PICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 Electronic Journal of Differential Equations, Vol (2013), No. 143, pp ISSN: URL: or ftp ejde.math.txstate.edu PICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS JAROSLAV JAROŠ Abstract. We established a Picone identit for sstems of nonlinear partial differential equations of first-order. With the help of this formula, we obtain qualitative results such as an integral inequalit of Wirtinger tpe and the existence of zeros for the first components of solutions in a given bounded domain. 1. Introduction The purpose of this article is to establish a Picone-tpe identit for the nonlinear differential sstem u = ua(x) + v q 2 v, div v = C(x) u p 2 u D(x) v, (1.1) where p > 1 is a constant, q = p/(p 1) is its conjugate, A(x), D(x) C(; R n ), C(x) C(, R), = diag{b 1 (x),..., B n (x)} is a diagonal matrix with the positive entries defined and continuous in a bounded domain R n with a piecewise smooth boundar and u and v denote real- and vector-valued functions of x = (x 1,..., x n ), respectivel, which are continuousl differentiable in their domains of definition. Here div and are the usual divergence and nabla operators, is the Euclidean length of a vector in R n and the dot is used to denote the scalar product of two vectors in R n. If the special case A(x) 0 in, the sstem (1.1) is equivalent with the secondorder half-linear partial differential equation where div ( P (x) u p 2 u ) + R(x) u p 2 u + Q(x) u p 2 u = 0, (1.2) P (x) = 1 p, R(x) = 1 p D(x), Q(x) = C(x). If the coefficient P (x) is a scalar function, then (1.2) reduces to the equation studied in 9 where the following theorem was proved Mathematics Subject Classification. 35B05. Ke words and phrases. Nonlinear differential sstem; Picone identit; Wirtinger inequalit. c 2013 Texas State Universit - San Marcos. Submitted March 31, Published June 22,

2 2 J. JAROŠ EJDE-2013/143 Theorem 1.1. Suppose that there exists a nontrivial function C 1 (; R) such that = 0 on and M P (x) 1 R(x) p P (x) p Q(x) p dx 0. (1.3) Then ever solution u of (1.2) must have a zero in. The proof of the above theorem was based on an identit which sas that if u is a solution of (1.2) satisfing u(x) 0 in and C 1 (; R) is not identicall zero in, then div p P (x) u p 2 u p 2 u u = P (x) { pp (x) R(x) p Q(x) p P (x) pp (x) R(x) p (1.4) p ( pp (x) R(x)) u u p 2 u u + (p 1) u u p}. Moreover, if D(x) 0 in, then (1.2) reduces to div P (x) u p 2 u + Q(x) u p 2 u = 0. (1.5) Identities of Picone tpe for (1.5) (or its special case where P (x) 1 in ) were established b several authors including Allegretto 1, Dunninger 3, Kusano et al 6 and Yoshida 10 who obtained a variet of qualitative results based on these formulas. For an extension of Picone s identit to the case of pseudo-p-laplacian and anisotropic p-laplacian see Došlý 2 and Fišnarová et al 4, respectivel. As was demonstrated in Mařík 7, an alternative approach to (1.2) and (1.5) can be based upon Riccati-tpe equations and inequalities. While comparison and oscillation theor for equations of the tpe (1.2) and (1.5) is well-developed, there appears to be little known for general sstems such as (1.1), particularl in the case where A(x) 0 or A(x) D(x) in (for some results concerning the case p = 2 see Wong 11). The purpose of this article is to generalize Picone s identit for nonlinear partial differential sstems of the form (1.1) and illustrate its applications b deriving Wirtinger-tpe inequalities formulated in terms of solutions of the sstem (1.1) and obtaining results about the existence and distribution of zeros of the first component of the solution of (1.1). Our results involve an arbitrar continuous vector-valued function G(x) and particular choices of this function lead to different integral inequalities or criteria for the existence of zeros of first components of solutions of (1.1). The are new even when the are specialized to the case of the damped equation (1.2). This article is organized as follows. In Section 2, the desired generalization of Picone s formula to nonlinear sstem (1.1) is derived and some particular cases of this new identit are discussed. Section 3 contains some applications of the basic formula which include the integral inequalities of the Wirtinger tpe and theorems about the existence of zeros for components of solutions of sstem (1.1). 2. Picone s identit Define ϕ p (s) := s p 2 s, s R, and Φ p (ξ) := ξ p 2 ξ, ξ R n. Let ξ, η R n and B be a diagonal matrix with positive entries B i, i = 1,..., n. Define the form

3 EJDE-2013/143 PICONE S IDENTITY FOR A SYSTEM OF PDES 3 F B b F B ξ, η = ξ B 1 p Φ p (ξ) pξ B 1 p Φ p (η) + (p 1)η B 1 p Φ p (η). (2.1) where B 1 p = diag{b 1 p 1,..., Bn 1 p }. The next lemma establishes the generalization of Picone s identit for the nonlinear sstem (1.1). Lemma 2.1. Let (u, v) be a solution of (1.1) with u(x) 0 in. Then, for an C 1 (; R) and G C(, R n ), div p v = G(x) 1 p Φ p ( G(x)) C(x) p p ( A(x) G(x) ) + D(x) A(x) p v (2.2) F B G(x), Φ q (v)/u. Proof. If (u, v) is a solution of (1.1) with u(x) 0 and C 1 (, R), then a direct computation ields div p v ϕ p () p = p v (p 1) u p u.v + p div v. (2.3) Using (1.1), adding and subtracting the terms G(x) 1 p Φ p ( G(x)) and pg(x). 1 p Φ p ( u Φ q(v)) (=pg(x) ϕp() ϕ v) p(u) on the right-hand side of (2.3), we obtain div p v = G(x) 1 p Φ p ( G(x)) C(x) p p ( A(x) G(x) ) + D(x) A(x) p v { G(x) 1 p ( ) Φ p G(x) which is the desired identit (2.2). p G(x) 1 p Φ p ( u Φ q(v) ) + (p 1) u Φ q(v) 1 p Φ p ( u Φ q(v) )}, Remark 2.2. If we put (x) 1 in (2.2) and denote w = v/, then (2.2) becomes the generalized Riccati equation div w + pg(x) + (p 1)Φ q (w) 1 p Φ p ( Φq (w) ) + p ( A(x) G(x) ) + D(x) A(x) w + C(x) = 0. (2.4) Moreover, if G(x) 0 and is a scalar function, then the Riccati-tpe equation (2.4) reduces to div w + (p 1) w q + (p 1)A(x) + D(x) w + C(x) = 0. (2.5) In the particular case where A(x) 0 and 1 in, Equation (2.5) has been emploed b Mařík 8 as a tool for studing oscillator properties of damped half-linear PDEs of the form (1.2).

4 4 J. JAROŠ EJDE-2013/143 Remark 2.3. If G(x) 0 in, then (2.2) simplifies to div p v = 1 p Φ p ( ) C(x) p (p 1)A(x) + D(x) p v F B, u Φ q(v). (2.6) In the particular case p = 2, the identit (2.6) reduces to the formula used (implicitl) b Wong 11 in establishing an integral inequalit of the Wirtinger tpe and comparison theorems based on this inequalit for the linear sstem and its Sturmian minorant u = ua(x) + v, div v = C(x)u D(x) v, (2.7) = a(x) + b(x)z, div z = c(x) d(x) z, (2.8) where the coefficient functions satisf the same assumptions as above with the onl difference that because of the linearit of the problem the matrices b(x) and are not necessaril diagonal, but are allowed to be an continuous smmetric and positive definite matrices. The choice G(x) = (1/q)A(x) + (1/p)D(x) in (2.2) ields div p v ( A(x) = + D(x) ) ( A(x) 1 p Φ p ( + D(x) )) q p q p ( A(x) C(x) p F B + D(x) ), q p u Φ q(v). (2.9) Under the further restriction A(x) 0 and B 1 (x) = = B n (x) =: in, the identit (2.9) reduces to the following Yoshida s formula for partial differential equations with p-gradient terms (see9, Theorem 8.3.1): div p v = 1 p p D(x) p C(x) p F B p D(x), u Φ q(v) (2.10) which was used in proving Theorem Applications In what follows, for simplicit we restrict our considerations to the isotropic case where B 1 (x) = = B n (x) =:. In this special case it follows from 6, Lemma 2.1 that the form F B ξ, η defined b (2.1) is positive semi-definite and the equalit in F B ξ, η 0 occurs if and onl if ξ = η. As the first application of the identit (2.2) we establish an inequalit of the Wirtinger tpe. Theorem 3.1. If there exists a solution (u, v) of (1.1) such that u(x) 0 in and p ( A(x) G(x) ) v + D(x) A(x) 0 (3.1)

5 EJDE-2013/143 PICONE S IDENTITY FOR A SYSTEM OF PDES 5 in, then the inequalit J := 1 p G(x) p C(x) p dx 0 (3.2) holds for an nontrivial function C 1 (; R) such that = 0 on. Moreover, if p(a G) + D A v/ 0 in, then equalit in (3.2) occurs if and onl if (x) is a solution of = G(x) + Φ q(v) u. (3.3) Proof. Assume that (1.1) has a solution (u, v) with u(x) 0 in which satisfies (3.1). Let (x) be a nontrivial continuousl differentiable real-valued function such that = 0 on. Integrating (2.2) on and using the divergence theorem we get ( ) 0 = J p A(x) G(x) + D(x) A(x) F B G(x), Φ q (v)/udx. p v dx Since the form F B is positive semi-definite and the condition (3.1) holds, we conclude that 0 J as claimed. Clearl, if p(a G)+D A v/ 0 in, then the equalit holds in (3.2) if and onl if F B G(x), Φ q (v)/u 0 in which is equivalent with the condition (3.3). As an immediate consequence of the above theorem we have the following result. Corollar 3.2. Let (u, v) be a solution of (1.1) such that u(x) 0 in and p ( A(x) G(x) ) v + D(x) A(x) 0 (3.4) in. Then, for ever nontrivial C 1 (; R) such that = 0 on, the inequalit (3.2) is valid. Moreover, the equalit holds in (3.2) if and onl if ( = u) ( ) G(x) A(x) (3.5) u in. Proof. We need to show onl that (3.5) is equivalent to (3.3). Using the first equation in (1.1), it is easil seen that G(x) + u Φ q(v) = u u + A(x) G(x) from which the assertion follows. ( = u + u) A(x) G(x) ( = u u) u( ) G(x) A(x), In the case where G(x) A(x) D(x) in, condition (3.4) is triviall satisfied and inequalit (3.2) reduces to 1 p A(x) p C(x) p dx 0.

6 6 J. JAROŠ EJDE-2013/143 Clearl, in this special case the equalit in (3.2) occurs if and onl if (x) is a constant multiple of u(x). Another choice of G(x) which guarantees the satisfaction of (3.4) is G(x) = The last result specializes as follows. (p 1)A(x) + D(x). p Corollar 3.3. If (u, v) is a solution of (1.1) with u(x) 0 in and a nontrivial C 1 (; R) is such that = 0 on, then J = 1 p (p 1)A(x) + D(x) p C(x) p dx 0. (3.6) p Furthermore, equalit in (3.6) occurs if and onl if (x) = Ku(x) exp{f(x)} on (3.7) for some constant K 0 and some continuous function f(x). Proof. It suffices to prove (3.7). If (3.5) holds, then from 5, Lemma 2.3 if follows that there exists a continuous function f(x) such that (x) is proportional to u(x) exp{f(x)}. The proof is complete. The above result can be reformulated as the following theorem which generalizes 9, Theorem Corollar 3.4. If for some nontrivial C 1 -function (x) defined on and satisfing = 0 on, the condition J = 1 p (p 1)A(x) + D(x) p C(x) p dx 0 (3.8) p holds, then for an solution (u, v) of (1.1) the first component u(x) must have a zero in. References 1 W. Allegretto, Y. X. Huang; A Picone s identit for the p-laplacian and applications, Nonlinear Anal. 32 (1998), O. Došlý; The Picone identit for a class of partial differential equations, Math. Bohem. 127 (2002), D. R. Dunninger; A Sturm comparison theorem for some degenerate quasilinear elliptic operators, Boll. Un. Mat. Ital. A (7) 9 (1995), S. Fišnarová, R. Mařík; Generalized Picone and Riccati inequalities for half-linear differential operators with arbitrar elliptic matrices, Electron. J. Differential Equations 2010 (2010), No. 111, J. Jaroš, T. Kusano, N. Yoshida; Picone-tpe inequalities for nonlinear elliptic equations with first order terms and their applications, J. Ineq. Appl. Vol (2006), T. Kusano, J. Jaroš, N. Yoshida; A Picone-tpe identit and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Anal. 40 (2000), R. Mařík; Oscillation criteria for PDE with p-laplacian via the Riccati technique, J. Math. Anal. Appl. 248 (2000), R. Mařík; Riccati-tpe inequalit and oscillation criteria for a half-linear PDE with damping, Electron. J. Differential Equations 2004 (2004), no. 11, N. Yoshida; Oscillation Theor of Partial Differential Equations, World Scientific, Singapore, Hackensack, London, 2008.

7 EJDE-2013/143 PICONE S IDENTITY FOR A SYSTEM OF PDES 7 10 N. Yoshida; A Picone identit for half-linear elliptic equations and its application to oscillation theor, Nonlinear Anal. 71 (2009), P. K. Wong; A Sturmian theorem for first order partial differential equations, Trans. Amer. Math. Soc. 166 (1972), Jaroslav Jaroš Department of Mathematical Analsis and Numerical Mathematics, Facult of Mathematics, Phsics and Informatics, Comenius Universit, Bratislava, Slovakia address: jaros@fmph.uniba.sk