¯v 2 (h + 1) 2 .

Characteristic Numbers of Emden – Fowler Equations 407Proposition 1. For h large the system (2.4) has exactly three solutions, whichhave the following characteristics.1. There exists a unique symmetric solution (u 0 , v 0 ), that is, u 0 = v 0 . Onehas that (u 0 , v 0 ) → (2A, 2A) as h → +∞.2. There exists a unique solution (u 1 , v 1 ) in the triangle {0 < u, v < 2A, v >u)} for h large. Moreover, (u 1 , v 1 ) → (0, 2A) as h → +∞.3. There exists a unique solution (u 2 , v 2 ) in the triangle {0 < u, v < 2A, v 1are depicted in the Figure 1. Notice that a set of zeros of Φ consists of thediagonal u = v and two symmetric branches.2A2.521.510.50.5 1 1.5 2 2.52AFigure 1. Zeros of Φ(u, v) (solid line) and Ψ(u, v) (dashed line), h = 2.Nehari’s numbers. The respective solutions of the boundary value problem(2.1), (2.2) look as shown in the picture.30252015105-1 -0.5 0.5 1Let H = 1 21∫−1x ′2 (t)dt. Denote by H sym and H asym the respective valuesof H for a symmetric solution (which is depicted by solid line), and for asymmetricsolutions (depicted by dashed lines). Notice that H(x) is the same forboth asymmetric solutions. The values of H sym and H asym are

408 A. Gritsans, F. Sadyrbaev( ( ) (H sym = 2 3 u 3 2 u 2 100 h+1 − sl′ u 2 10h+1sl( ( ) (H asym = 1 3 u 3 2 u 2 111 h+1 − sl′ u 2 11h+1sl( ( ) (+ 1 3 v 3 2 v 2 111 h+1 − sl′ v 2 11h+1slProposition 2.H symH asym−−−−−→h→+∞ 2.u 1 20h+1u 1 21h+1v 1 21h+1)),)))).Therefore for h large two asymmetric solutions are the Nehari’s solutions.References[1] A. Gritsans and F. Sadyrbaev. Lemniscatic functions in the theory of theEmden-Fowler differential equation. Mathematics. differential equations (Univ.of Latvia, Institute of Math. and Comp. Sci.), 3(3), 5 – 27, 2003. (electr. versionhttp://www.lumii.lv/sbornik1/Sbornik-2003-english.htm)[2] Z. Nehari. Characteristic values associated with a class of nonlinear second-orderdifferential equations. Acta Math., 105(3), 141 – 175, 1961.[3] F. Sadyrbaev. On solutions of the Emden-Fowler type equation. Differentialequations, 25(5), 799 – 805 (in Russian), 1989. English translation:F.Zh.Sadyrbaev, Solutions of the Emden-Fowler type equation., Differential Equations(ISSN:0012-2661, by Kluwer Academic/Plenum Publisher) 25, No.5, 560-565 (1989)