Extension of Díaz-Saá's inequality in RN and application to a system of p-Laplacian

Publ. Mat. 46 (2002), 473–488 EXTENSION OF D´ IAZ-SA´ A’S INEQUALITY IN RNAND APPLICATION TO A SYSTEM OF p-LAPLACIAN Karim Cha¨ ıb Abstract The purpose of this paper is to extend the D´ıaz-Sa´a’s inequality for the unbounded domains as RN: RN−∆pu up−1+∆pv vp−1(up−vp)dx ≥0 with ∆pu= div |∇u|p−2∇u. The proof is based on the Picone’s identity which is very useful in problems involving p-Laplacian. In a second part, we study some properties of the first eigenvalue for a system of p-Laplacian. We use D´ıaz-Sa´a’s inequality to prove uniqueness and Egorov’s theorem for the isolation. These results generalize J. Fleckinger, R. F. Man´asevich, N. M. Stavrakakis and F. de Th´elin’s work [9] for the first property and A. Anane’s one for the isolation. In a well-known paper [7]H.Br´ezis and L. Oswald obtain some necessary and sufficient conditions for the existence and uniqueness of a positive solution of the equation: −∆u=f(x, u)inΩ,u=0on∂Ω when Ω is a bounded open set in RN. Later J. I. D´ıaz and J. E. Sa´a[8] extend these results to the case of an equation involving the p-Laplacian, ∆pu= div |∇u|p−2∇u(p>1). In their proof a fundamental ingredient is the so called D´ıaz-Sa´a inequality: (1) Ω−∆pz1 zp−1 1 +∆pz2 zp−1 2(zp 1−zp 2)dx ≥0 where ∆pu= div(|∇u|p−2∇u). 2000 Mathematics Subject Classification. 35J65 (35P30, 35B05). Key words. D´ıaz-Sa´a inequality, Picone identity, p-Laplacian, elliptic system, nonlinear eigenvalue, unbounded domain. 474 K. Cha¨ ıb This inequality expresses the monotony of the operator w→ −∆pw 1 p w p−1 p .It has been proved in [8] under the following hypotheses: For i=1,2, zi∈L∞(Ω) ∩W1,p(Ω) such that zi≥0a.e.onΩand ∆pzi∈L∞(Ω); For i=j,i, j =1,2, zi zj∈L∞(Ω). In the case of bounded domains, the Hopf’s Maximum Principle gives them the hypothesis zi zj∈L∞(Ω) which is necessary in the original proof of (1). But when we consider unbounded domains such as RNthis condition is not verified in general. Here, under some adapted hypotheses for solutions of elliptic problems, we establish an inequality of D´ıaz-Sa´a type which remains true in whole RN. When we say the hypotheses are well adapted, it means that they naturally appear in weighted problems involving the p-Laplacian on RN. In the second part, we apply this D´ıaz-Sa´a type inequality to obtain an uniqueness result of the first eigenvalue for a system of p-Laplacian in RN (Sλ)       −∆pu=λb(x)|u|α|v|βvx∈RN −∆qv=λb(x)|u|α|v|βux∈RN lim|x|→+∞u(x)=0=lim |x|→+∞v(x). This work follows an A. Anane’s paper [3] where he studies the case of one equation and J. Fleckinger, R. F. Man´asevich, N. M. Stavrakakis and F. de Th´elin [9] where a system quite different is considered. In [9], a local method is presented to control a possible blowing-up of the quotient zi zjin RN. But, in the integration by parts, some integrals on the boundary appear which make the demonstration quite boring. Finally, following A. Anane’s article [3], we will conclude this paper by showing the isolation of the first eigenvalue of the system (Sλ). As in this article, we will use the Egorov’s theorem. This property was not proved in [9]; besides it seems that the method used here cannot be applied for their system because intuitively the sign of umust depend on the sign of vand vice versa. Extension of D´ ıaz-Sa´ a’s Inequality in RN475 1. D´ıaz-Sa´a inequality Denote by D1,p(RN) the closure of C∞ 0(RN) for the Lp(RN) norm of the gradient ||u||p D1,p(RN)=RN |∇u|pdx. The following results are the consequences of the principal theorems which will be presented later. Proposition 1 (D´ıaz-Sa´a inequality on RNin the case 1<p<N).For i=1,2, let zi∈D1,p(RN)such that zi≥0(≡ 0) and differentiable. Then we have RN−∆pz1 zp−1 1 +∆pz2 zp−1 2(zp 1−zp 2)dx ≥0 if we assume that ∆pzi zp−1 i ∈LN p(RN)∩L∞ loc(RN)for i=1,2. If we have equality then there exists a constant Csuch that z1=Cz2. Proposition 2 (D´ıaz-Sa´aonRNin the case p≥N).For i=1,2, let zi∈W1,p(RN)such that zi≥0(≡ 0) and differentiable. Then the assertions above remain true if we assume that for N=p, there exists some s>1such that ∆pzi zp−1 i ∈Ls(RN)∩L∞ loc(RN) or for N<p, ∆pzi zp−1 i ∈L1(RN)∩L∞ loc(RN) with i=1,2. The proof of the theorems that we will present here needs Picone’s identity for the p-Laplacian described for example in [2] by W. Allegretto and Y. X. Huang. Picone’s result is an equality true almost everywhere which avoids some integration problems when we work in unbounded open sets. Now, we present a Picone’s identity for the p-Laplacian quite more general than the W. Allegretto and Y. X. Huang’s one because we only assume a sign condition for one of the two equations. 476 K. Cha¨ ıb Proposition 3 (Generalized Picone’s identity for the p-Laplacian).Let u,vdifferentiable and v>0a. e. on Ωa subset of RN. Note L(u, v)=|∇u|p+(p−1)|u|p vp|∇v|p−p|u|p−2u vp−1∇u·∇v|∇v|p−2a. e. on Ω R(u, v)=|∇u|p−∇|u|p vp−1|∇v|p−2∇va. e. on Ω. Then L(u, v)=R(u, v)≥0. Moreover, L(u, v)=0a. e. on Ωif and only if ∇u v=0a. e. on Ω. Proof of Proposition 3: By a simple calculation, we show that R(u, v)= L(u, v), because ∇|u|p vp−1=p|u|p−2u∇u vp−1−(p−1)|u|p∇v vp. To prove positivity, we observe that |u|p−2u vp−1∇u·∇v|∇v|p−2≤|u|p−1 vp−1|∇v|p−1|∇u|(2) and by Young’s inequality p|u|p−1 vp−1|∇v|p−1|∇u|≤(p−1)|u|p vp|∇v|p+|∇u|p.(3) Hence by (2) and (3) |∇u|p+(p−1)|u|p vp|∇v|p−p|u|p−2u vp−1∇u·∇v|∇v|p−2=L(u, v)≥0. Moreover, if L(u, v) = 0 then by (2) and (3) |∇u|p+(p−1)|u|p vp|∇v|p−p|u|p−1 vp−1|∇v|p−1|∇u|= 0 a. e. on Ω.(4) We define N:= x∈Ω such that |u| v|∇v|=0 ⊂Ω. On N, by the equation (4) we have |u| v|∇v|=|∇u|=0 ae. onN and hence u v∇v=∇u= 0 a. e. on N.(5) Extension of D´ ıaz-Sa´ a’s Inequality in RN477 On Nc, we note Q:= |∇u| |∇v||u| v and substituting in (4) we obtain Qp−pQ +p−1=0 iffQ= 1 because p>1, i. e. |∇u|=|u| v|∇v|a. e. on Nc.(6) Using (6) in L(u, v) = 0, it follows that |∇u|p+(p−1)|∇u|p−2|∇u|2−p∇u·∇vu v|∇u|p−2= 0 a. e. on Nc, so ∇u·∇u−∇vu v= 0 a. e. on Nc, and u v∇v=∇uae.onNcbecause |∇u|=|u| v|∇v|.(7) Indeed ∇vcannot be perpendicular to u v∇v−∇ubecause it would signify that ∇uis the hypotenuse of a right triangle with edges u v∇vand u v∇v− ∇u, that is not possible because u v|∇v|=|∇u|. By (5) and (7) we have u v∇v=∇ua. e. on Ω and finally, ∇u v= 0 a. e. on Ω. We can easily remark that in Picone’s identity, we can take any set Ω, for example non connected and unbounded. Now, we can establish the following theorem whose principal argument of the proof is the above identity. Theorem 1. For 1<p<N.LetΦin D1,p(RN)and z≥0(≡ 0) in D1,p(RN)both differentiable. Then we have RN |∇Φ|pdx ≥RN−∆pz zp−1|Φ|pdx, if we assume that ∆pz zp−1∈LN p(RN)∩L∞ loc(RN). Moreover, in the equality case there exists Csuch that z=CΦon RN. Proof of Theorem 1: First we remark that z∈D1,p(RN) is a non trivial solution of the problem −∆pv=−∆pz zp−1vp−1 v≥0in RN. 478 K. Cha¨ ıb Seeing that ∆pz zp−1∈L∞ loc(RN), we can apply V´azquez’s Strong Maximum Principle [16] to prove that z>0onRN. Moreover, we use P. Tolksdorf’s regularity theorem [15] to show that for all r>0, there exists α(r)>0 such that z∈C1,α(Br). In particular for Ω0a bounded domain of RN, there exists α0>0 such that z∈C1,α0(Ω0). Let (Φn)n∈Na sequence of functions in C∞ 0(RN) such that (Φn)n∈N converges to Φ in D1,p(RN). We apply Picone’s identity to the functions Φnand z 0≤RN L(Φn,z)dx ≤RN R(Φn,z)dx ≤RN |∇Φn|pdx −RN ∇|Φn|p zp−1|∇z|p−2∇z dx. But Φn∈C∞ 0(RN) and z>0 then |Φn|p zp−1is an admissible function test, integrating by parts we obtain 0≤RN |∇Φn|pdx +RN ∆pz zp−1|Φn|pdx. (Φn)n∈Nconverges to Φ in D1,p(RN), so we have that (Φn)n∈Nconverges to Φ in Lp∗(RN) and (|Φn|p)n∈Nconverges to |Φ|pin Lp∗ p(RN). Consequently, RN ∆pz zp−1(|Φn|p−|Φ|p)dx≤    ∆pz zp−1   L N p(RN) |Φn|p−|Φ|p L p∗ p(RN)    tends to 0 .(8) And the result follows: RN |∇Φ|pdx ≥RN−∆pz zp−1|Φ|pdx.(9) We now consider the equality case RN |∇Φ|pdx =RN−∆pz zp−1|Φ|pdx. Let Ω0a bounded domain in RNand (Φn)n∈Ndefined as before. 0≤Ω0 L(Φn,z)dx ≤RN L(Φn,z)dx ≤RN |∇Φn|pdx +RN ∆pz zp−1|Φn|pdx tends to 0 when n→∞. Extension of D´ ıaz-Sa´ a’s Inequality in RN479 But Ω0L(Φn,z)dx converges to Ω0L(Φ,z)dx because Ω0is bounded and z∈C1,α0(Ω0). So L(Φ,z)=0a.e.onΩ 0, the set Ω0being taken arbitrary in RNwe can conclude that L(Φ,z) = 0 a. e. on RN. And, by Picone’s identity (Proposition 3), there exists C>0 such that Φ = Cz. Theorem 2. For p≥N.LetΦin W1,p(RN)and z≥0(≡ 0) in W1,p(RN)both differentiable. Then we have RN |∇Φ|pdx ≥RN−∆pz zp−1|Φ|pdx, if we assume that for p=N, there exists some s>1such that ∆pz zp−1∈Ls(RN)∩L∞ loc(RN) or for p>N, ∆pz zp−1∈L1(RN)∩L∞ loc(RN). Moreover, if the above integral is zero then there exists Csuch that z=CΦon RN. The proof of this theorem is quite similar to Theorem 1. We solve the problem of convergence in (8) by the particular embeddings of the space W1,p(RN) in the case p≥N(see [6]): If p=N,W1,p(RN) is continuously embedded in any Lq(RN) where q∈[p, +∞); If p>N,W1,p(RN) is continuously embedded in L∞(RN). The existence result of a solution in W1,p(RN) for the p-Laplacian where p≥Nhas been established by W. Allegretto and Y. X. Huang in [1]. Proofs of Proposition 1 and Proposition 2 (D´ıaz-Sa´a inequality): We only have to apply the preceding results for the couple of the functions (z1,z 2) and we have RN |∇z1|pdx ≥RN−∆pz2 zp−1 2zp 1dx and RN−∆pz1 zp−1 1 +∆pz2 zp−1 2zp 1dx ≥0. 480 K. Cha¨ ıb Doing the same work for the couple of the functions (z2,z 1) and adding the two inequalities obtained we arrive to the expected result. We note that in D´ıaz-Sa´a inequality, we assume functions are positives so we needn’t the absolute values as in the theorems. Following a discussion with P. Tak´aˇc, it appears that D´ıaz-Sa´a inequality in RNand both theorems can be proved using very carefully J. I. D´ıaz and J. E. Sa´a’s method. For this, we have to remark first that if this inequality is true for positive functions Φ it remains true for functions Φ changing sign [11, “Chain rule”, Lemma 7.6]. After that, we just have to prove the convexity of the application w−→ |∇w1 p|p and compute the directional derivative of J(w)=Ω|∇w1 p|pdx which is formally: J(w)v=1 pΩ |∇w1 p|p−2∇w1 p∇(w1 p−1v)dx. To have more details on this method we can see J. Fleckinger, J. Hern´andez, P. Tak´aˇc and F. de Th´elin’s article [10]. But that proof needs a good attention in the computation because some terms, in particular the derivative of J, have to be defined correctly, and we think that our proof using Picone’s identity is easier. 2. Property of first eigenvalue for a system of p-Laplacian In this second part, we study some properties of the first eigenvalue for a potential system of p-Laplacian: (Sλ)       −∆pu=λb(x)|u|α|v|βvx∈RN −∆qv=λb(x)|u|α|v|βux∈RN lim|x|→+∞u(x)=0=lim |x|→+∞v(x). And we assume: (H1) N>p>1,N>q>1,α≥0,β≥0,α+1 p+β+1 q=1 and α+β+2<N. (H2) b∈C0,γ loc (RN) with γ∈(0,1), b∈LN α+β+2 (RN)∩L∞(RN) and b≥0(≡ 0). For a start, we establish existence of a first eigenvalue for (Sλ) and the regularity of the associated eigenfunctions. Extension of D´ ıaz-Sa´ a’s Inequality in RN481 Theorem 3. We suppose that Hypotheses (H1) and (H2) are satisfied. (i) System (Sλ)admits a first eigenvalue λ1which is positive and defined by λ1= inf Γα+1 pRN |∇u|pdx +β+1 qRN |∇v|qdx where Γ=x∈RNsuch that RN b(x)|u|α|v|βuv dx =1 ; (ii) If (u, v)is a couple of eigenfunctions solution of (Sλ1)then for all r>0,u∈C1,ρ(Br)and v∈C1,γ(Br)where ρ=ρ(r)>0and γ=γ(r)>0; (iii) There exists a couple of eigenfunctions solution of (Sλ1)which are positive on RN. The proof of this theorem is more or less the same as J. Fleckinger, R. F. Man´asevich, N. M. Stavrakakis and F. de Th´elin’s one in [9]for the system: −∆pu=λb(x)|u|α−1u|v|β+1 x∈RN −∆qv=λb(x)|u|α+1|v|β−1vx∈RN. The approach done in [9] is standard because their problem as (Sλ)is variational. We present now an uniqueness and isolation result for the first eigenvalue. The proof is an interesting application of Theorem 1 and is simpler than in [9]. Theorem 4. We suppose that Hypotheses (H1) and (H2) are satisfied. (i) In the set of continuous functions, the dimension of eigenspace corresponding to principal eigenvalue λ1is 1; (ii) λ1is the only one eigenvalue of (Sλ1)which corresponds to a constant sign eigenvector; (iii) λ1is isolated i. e. there exists #>0such that for all λ∈(λ1,λ 1+#] the system (Sλ)has no solution. 488 K. Cha¨ ıb [6] H. Br´ ezis,“Analyse fonctionnelle. Th´eorie et applications”, Collection Math´ematiques Appliqu´ees pour la Maˆıtrise, Masson, Paris, 1983. [7] H. Br´ ezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10(1) (1986), 55–64. [8] J. I. D´ ıaz and J. E. Sa´ a, Existence et unicit´e de solutions positives pour certaines ´equations elliptiques quasilin´eaires, C. R. Acad. Sci. Paris S´er. I Math. 305(12) (1987), 521–524. [9] J. Fleckinger, R. F. Man´ asevich, N. M. 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Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51(1) (1984), 126–150. [16] J. L. V´ azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12(3) (1984), 191–202. Math´ematiques pour l’Industrie et la Physique UMR CNRS 5640 Universit´e Paul Sabatier UFR MIG 118, route de Narbonne 31062 Toulouse Cedex 4 France E-mail address:[email protected] Primera versi´o rebuda el 14 de desembre de 2001, darrera versi´o rebuda el 23 de gener de 2002.