THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS D. A. KANDILAKIS, M. pptx

THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS D. A. KANDILAKIS, M. MAGIROPOULOS, AND N. B. ZOGRAPHOPOULOS Received 12 October 2004 and in revised form 21 January 2005 We study the properties of the positive principal eigenvalue and the corresponding eigenspaces of two quasilinear elliptic systems under nonlinear boundary conditions. We prove that this eigenvalue is simple, unique up to positive eigenfunctions for both sys- tems, and isolated for one of them. 1. Introduction Let Ω be an unbounded domain in R N , N ≥2, with a noncompact and smooth boundary ∂Ω. In this paper we prove certain properties of the principal eigenvalue of the following quasilinear elliptic systems −∆ p u =λa(x)|u| p−2 u + λb(x)|u| α−1 |v| β+1 u,inΩ, −∆ q v = λd(x)|v| q−2 v + λb(x)|u| α+1 |v| β−1 v,inΩ, (1.1) −∆ p u = λa(x)|u| p−2 u + λb(x)|u| α |v| β v in Ω, −∆ q v = λd(x)|v| q−2 v + λb(x)|u| α |v| β u in Ω (1.2) satisfying the nonlinear boundary conditions |∇u| p−2 ∇u ·η + c 1 (x)|u| p−2 u = 0on∂Ω, |∇v| q−2 ∇v ·η + c 2 (x)|v| q−2 v = 0on∂Ω, (1.3) where η is the unit outward normal vector on ∂Ω. As it will be clear later, under condition (H1), 1 <p,q<N, α,β ≥ 0and α +1 p + β +1 q = 1, α +1< pq ∗ N , β +1< p ∗ q N , (1.4) systems (1.1), (1.2) are in fact nonlinear eigenvalue problems. Our procedure here will be based on the proper space setting provided in [14], (see Section 2). In this section, we also state the assumptions on the coefficient functions. Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 307–321 DOI: 10.1155/BVP.2005.307 308 The first eigenvalue of p-Laplacian systems Problems of such a type arise in a variety of applications, for example, non-Newtonian fluids, reaction-diffusion problems, theory of superconductors, biology, and so forth, (see [2, 15] and the references therein). As a consequence, there are many works treating non- linear systems from different points of view, for example, [4, 7, 9, 11, 13]. Properties of the pr i ncipal eigenvalue are of prime interest since for example they are closely associated with the dynamics of the associated evolution equations (e.g., global bi- furcation, stability) or with the description of the solution set of corresponding perturbed problems (e.g., [17]). These properties are: existence, positivit y, simplicity, uniqueness up to eigenfunctions which do not change sign and isolation, which hold in the case of the Lapla- cian operator in a bounded domain. It is well known that these properties also hold for the p-Laplacian scalar eigenvalue problem (in both bounded and unbounded domains) and were recently obtained in [12] under nonlinear boundary conditions while the case of some (p,q)-Laplacian systems with Dirichlet boundary conditions was also successfully treated in [1, 10, 16, 18]. Note that we discuss the case of a potential (or gradient) system,whichisarestriction. However, this is in some sense natural because the aforementioned properties of the prin- cipal eigenvalue are stronger than in the scalar equation case; for example the principal eigenvalue of the system is the only eigenvalue which a dmits a nonnegative eigenfunc- tion in the sense that both components do not change sign. It is also remarkable that the associated “eigenspaces” are generally not linear subspaces. Starting with the system (1.1)–(1.3), we proceed as follows: in Section 2,wegivethe space setting and the assumptions on the coefficient functions. In Section 3, using the compactness of the corresponding operators we prove the existence and positivity of λ 1 and we state a regularity result based on the iter ative procedure of [5]. In Section 4, we prove the simplicity and the uniqueness up to positive (componentwise) eigenfunc- tions. This is done by using the Picone’s identity (see [1]). Finally, in Section 5,weprove Theorem 2.3 by establishing the connection between the two systems with respect to ex- istence and simplicity of the common principal eigenvalue λ 1 as well as the regularity of the eigenfunctions. In addition, we show that λ 1 is isolated for the system (1.2)-(1.3). 2. Preliminaries and statement of the results Let Ω be an unbounded domain in R N , N ≥ 2, with a noncompact and smooth bound- ary ∂Ω.Form>0andr ∈ (1,+∞)letw m (x) = 1/(1 + |x|) m and assume that the space L r (w m ,Ω):={u :  Ω (1/(1 + |x|) m )|u| r < +∞} is supplied with the norm u w m ,r =   Ω 1  1+|x|  m |u| r  1/r . (2.1) We require the following hypotheses: (H1) 1 <p,q<N, α,β ≥ 0with(α +1)/p+(β +1)/q = 1, α +1<pq ∗ /N and β +1< p ∗ q/N. Here p ∗ and q ∗ are the critical Sobolev exponents defined by p ∗ = pN N − p , q ∗ = qN N −q . (2.2) D. A. Kandilakis et al. 309 (H2) (i) There exists positive constants α 1 , A 1 with α 1 ∈ (p +((β +1)(N − p)/q ∗ ),N)and 0 <a(x) ≤A 1 w α 1 (x)a.e.inΩ, (2.3) (ii) there exists positive constants α 2 , D 1 with α 2 ∈ (q +((α +1)(N −q)/p ∗ ),N)and 0 <d(x) ≤D 1 w α 2 (x)a.e.inΩ, (2.4) (iii) m{x ∈Ω : b(x) > 0} > 0and 0 ≤ b(x) ≤ B 1 w s (x)a.e.inΩ, (2.5) where B 1 > 0ands ∈ (max{p, q},N). (H3) c 1 (·)andc 2 (·) are positive and continuous functions defined on R N with k 1 w p−1 (x) ≤c 1 (x) ≤K 1 w p−1 (x), l 1 w q−1 (x) ≤c 2 (x) ≤L 1 w q−1 (x), (2.6) for some positive constants k 1 , K 1 , l 1 , L 1 . Let C ∞ δ (Ω) be the space of C ∞ 0 (R N )-functions restricted to Ω.Form ∈ (1,+∞), the weighted Sobolev space E m is the completion of C ∞ δ (Ω)inthenorm |||u||| m =   Ω |∇u| m +  Ω 1 (1 + |x|) m |u| m  1/m . (2.7) By [14, Lemma 2] we see that if c(·) is a positive continuous function defined on R n then the norm u 1,m =   Ω |∇u| m +  ∂Ω c(x)|u| m  1/m (2.8) is equivalent to ||| · ||| m . The proof of the following lemma is also provided in [14]. Lemma 2.1. (i) If p ≤r ≤ pN N − p , N>α≥ N −r N − p p , (2.9) 310 The first eigenvalue of p-Laplacian systems then the embedding E ⊆ L r (w α ,Ω) is continuous. If the upper bound for r in the first in- equality and the lower bound in the second is strict, then the embedding is compact. (ii) If p ≤m ≤ p(N −1) N − p , N>β≥ N −1 −m N − p p , (2.10) then the embedding E ⊆ L m (w β ,∂Ω) is continuous. If the upper bounds for m are str ict, then the embedding is compact. It is natural to consider our systems on the space E =E p ×E q supplied with the norm   (u,v)   pq =u 1,p + v 1,q . (2.11) We now define the functionals Φ, I, J : E → R as follows: Φ(u,v) = α +1 p  Ω |∇u| p + α +1 p  ∂Ω c 1 (x)|u| p + β +1 q  Ω |∇v| p + β +1 q  ∂Ω c 2 |v| q −λ α +1 p  Ω a(x)|u| p −λ β +1 q  Ω d(x)|v| q −λ  Ω b(x)|u| α+1 |v| β+1 , I(u,v) = α +1 p  Ω |∇u| p + β +1 q  Ω |∇v| p + α +1 p  ∂Ω c 1 (x)|u| p + β +1 q  ∂Ω c 2 |v| q , J(u,v) = α +1 p  Ω a(x)|u| p + β +1 q  Ω d(x)|v| q +  Ω b(x)|u| α+1 |v| β+1 . (2.12) In view of (H1)–(H3), the functionals Φ, I, J are well defined and continuously differen- tiable on E.Byaweak solution of (1.1)wemeananelement(u 0 ,v 0 )ofE whichisacritical point of the functional Φ. The main results of this work are the following theorems. Theorem 2.2. Let Ω be an unbounded domain in R N , N ≥ 2, with a noncompact and smooth boundary ∂Ω. Assume that the hypotheses (H1), (H2), and (H3) hold. Then (i) System (1.1)–(1.3) admits a positive principal eigenvalue λ 1 given by λ 1 = inf  I(u,v):J(u,v) =1  . (2.13) Each component of the associated normalized eigenfunction (u 1 ,v 1 ) is positive in Ω and of class C 1,δ loc (Ω) for some δ ∈(0,1). (ii) The set of eigenfunctions corresponding to λ 1 forms a one dimensional manifold E 1 ⊆ E defined by E 1 =  cu 1 ,±|c| p/q v 1  : c ∈ R\{0}  . (2.14) Furthermore, a componentwise positive eigenfunction always corresponds to λ 1 . D. A. Kandilakis et al. 311 Theorem 2.3. Assume that the hypotheses of Theorem 2.2 hold. (a) System (1.2)-(1.3) shares the same positive principal eigenvalue λ 1 and the same prop- erties of the associated eigenfunctions with (1.1)–(1.3). (b) The set of eigenfunctions corresponding to λ 1 forms a one dimensional manifold E 2 ⊆ E defined by E 2 =  ±  cu 1 ,c p/q v 1  : c>0  . (2.15) (c) λ 1 is isolated for the system (1.2)-(1.3), in the sense that there exists η>0 such that the interval (0,λ 1 + η) does not contain any other eigenvalue than λ 1 . 3. Existence and regularity In this section, we prove the existence of a positive principal eigenvalue and the regularity of the corresponding eigenfunctions for the system (1.1)–(1.3). Existence. The operators I, J are continuously Fr ´ echet differentiable, I is coercive on E ∩ {J(u,v) ≤ const}, J is compact and J  (u,v) = 0onlyat(u,v) = 0. So the assumptions of Theorem 6.3.2 in [3] are fulfilled implying the existence of a pri ncipal eigenvalue λ 1 , satisfying λ 1 = inf J(u,v)=1 I(u,v). (3.1) Moreover, if (u 1 ,v 1 ) is a minimizer of (2.13)then(|u 1 |,|v 1 |) should be also a minimizer. Hence, we may assume that there exists an eigenfunction (u 1 ,v 1 ) corresponding to λ 1 , such that u 1 ≥ 0andv 1 ≥ 0, a.e. in Ω. Regularity. We show first that w p u 1 and w q v 1 are essentially bounded in Ω.Tothatpur- pose define u M (x):= min{u 1 (x), M}. It is clear that u kp+1 M ∈ E p ,fork ≥0. Multiplying the first equation of (1.1)byu kp+1 M and integrating over Ω,weget  Ω   ∇u 1   p−2 ∇u 1 ·∇  u kp+1 M  dx +  ∂Ω c 1 (x) u p−1 1 u kp+1 M dx ≤ λ 1  Ω a(x) u (k+1)p 1 dx + λ 1  Ω b(x)v β+1 1 u 1 kp+α+1 dx. (3.2) Note that  Ω   ∇u 1   p−2 ∇u 1 ·∇  u kp+1 M  dx = (kp+1)  Ω   ∇u M   p u kp M dx = kp+1 (k +1) p  Ω   ∇u k+1 M   p dx, (3.3) so since (kp+1)/(k +1) p ≤ 1, then  Ω   ∇u 1   p−2 ∇u 1 ·∇  u kp+1 M  dx +  ∂Ω c 1 (x) u p−1 1 u kp+1 M dx ≥ c 3 kp+1 (k +1) p   Ω 1 (1 + |x|) p u (k+1)p ∗ M dx  p/p ∗ (3.4) 312 The first eigenvalue of p-Laplacian systems due to Lemma 2.1(i) and (2.8). Let t = p(1 −(β +1/q ∗ )) −1 , which is less than p ∗ because of H(1). Then H(2)(i) and H ¨ older inequality imply that  Ω a(x) u (k+1)p 1 dx ≤ A 1  Ω 1  1+|x|  α 1 u (k+1)p 1 dx = A 1  Ω 1  1+|x|  α 1 −p 2 /t u (k+1)p 1  1+|x|  p 2 /t dx ≤ A 1   Ω 1 (1 + |x|) (tα 1 −p 2 )/(t−p) dx  (t−p)/t   Ω 1 (1 + |x|) p u (k+1)t 1 dx  p/t (3.5) (observe that (tα 1 − p 2 )/(t − p) >N by H(2)(i)). Also, because of (H1), we may assume that  Ω b(x)v β+1 1 u kp+α+1 1 dx ≤  Ω b(x)v β+1 1 u (k+1)p 1 dx, (3.6) otherwise we could consider u M (x) =    min  u 1 (x), M  , u 1 (x) ≥1, 0, u 1 (x) < 1 (3.7) as a test function. So  Ω b(x)v β+1 1 u (k+1)p 1 dx ≤ B 1  Ω 1  1+|x|  s v β+1 1 u (k+1)p 1 dx = B 1  Ω v β+1 1  1+|x|  s(1−(p/t)) u (k+1)p 1  1+|x|  s(p/t) dx ≤ B 1   Ω v (β+1)(t/t−p) 1 (1 + |x|) s dx  (t−p)/t   Ω u (k+1)t 1 (1 + |x|) s dx  p/t ≤ B 1   Ω 1 (1 + |x|) q v q ∗ 1 dx  (t−p)/t   Ω 1 (1 + |x|) p u (k+1)t 1 dx  p/t , (3.8) by H(2)(iii). On combining (3.2)–(3.8), we conclude that   u M   w p ,(k+1)p ∗ ≤ C 1/(k+1)  k +1 (kp+1) 1/p  1/(k+1)   u 1   w p ,(k+1)t , (3.9) where C is independent of M and k. We now follow the same steps as in the proof of [8, Theorem 2] or [5, Lemma 3.2]. Let k 1 = (p ∗ /t) −1. Since (k 1 p +1)/(k 1 +1) p ≤ 1, we can D. A. Kandilakis et al. 313 choose k = k 1 in (3.9)toget   u M   w p ,(k 1 +1)p ∗ ≤ C 1/(k 1 +1)  k 1 +1  k 1 p +1  1/p  1/(k 1 +1)   u 1   w p , p ∗ , (3.10) while by letting M →∞we obtain that   u 1   w p ,(k 1 +1)p ∗ ≤ C 1/(k 1 +1)  k 1 +1  k 1 p +1  1/p  1/(k 1 +1)   u 1   w p , p ∗ . (3.11) Hence, u 1 ∈ L (k 1 +1)p ∗ (w p ,Ω). Note that if k ≥ k 1 then (kp+1)/(k +1) p ≤ 1. Choosing in (1.1) k = k 2 with (k 2 +1)t = (k 1 +1)p ∗ , that is, k 2 = (p ∗ /t) 2 −1, we have   u 1   w p ,(k 2 +1)p ∗ ≤ C 1/(k 1 +1)  k 2 +1  k 2 p +1  1/p  1/(k 2 +1)   u 1   w p ,(k 1 +1)p ∗ . (3.12) Hence, u 1 ∈ L (k 2 +1)p ∗ (w p ,Ω). Proceeding by induction we arrive at   u 1   w p ,(k n +1)p ∗ ≤ C 1/(k n +1)  k n +1  k n p +1  1/p  1/(k n +1)   u 1   w p ,(k n−1 +1)p ∗ . (3.13) From (3.10)and(3.13)weconcludethat   u 1   w p ,(k n +1)p ∗ ≤ C  n i=1 1/(k i +1) n  i=1  k i +1  k i p +1  1/p  1/(k i +1)   u 1   w p , p ∗ = C  n i=1 1/(k i +1) n  i=1     k i +1  k i p +1  1/p  1/ √ k i +1    1/ √ k i +1   u 1   w p , p ∗ . (3.14) Since (y +1/(yp+1) 1/p ) 1/ √ y+1 > 1fory>0, and lim y→∞ (y +1/(yp+1) 1/p ) 1/ √ y+1 = 1, there exists K>1 independent of k n such that   u 1   w p ,(k n +1)p ∗ ≤ C  n i=1 1/(k i +1) K  n i=1 1/ √ k i +1   u 1   w p , p ∗ , (3.15) where 1/(k i +1)= (t/p ∗ ) i and 1/  k i +1= (  t/p ∗ ) i . Letting now n →∞we conclude that   u 1   w p ,∞ ≤ c   u 1   w p , p ∗ , (3.16) for some positive constant c.By[8], u 1 ∈ C 1,δ loc (Ω). Similarly v 1 ∈ C 1,δ loc (Ω). Finally, we notice that for the principal eigenvalue, each component of an eigenfunc- tion is either positive or negative in Ω due to the Harnack inequality [8]andifweassume that u 1 (x 0 ) = 0forsomex 0 ∈ ∂Ω then by [19, Theorem 5] we have |∇u 1 (x 0 )| p−2 ∇u 1 (x 0 ) · η(x 0 ) < 0, contradicting (1.3). Thus u 1 > 0(oru 1 < 0) on Ω. Similarly v 1 > 0(orv 1 < 0) on Ω. 314 The first eigenvalue of p-Laplacian systems 4. The eigenfunctions corresponding to λ 1 In this section, we complete the proof of Theorem 2.2 establishing the simplicity of λ 1 . More precisely, we show that if (u 2 ,v 2 ) is another pair of eigenfunctions corresponding to λ 1 , then there exists c ∈ R\{0} such that (u 2 ,v 2 ) = (cu 1 ,±|c| p/q v 1 ). To that end, we employ a technique similar to the one described in [1]. Namely, we will prove that if (w 1 ,w 2 ) is a positive on ¯ Ω solution of the problem −∆ p u ≤ λa(x)|u| p−2 u + λb(x)|u| α−1 |v| β+1 u,inΩ, −∆ q v ≤ λd(x)|v| q−2 v + λb(x)|u| α+1 |v| β−1 v,inΩ, |∇u| p−2 ∇u ·η + c 1 (x)|u| p−2 u = 0, on ∂Ω, |∇v| q−2 ∇v ·η + c 2 (x)|v| q−2 v = 0, on ∂Ω, (4.1) for some λ>0, and (w  1 ,w  2 )isapositiveon ¯ Ω solution of −∆ p u ≥ λa(x)|u| p−2 u + λb(x)|u| α−1 |v| β+1 u in Ω, −∆ q v ≥ λd(x)|v| q−2 v + λb(x)|u| α+1 |v| β−1 v in Ω, |∇u| p−2 ∇u ·η + c 1 (x)|u| p−2 u = 0on∂Ω, |∇v| q−2 ∇v ·η + c 2 (x)|v| q−2 v = 0on∂Ω (4.2) then (w  1 ,w  2 ) = (cw 1 ,c p/q w 2 ) for a constant c>0. Let ϕ ∈C ∞ δ (Ω), ϕ>0, then ϕ p /(w  1 ) p−1 ∈ E p . By Picone’s identity [1], we get 0 ≤  Ω R  ϕ,w  1  =  Ω |∇ϕ| p −  Ω ∇   ϕ p  w  1  p−1   ·   ∇w  1   p ∇w  1 =  Ω |∇ϕ| p +  Ω ϕ p  w  1  p−1 ∆ p w  1 −  ∂Ω ϕ p  w  1  p−1   ∇w  1   p ∇w  1 ·η ≤  Ω |∇ϕ| p −λ  Ω ϕ p  w  1  p−1  a(x)  w  1  p−1 + b(x)  w  1  α  w  2  β+1  −  ∂Ω ϕ p  w  1  p−1   ∇w  1   p ∇w  1 ·η =  Ω |∇ϕ| p −λ  Ω a(x) ϕ p  w  1  p−1  w  1  p−1 −λ  Ω b(x)ϕ p  w  1  α  w  1  p−1  w  2  β+1 −  ∂Ω ϕ p  w  1  p−1   ∇w  1   p ∇w  1 ·η, (4.3) while the boundary conditions imply that 0 ≤  Ω |∇ϕ| p −λ  Ω a(x) ϕ p  w  1  p−1  w  1  p−1 −λ  Ω b(x)ϕ p  w  1  α  w  1  p−1  w  2  β+1 +  ∂Ω c 1 (x) ϕ p  w  1  p−1  w  1  p−1 . (4.4) D. A. Kandilakis et al. 315 Letting ϕ → w 1 in E p we obtain 0 ≤  Ω   ∇w 1   p −λ  Ω a(x) w p 1 −λ  Ω b(x)w p 1  w  1  α−p+1  w  2  β+1 +  ∂Ω c 1 (x)w p 1 . (4.5) Note also that  Ω   ∇w 1   p +  ∂Ω c 1 (x) w p 1 ≤ λ  Ω a(x) w p 1 + λ  Ω b(x)w α+1 1 w β+1 2 . (4.6) On combining (4.5)and(4.6)weget 0 ≤  Ω b(x)  w α+1 1 w β+1 2 −w p 1  w  1  α−p+1  w  2  β+1  . (4.7) Similarly, 0 ≤  Ω b(x)  w α+1 1 w β+1 2 −w q 2  w  2  β+1−q  w  1  α+1  . (4.8) We can now work as in Theorem 2.7 in [1] to get the desired result. Returning to our problem, we obtain E 1 as the set of eigenfunctions corresponding to λ 1 , simply by applying the previous result to the case of our system with λ = λ 1 ,and taking (u 1 ,v 1 )insteadof(w 1 ,w 2 ). One has now to combine the fact that the nonnegative solutions are given by (cu 1 ,c p/q v 1 ), c>0, with the trivial observation that if (u,v)isan eigenfunction then (−u,v), (u, −v), (−u,−v) are also eigenfunctions. The same technique can be used for proving that nonnegative solutions in Ω cor- respond only to the first eigenvalue. Assume, for instance, that there exists an eigen- pair (λ ∗ ,u 2 ,v 2 )fortheproblem(1.1)suchthatλ ∗ >λ 1 , u 2 ≥ 0andv 2 ≥ 0, a.e. in Ω. Then (u 1 ,v 1 ) is a solution of (1.2)withλ = λ ∗ and (u 2 ,v 2 )isasolutionof(1.3). Then (u 2 ,v 2 ) = (cu 1 ,c p/q v 1 ), for some c>0, which is a contradiction. 5. The second system In this section, we present the proof of Theorem 2.3. (a) Since for positive solutions systems (1.1)and(1.2) coincide, we deduce that (λ 1 ,u 1 , v 1 ) is also an eigenpair for the system (1.2). Assume that there exists another nontrivial eigenpair (λ ∗ ,u ∗ ,v ∗ )of(1.2), such that 0 <λ ∗ <λ 1 . Then the following equalit y must be satisfied λ ∗ = I  u ∗ ,v ∗  ˜ J  u ∗ ,v ∗  , (5.1) with ˜ J(u ∗ ,v ∗ ) > 0, where ˜ J(·,·)isdefinedby ˜ J(u,v) = α +1 p  Ω a(x)|u| p + β +1 q  Ω d(x)|v| q +  Ω b(x)|u| α |v| β uv. (5.2) 316 The first eigenvalue of p-Laplacian systems Note that ˜ J(·,·)isalsocompact.From(5.1)wealsohavethat λ ∗ = I  u ∗ ,v ∗  J  u ∗ ,v ∗  J  u ∗ ,v ∗  ˜ J  u ∗ ,v ∗  ≥ I  u ∗ ,v ∗  J  u ∗ ,v ∗  , (5.3) since J  u ∗ ,v ∗  ˜ J  u ∗ ,v ∗  ≥ 1. (5.4) Normalizing (u ∗ ,v ∗ )bysetting u ∗ =:   u ∗    J  u ∗ ,v ∗  1/p , v ∗ =:   v ∗    J  u ∗ ,v ∗  1/q , (5.5) we get t hat I  u ∗ ,v ∗  = I  u ∗ ,v ∗  J  u ∗ ,v ∗  , (5.6) J  u ∗ ,v ∗  = 1. (5.7) From relations (5.3)–(5.7)weconcludethat λ ∗ ≥ I  u ∗ ,v ∗  J  u ∗ ,v ∗  = I  u ∗ ,v ∗  ≥ λ 1 , (5.8) a contradiction. (b) Let (u,v)beaneigenfunctionof(1.2) corresponding to λ 1 .Ifuv ≥0 a .e., then the right-hand sides of (1.1)and(1.2) are equal, so (u,v) is an eigenfunction of (1.1), and we are done. On the other hand we cannot have uv < 0 on a set of positive measure, because then λ 1 = I(u,v) ˜ J(u,v) > I(u,v) J(u,v) = λ 1 , (5.9) a contradiction. (c) Suppose that there exists a sequence of eigenpairs (λ n ,u n ,v n )of(1.2)withλ n → λ 1 . By the variational characterization of λ 1 we know that λ n ≥ λ 1 .Sowemayassumethat λ n ∈ (λ 1 ,λ 1 + η)foreachn ∈ N. Furthermore, without loss of generality, we may assume that (u n ,v n )=1, for all n ∈N. Hence, there exists ( ˜ u, ˜ v) ∈E such that (u n ,v n )  ( ˜ u, ˜ v). The simplicity of λ 1 implies that ( ˜ u, ˜ v) = (u 1 ,v 1 )or( ˜ u, ˜ v) = (−u 1 ,−v 1 ). Let us suppose [...]... contradicts (5.25) and the conclusion follows The proof is complete Acknowledgments The first author is supported by the Greek Ministry of Education at the University of the Aegean under the Project EPEAEK II-PYTHAGORAS with title “Theoretical and Numerical Study of Evolutionary and Stationary PDEs Arising as Mathematical Models in Physics and Industry.” The third author acknowledges support by the Operational... lin, Principal eigenvalues for a e some quasilinear elliptic equations on RN , Adv Differential Equations 2 (1997), no 6, 981– 1003 Y Hu, Multiplicity of solutions for some semi-linear elliptic systems with non-homogeneous boundary conditions, Appl Anal 83 (2004), no 1, 77–96 S Mart´nez and J D Rossi, Isolation and simplicity for the first eigenvalue of the p-Laplacian ı with a nonlinear boundary condition,... (EPEAEK II) and particularly by the PYTHAGORAS Program no 68/831 of the Ministry of Education of the Hellenic Republic References [1] [2] [3] W Allegretto and Y X Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal 32 (1998), no 7, 819–830 A Bensoussan and J Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences, vol 151,... q−2 α β vm um vn − vm dx vm vn − vm dx + Ω b(x) um α β vm um dx (5.11) From (5.10) and (5.11), by using the compactness of the operator J˜ and the monotonicity of the p-Laplacian operator [6], we obtain Ω Ω ∇u n p ∇vn q dx −→ dx −→ Ω Ω ∇u 1 p ∇v1 q dx, (5.12) dx Exploiting the strict convexity of E p and Eq we get that (un ,vn ) → (u1 ,v1 ) in E For a fixed n ∈ N and for every (φ,ψ) ∈ E we have Ω ∇u... 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Mathematical Sciences, vol 151, Springer, Berlin, 2002 M S Berger, Nonlinearity and Functional Analysis, Pure and Applied Mathematics, Academic Press, New York, 1977 D A Kandilakis et al 321 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] J F Bonder and J D Rossi, Existence results for the p-Laplacian with nonlinear boundary conditions, J Math Anal Appl 263 (2001), no 1, 195–223 P... elliptic systems, Differential Integral Equations 16 (2003), no 12, 1519–1531 P Dr´ bek, A Kufner, and F Nicolosi, Quasilinear Elliptic Equations with Degenerations and a Singularities, de Gruyter Series in Nonlinear Analysis and Applications, vol 5, Walter de Gruyter, Berlin, 1997 J Fern´ ndez Bonder, J P Pinasco, and J D Rossi, Existence results for Hamiltonian elliptic a systems with nonlinear boundary conditions, ... 0, n (5.25) where the constant c16 is independent of λn and un Since un → u1 in E p and vn → v1 in ∗ ∗ Eq , we have that un → u1 in L p (w1 ,Ω) and vn → v1 in Lq (w2 ,Ω) Consequently, un → u1 ∗ ∗ in L p (w1 ,BK (0)) and vn → v1 in Lq (w2 ,BK (0)) By Egorov’s theorem we conclude that un (x) (vn (x)) converges uniformly to u1 (x) (resp., v1 (x)) on Brε (0) with the exception of a set with arbitrarily... 1, 195–223 P Dr´ bek, S El Manouni, and A Touzani, Existence and regularity of solutions for nonlinear a elliptic systems in RN , Atti Sem Mat Fis Univ Modena 50 (2002), no 1, 161–172 P Dr´ bek and J Hern´ ndez, Existence and uniqueness of positive solutions for some quasilinear a a elliptic problems, Nonlinear Anal Ser A: Theory Methods 44 (2001), no 2, 189–204 P Dr´ bek, N M Stavrakakis, and N B... λn ∂Ω p−2 c1 (x) un Ω vn ψ dx + λn Ω α b(x) un p−2 c1 (x) vn un φ dx β vn vn φ dx, (5.13) vn ψ dx b(x) un α β vn un ψ dx, 318 The first eigenvalue of p-Laplacian systems − − Let ᐁn =: {x ∈ Ω : un (x) n − − − − 0, with Ωn = ᐁn ∪ ᐂn Denoting by u− = min{0,un } and vn = min{0,vn } and choosing n − φ ≡ u− and ψ ≡ vn , it follows that n ᐁ− ∇u . problem (in both bounded and unbounded domains) and were recently obtained in [12] under nonlinear boundary conditions while the case of some (p,q)-Laplacian systems with Dirichlet boundary conditions. THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS D. A. KANDILAKIS, M. MAGIROPOULOS, AND N. B. ZOGRAPHOPOULOS Received 12 October 2004 and in revised form 21 January. University of the Aegean under the Project EPEAEK II-PYTHAGORAS with title “Theoretical and Numerical Study of Evolutionary and Stationar y PDEs Arising as Mathematical Mod- els in Physics and Industry.”