PRÜFER SUBSTITUTIONS ON A COUPLED SYSTEM INVOLVING THE p-laplacian

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1 Electronic Journal of Differential Equations, Vol. 23 (23), No. 23, pp. 9. ISSN: URL: or ftp ejde.ath.txstate.edu PRÜFER SUBSTITUTIONS ON A COUPLED SYSTEM INVOLVING THE p-laplacian WEI-CHUAN WANG Abstract. In this article, we eploy a odified Prüfer substitution acting on a coupled syste involving one-diensional p-laplacian equations. The basic properties for the initial valued proble and soe estiates are obtained. We also derive an analogous Sturian theory and give a reconstruction forula for the potential function.. Introduction There has been recently a lot of interest in the study of the p-laplacian eigenvalue proble p y + q y p 2 y = λ y p 2 y, y Ω =, where p > and q C(Ω), Ω R n. This is a quasilinear partial differential equation when p 2. The ost cited application is the highly viscid fluid flow (cf. Ladyzhenskaya [4] and Lions [6]). When p = 2, q and λ both vanish, it becoes the Laplacian equation. The p-laplacian operator has the originally physical eaning, and can also be treated as a generalization of the Laplacian operator. For the one-diensional case, the p-laplacian eigenvalue proble becoes, after scaling, (y (p ) ) = (p )(λ q(x))y (p ), (.) y() = y() =, (.2) where p >, f (p ) f p sgn f = f p 2 f, and q is a continuous function defined on [, ]. The following Stur-Liouville property for the one-diensional p- Laplacian operator is well-known now (cf. Binding & Drabek [2], Reichel & Walter [8], Walter [], etc.). Theore.. For (.)-(.2), there exists a sequence of eigenvalues {λ n } n= such that < λ < λ 2 < λ 3 < < λ n <..., and the eigenfunction corresponding to λ n has exactly n zeros in (, ). 2 Matheatics Subject Classification. 34A55, 34B24, 47A75. Key words and phrases. Coupled syste; p-laplacian; Prüfer substitution. c 23 Texas State University - San Marcos. Subitted July 2, 23. Published Septeber 25, 23.

2 2 W.-C. WANG EJDE-23/23 In this article, we consider the coupled one-diensional p-laplacian proble (u (x) (p ) ) + (p )λu(x) (p ) (p )q(x)v(x) (p ) =, (v (x) (p ) ) + (p )λv(x) (p ) + (p )q(x)u(x) (p ) =, with the initial conditions (.3) u() = v() =, u () = v () = λ /p, (.4) where λ is soe positive paraeter, p >, and q is a continuous function defined in R. When p = 2, (.3) reduces to u (x) + λu(x) q(x)v(x) =, v (x) + λv(x) + q(x)u(x) =, (.5) which is a linear coupled syste. One can treat (.5) as a steady state reaction diffusion odel. Define H(u, v) = λ 2 u2 λ 2 v2 q(x)uv. Then H = λu q(x)v, u H = λv + q(x)u. v Equation (.5) can be viewed as a siplest odel of diffusion systes with skewgradient structure (cf. [, 2]). Here we intend to study the existence of sign-changing solutions (or nodal solutions) of (.3)-(.4) and try to derive an analog of Theore.. Eploying the inforation of solutions, a reconstruction forula for q(x) is given. Such a procedure is called an inverse nodal proble. An inverse proble of this type was designated by McLaughlin [7] in 988. When one studies the inverse nodal proble of (.3)-(.4), an interesting observation arises. The asyptotic forula given in Theore.3 (see the following) coincides with the one of the classical Stur- Liouville eigenvalue proble y + w (x)y = µy, y() = y() = (cf. [7, 9, 5]). Besides, the Prüfer substitution is an efficient ethod in showing the oscillation property for solutions (cf. []). In this article we utilize a odified Prüfer substitution to treat this proble. Fortunately we can tackle the effect of the two coupled functions in (.3)-(.4), and obtain the detailed estiates of paraeters λ and nodal points. The following are our ain results. Theore.2. There exists a sequence of real paraeters {λ k } k= of the onediensional coupled syste (.3)-(.4), where N such that < λ < λ + < λ +2 < λ +3 <, and the corresponding solution u(x, λ k ) has exactly k zeros in (, ) for k. In particular, the solution pair {u(x, λ k ), v(x, λ k )} satisfies the following boundary condition u(, λ k ) = v(, λ k ) =, u(, λ k ) = for every k. Define the zero set (or nodal set) {x (k) i } k and the index i k (x) = ax{i : x (k) i where x (k) = and x (k) k q(x). i= of the solution u(x, λ k) to (.3)-(.4) x}. Let l (k) i = x (k) i+ x(k) i for i k, =. We obtain an asyptotic forula for the function

3 EJDE-23/23 PRÜFER SUBSTITUTIONS 3 Theore.3. Suppose that the above assuptions hold. forula for q(x) in (.3) is for all x [, ]. q(x) Then an asyptotic q(t)dt = li [p() p (l () i (x) )], (.6) We reark that in Theore.2, the right endpoint conditions v(, λ k ) also vanish when λ k tends to the infinity. Siultaneously, one can show an analogous Sturian theory for v(x, λ). Then the data coing fro v(x, λ k ) also ake the asyptotic forula (.6) valid. This article is organized as follows. After the introduction, we eploy a odified Prüfer substitution to show the local solution of the initial value proble (.3)-(.4) is unique and can be extended to the whole interval [, ]. In section 3, we derive several leas to coplete the proof of Theore.2. In section 4, soe detailed estiates and the proof of Theore.3 are given. 2. Preliinaries - A odified Prüfer substitution To discuss the existence and uniqueness of the local solution of (.3)-(.4). We need the following lea. Lea 2. ([, p. 8]). Let W C (I), x I and W (x ) =, where I is a copact interval containing x. Denote by W x the axiu of W in the interval fro x to x. Then W (x) K W x in I iplies W = for x x, x I. (2.) K Proposition 2.2. For any fixed λ R +, the proble (.3)-(.4) has a unique local solution which exists on an open interval I containing zero. Proof. Syste (.3) can be written as u = U (p ), U = (p )[qv (p ) λu (p ) ], v = V (p ), V = (p )[qu (p ) λv (p ) ], (2.2) with u() = v() = and U() = V () = λ /p, where p = p/(p ) is the conjugate exponent of p. Then the local existence of a solution is valid by the Cauchy-Peano theore. Now it suffices to prove the uniqueness. By (.4), we ay assue that λ /p 2 x < u(x), v(x) < 2λ/p x for x I. (2.3) Suppose that {u (x), v (x)} and {u 2 (x), v 2 (x)} are two distinct local solutions of (.3)-(.4). Without loss of generality, we assue that u (x) u 2 (x) and v (x) v 2 (x) in soe sall interval I which contains zero. By (2.2), for x I we have u (x) (p ) u 2(x) (p ) { = (p ) q(t)[v (t) (p ) v 2 (t) (p ) ]dt λ } [u (t) (p ) u 2 (t) (p ) ]dt,

4 4 W.-C. WANG EJDE-23/23 v (x) (p ) v 2(x) (p ) { = ( p) q(t)[u (t) (p ) u 2 (t) (p ) ]dt λ i.e., u (x) (p ) u 2(x) (p ) + v (x) (p ) v 2(x) (p ) = (p ) } [v (t) (p ) v 2 (t) (p ) ]dt ; (q(t) + λ)[v (t) (p ) v 2 (t) (p ) u (t) (p ) + u 2 (t) (p ) ]dt. (2.4) It follows fro the ean value theore, that for a and a 2 of the sae sign, a (p ) a (p ) 2 = (p )(a a 2 ) ā p 2, (2.5) where ā lies between a, a 2. Note that there exists soe c such that the left hand side of (2.4) is greater than or equal to c u (x) + v (x) u 2(x) v 2(x). On the other hand, by (2.3) the right hand side of (2.4) is less than or equal to (p + )( q x + λ) u (t) + v (t) u 2 (t) v 2 (t) 2λ /p t p 2 dt, where the notation x is defined as in Lea 2.. Now set W (x) = u (x) + v (x) u 2 (x) v 2 (x). By Lea 2., we can obtain that W (x) = in I. This proves the uniqueness of the local solution. Now we introduce a odified Prüfer substitution for the local solution {u(x), v(x)} using the generalized sine function S p (x). The S p (x) function is well known now (cf. [2, 3, 8]), and satisfies and S p (x) p + S p(x) p =, (2.6) (S p ) = S(p ) p S p = S(p ) p (S p) (p ) S p. (2.7) p 2 Thus one has S p (π p /2) =, and by (2.6), S p() =, S p(π p /2) =. Define u(x, λ) = R(x, λ)s p (λ /p θ(x, λ)), u (x, λ) = λ /p R(x, λ)s p(λ /p θ(x, λ)), (2.8) v(x, λ) = r(x, λ)s p (λ /p φ(x, λ)), v (x, λ) = λ /p r(x, λ)s p(λ /p φ(x, λ)). (2.9) Then, we obtain λr(x, λ) p = λ u(x, λ) p + u (x, λ) p, λr(x, λ) p = λ v(x, λ) p + v (x, λ) p, (2.) where R(x, λ) and r(x, λ) are the Prüfer aplitude functions; and θ(x, λ) and φ(x, λ) are the Prüfer phase angles of {u(x), v(x)}, respectively. By a direct coputation, we have the following lea. Lea 2.3. For the odified Prüfer substitution (2.8)-(2.9), one has θ (x, λ) = q(x) r(x, λ) ( λ R(x, λ) )p S p (λ /p θ(x, λ))s p (λ /p φ(x, λ)) (p ), (2.) R (x, λ) = q(x) λ p p r(x, λ) p R(x, λ) p 2 S p(λ /p φ(x, λ)) (p ) S p(λ /p θ(x, λ)), (2.2) φ (x, λ) = + q(x) λ) (R(x, λ r(x, λ) )p S p (λ /p φ(x, λ))s p (λ /p θ(x, λ)) (p ), (2.3) r (x, λ) = q(x) R(x, λ) p λ p p r(x, λ) p 2 S p(λ /p θ(x, λ)) (p ) S p(λ /p φ(x, λ)), (2.4)

5 EJDE-23/23 PRÜFER SUBSTITUTIONS 5 where = d dx. Proof. Here we prove the first two equations, and the rest is siilar. For the sake of siplicity, we drop the function variable λ in the proof. By (2.8), u (x) (p ) u(x) (p ) = λ p p S p (λ /p θ(x)) (p ). S p (λ /p θ(x)) (p ) Differentiating the above equation on both sides and applying (.3) and (2.7), we obtain [λ + u (x) u(x) p q(x)( v(x) u(x) )(p ) ] = λθ (x)[ + S p(λ /p θ(x)) S p (λ /p θ(x)) p ]. Multiplying by S p (λ /p θ(x)) p, fro (2.6), it follows that (2.) holds. Next, differentiate u(x) = R(x)S p (λ /p θ(x)) with respect to x and eploy (2.), to obtain (2.2). Applying Lea 2.3, we find that {u(x), v(x); λ} is a solution of (.3)-(.4) if and only if {θ(x), R(x), φ(x), r(x); λ} is a solution of (2.)-(2.4) coupled with the following conditions θ(, λ) = φ(, λ) =, and R(, λ) = r(, λ) =. (2.5) Next we derive soe properties for the phase and aplitude functions. Lea 2.4. (i) For x >, the aplitude functions satisfy that 2 exp[ c λ p p x] R(x, λ) p + r(x, λ) p 2 exp[c 2 λ p p x], (2.6) where c, c 2 are soe positive constants. (ii) For fixed x > and sufficiently large λ, we have Moreover, R(x,λ) r(x,λ) r(x, λ) = + o(). (2.7) R(x, λ) has the sae asyptotic estiate as in (2.7). Proof. (i) By assuption and (2.2) and (2.4), there exist soe positive constants c and c 2 such that c λ p p [R(x) p + r(x) p ] R(x) p 2 R (x) + r(x) p 2 r (x) c 2 λ p p [R(x) p + r(x) p ]. Solving the above differential inequality and applying the initial condition (2.5), we obtain the inequality (2.6). (ii) As in (i), there exists soe positive constant c 3 such that R(x)r (x) r(x)r (x) R(x) 2 Letting y(x) = r(x) R(x), we have Note that c 3 λ p R(x) p 2 p [ r(x) p 2 + r(x)p R(x) p ]. y (x) c 3 λ p p [y(x) 2 p + y(x) p ]. dy dx c 3λ p + y 2p 2 p ( y p 2 ); i.e., y p 2 dy + y 2p 2 c 3λ p p dx.

6 6 W.-C. WANG EJDE-23/23 Let z = y p and integrate the above inequality; we obtain i.e., So tan (y(x) p ) tan (y() p ) (p )c 3 λ p p x; < tan (y(x) p ) tan () + (p )c 3 λ p p x. as λ is sufficiently large. This copletes the proof. y(x) p + o() (2.8) Fro Proposition 2.2 and (2.6), we have the following property. Proposition 2.5. For any fixed λ R +, proble (.3)-(.4) has a unique solution which exists over the whole interval [, ]. 3. The Sturian property In this section, we first derive the following lea for the proof of Theore.2. Lea 3.. For λ >, the phase angle function θ(x, λ) satisfies the following properties. (i) θ(, λ) is continuous in λ and satisfies θ(, λ) =. (ii) If λ /p θ(x n, λ) = nπ p for soe x n (, ), then λ /p θ(x, λ) > nπ p for every x > x n. (iii) li λ λ/p θ(, λ) =. (3.) Proof. For (i), θ(, λ) is continuous in λ followed by the continuous dependence on paraeters. And θ(, λ) = is valid by (2.5). Also if λ /p θ(x n, λ) = nπ p for soe x n (, ), then by (2.) and Lea 2.4, we have θ (x n, λ) = >. (3.2) This proves (ii). For (iii), integrating (2.) over [, ] and applying (i), one obtains λ /p θ(, λ) = λ /p λ p p By (2.7), one has r(t, λ) q(t)( R(t, λ) )p S p (λ /p θ(t, λ))s p (λ /p φ(t, λ)) (p ) dt. λ /p θ(, λ) = λ /p + O( for sufficiently large λ. This copletes the proof. λ p ) (3.3) We reark that using (2.3) and Lea 2.4, one can apply the siilar arguents as in the above proof to obtain the conclusions in Lea 3. for the phase function φ(x, λ). Proof of Theore.2. By Lea 3., for every sufficiently large k N, there exists λ k > satisfies λ /p k θ(, λ k) = kπ p. This iplies that there exists N such that λ /p k θ(, λ k) = kπ p for every k. In this case, λ < λ + < < λ k+ <, and {θ(x, λ k ), φ(x, λ k )} k satisfy (2.)-(2.5). Hence, {u(x, λ k ), v(x, λ k )} k are solutions of (.3)-(.4) and satisfy u(, λ k ) = v(, λ k ) =, u(, λ k ) = for every k.

7 EJDE-23/23 This copletes the proof. PRÜFER SUBSTITUTIONS 7 4. Soe detailed estiates - Proof of Theore.3 Theore 4.. The paraeter λ of (.3)-(.4) satisfies as. λ /p = + p( ) p q(t)dt + o( ) (4.) p Proof. First, integrating (2.) over [, x], with the associated λ, θ(x, λ ) θ(, λ ) = x λ r(t, λ) q(t)( R(t, λ) )p S p (λ /p θ(t, λ))s p (λ /p φ(t, λ)) (p ) dt. (4.2) Letting x =, by Theore.2, Lea 2.4 and the initial condition (.4), one obtains = + O( ); (4.3) λ /p p p λ i.e., λ /p = + O( ). (4.4) p Again, integrating (2.) over [, x], with the associated λ, and applying (4.4), one gets λ /p θ(x, λ ) = x + O( ). (4.5) p Siilarly, fro (2.3), we obtain Hence, by (4.5), i.e., S p (λ /p φ(x, λ ) = x + O( ). (4.6) p λ /p θ(x, λ )) = S p ( x) + S p(π p x)o( p ) + o( ); (4.7) p S p (λ /p θ(x, λ )) = S p ( x) + o(). (4.8) And the sae asyptotic forula is true for S p (λ /p φ(x, λ )). Now substituting (4.8) into (4.2) and taking x =, one obtains λ /p = λ = pλ q(t) S p ( t) p dt + o( ) λ q(t)dt λ q(t)[ S p ( t) p ]dt + o( ), p λ (4.9) for sufficiently large. By a generalized Rieann-Lebesgue lea, the asyptotic estiate (4.) is valid. Hence, the asyptotic forula for λ is λ = ( ) p + Next we derive the asyptotic forula of the nodal length. q(t)dt + o(). (4.)

8 8 W.-C. WANG EJDE-23/23 Lea 4.2. For, the nodal length of the solution u(x, λ ) satisfies l () i = p p+ π p p q(t)dt + ( ) p () i+ x () i Proof. Letting λ = λ and integrating (2.) fro x () i π p = l () λ /p i () i+ x () i q(t) λ ( r(t, λ ) R(t, λ ) )p S p (λ /p By the paraeter estiates (4.), we have λ /p = p( ) p+ q(t) S p ( t) p dt + o( ). p+ (4.) to x () i+, we obtain θ(t, λ ))S p (λ /p φ(t, λ )) (p ) dt. (4.2) q(t)dt + o( ). (4.3) p+ Substituting (2.7), (4.5)-(4.6), (4.) and (4.3) into (4.2), one can obtain (4.). As in the proof of Lea 4.2, one can obtain the asyptotic estiate, for the nodal points x () i, x () i = i i p p+ π p p q(t)dt+ () x i ( ) p q(t) S p ( t) p dt+o( ), (4.4) p which show the existence of a dense subset of nodal points in [, ]. Proof of Theore.3. For any x (, ), write i (x) = i for the sake of siplicity. Recall an easy identity, (S p (t)s p(t) (p ) ) = p S p (t) p. It follows fro the ean value theore for integrals, (4.4), the above identity and a change of variables. Then, () i+ x () i q(t) S p ( t) p dt = q(x) = q(x) = q(x) πpx () i+ x () i πp πp Substituting (4.5) into (4.), one obtains p( ) p (l () i S p (σ) p dσ S p (σ) p dσ( + o()) [ p p (S p(σ)s p(σ) (p ) ) ]dσ( + o()) = q(x) p ( + o()). (4.5) ) = q(x) Therefore, the asyptotic forula is valid. q(t)dt + o(). (4.6) Acknowledgeents. This research was partially supported by the National Science Council of Taiwan, under contract NSC 2-25-M-57 -.

9 EJDE-23/23 PRÜFER SUBSTITUTIONS 9 References [] J. Benedikt, P. Girg; Prüfer transforation for the p-laplacian, Electronic J. Differential Equations, Conference 8 (2), -3. [2] P. Binding, P. Drabek; Stur Liouville theory for the p-laplacian, Studia Scientiaru Matheaticaru Hungarica 4 (23), [3] A. Elbert; A half-linear second order differential equation, Colloqia atheatica Societatis Jónos Bolyai, 3. Qualitiative Theory of Differential Equations, Szeged (Hungary) (979), [4] O. A. Ladyzhenskaya; The Matheatical Theory of Viscous Incopressible Flow, 2nd ed., Gordon and Breach (New York), 969. [5] C. K. Law, C. L. Shen, C. F. Yang; The inverse nodal proble on the soothness of the potential function, Inverse Probles 5 (999), ; Errata, 7 (2), [6] J. L. Lions; Quelques ethodes de resolution des problees aux liites non lineaires, Dunod (Paris) 969. [7] J. R. McLaughlin; Inverse spectral theory using nodal points as data a uniqueness result, J. Diff. Eqns. 73 (988), [8] W. Reichel, W. Walter; Stur-Liouville type probles for the p-laplacian under asyptotic non-resonance conditions, J. Differential Equations 56 (999), 5-7. [9] C. L. Shen; On the nodal sets of the eigenfunctions of the string equation, SIAM J. Math. Anal. 6 (988), [] W. Walter; Stur-Liouville theory for the radial p-operator, Math. Z. 227 (998), [] E. Yanagida; Mini-axiizers for reaction-diffusion systes with skew-gradient structure, J. Differential Equations 79 (22), [2] E. Yanagida; Standing pulse solutions in reaction-diffusion systes with skew-gradient structure, J. Dyna. Differential Equations 4 (22), Wei-Chuan Wang Center for General Education, National Queoy University, Kinen, 892, Taiwan E-ail address: wangwc72@gail.co