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Finite-dimensional approximation for a class of elliptic obstacle problems
Nonlinear Analysis 52 (2003) 1745 – 1754
www.elsevier.com/locate/na
Finite-dimensional approximation for a class of elliptic obstacle problems Yisheng Huanga;∗ , Yuying Zhoub a Department
of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, People’s Republic of China
b Department
Received 27 November 2001; accepted 13 May 2002
Abstract In this paper we study the /nite-dimensional approximation for a class of elliptic obstacle problems involving unbounded domains by using the Galerkin method. The convergence of the approximation is deduced. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Obstacle problem; Finite-dimensional approximation; Galerkin cone
1. Introduction and preliminaries p ∞ Let be an arbitrary open set in RN , D1; 0 () the completion of C0 () in the p 1=p and 1 ¡ p ¡ N . Let P(x) be a sign-changing function with norm { |∇u| d x} P = P + − P − (as usual, P ± (x) := max{± P(x); 0}). We assume that P satis/es the hypothesis:
< lim x→y |x−y|p P2 (x) = (P) P ∈ L1 (); P + =P1 +P2 = 0, P1 ∈ LN=p () for every y ∈ , 0 and lim |x|→∞ |x|p P2 (x) = 0.
x∈
x∈
Denote E the space {u ∈ D01; p (): u E ¡ ∞}, where u pE = − by p P |u| d x. Let K be the cone {u ∈ E: u(x) ¿ 0 a.e. for x ∈ }.
|∇u|p d x +
Supported by NNSF of China (10161010) and NSF of Yunnan Province (1999A0003R).
∗
Corresponding author. Tel.: +86-512-625-27063; fax: +86-512-6522-2691. E-mail address: [email protected] (Y. Huang).
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 9 2 - 4
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Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
In this paper we are concerned with the following obstacle problem: /nd u ∈ K such that f(x; u)(v − u) (1) |∇u|p−2 ∇u · ∇(v − u) ¿ P(x)|u|p−2 u(v − u) +
∀u ∈ K, where is the real parameter and f(x; t) is the CarathEeodory function satisfying some conditions to be speci/ed later. The solvability of variational inequalities like (1), relating to an elliptic operator, has been studied extensively in recent years by various methods, such as bifurcation, variational, super-subsolution approaches and the topological /xed point method, see for example [7–9,11] and the references therein. In this paper we study the solvability of (1) in another way, which is the Galerkin method, i.e., the /nite-dimensional approximation of problem (1). This method has been used by Isac [3,4] to consider the solvability of a class of abstract nonlinear complementarity problems and in [5] to the solvability of equations involving the p-Laplacian. However, the arguments used in [5] cannot be applied to our situation even if K = E (in this case (1) is in fact an equation) since may not be bounded. Motivated by Isac [3,4], we /rst study the /nite-dimensional approximation for an abstract variational inequality involving an operator satisfying the condition (S)+ (see the de/nition below), and deduce the convergence of the approximation, especially our results extend and improve corresponding ones in [3,4]; then we prove our main theorem by applying the preceding results. Before presenting our main result, we make some assumptions on f in problem (1). Suppose that f is a CarathEeodory function satisfying (f1 ) there exist R ¿ 0 and 0 : 0 6 0 ¡ 1 such that f(x; s)s 6 0 P(x)|s|p holds ∀x ∈ and ∀s: |s| ¿ R, where 1 will be given in the following; ∗ (f2 ) |f(x; s)| 6 (x) + (x)|s|p−1 for a.e. (x; s) ∈ × R, where 0 6 (x) ∈ L(p ) () with p∗ = Np=(N − p) (the critical Sobolev exponent), (p∗ ) = p∗ =(p∗ − 1), 0 6 (x) ∈ LN=p () ∩ L∞ loc (). It was shown in [12], under the above condition (P), that the eigenvalue problem u ∈ D01; p () has the least eigenvalue 1 := inf { |∇u|p d x: u ∈ E; P|u|p d x = 1} satisfying 1 ¿ 0, where p u := div(|∇u|p−2 ∇u) is the p-Laplacian. Let {wn } be a sequence of linear independent elements of E such that n∈N Vn =E, where Vn = span{w1 ; w2 ; : : : ; wn } for each n ∈ N. Then the main result obtained in this paper is as follows. −p u = P(x)|u|p−2 u;
Theorem 1.1. Let Kn := K ∩ Vn ∀n ∈ N. Suppose that f and P satisfy conditions (f1 ) and (f2 ) and condition (P), respectively. Then, (1) ∀: 0 ¡ ¡ 1 − 0 and ∀n ∈ N, problem (1) has a solution un ∈ Kn ; (2) there exists a subsequence {unj } of {un } such that unj → u0 ∈ K as j → ∞; moreover u0 is a solution of problem (1) for each : 0 ¡ ¡ 1 − 0 .
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
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2. Preliminaries In this section we will present some existence results for the abstract variational inequalities, which we shall use to prove our main theorem. Hence we give some notation /rst. Let X be a real Banach space and X∗ its dual space. Denote by · ; · the duality pairing between X and X∗ , and K is a cone of X (i.e., K is a closed convex subset of X such that K ⊂ K for all ∈ R+ and K ∩ (−K) = {0}). Given a nonlinear mapping g : K → X∗ , the variational inequality problem VIP(g; K) is /nd x ∈ K; such that g(x); y − x ¿ 0 holds for all y ∈ K: If g(x) = x − T1 (x) + T2 (x), where T1 is a linear mapping and T2 is a nonlinear mapping, then the VIP(g; K) contains as a particular case the mathematical model used in the study of the post-critical equilibrium state of a thin plate subjected to unilateral conditions, see for example [3,6,10] and references therein. Denition 2.1. A Galerkin approximation of the cone K is a countable family of cones {Kn }n∈N satisfying the following properties: (1) Kn ⊂ K∀n ∈ N; (2) dim(Kn ) ¡ + ∞ ∀n ∈ N; (3) limn→∞ [Proj|Kn (x)] = x; ∀x ∈ K. We call a cone K to be a Galerkin cone if K has a Galerkin approximation. Denition 2.2. Let K be a cone of the Banach space X. A mapping f: K → X∗ is called to satisfy the weak Karamandian condition ((WK) for short) if, there exists a bounded closed convex subset A ⊂ K, such that for each x ∈ K \ A there exists z ∈ A satisfying f(x); x − z ¿ 0: Denition 2.3. Let {Kn }n∈N be a Galerkin approximation of a cone K ⊂ X, a mapping f : K → X∗ is called to satisfy the generalized Karamardian condition ((GK) for short) if, for any n ∈ N, there exists a bounded closed convex set Dn ⊂ Kn , such that for each x ∈ Kn \ Dn there exists y ∈ Dn satisfying f(x); x − y ¿ 0: Remark 2.1. De/nitions 2.1–2.3 can be seen in [3] if X is a Hilbert space or in [13] for X being a Banach space. Let X and K be as in De/nition 2.2. Recall that a mapping f : X → X∗ is called to satisfy the Karamardian condition if there exists a compact convex set C ⊂ K such that for each x ∈ K \ C there exists z ∈ C satisfying f(x); x − z ¿ 0, see [3] for more details. The following de/nition can be seen in [4].
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Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
Denition 2.4. A mapping f : X → X∗ is called to satisfy the condition (S)+ if, {un }n∈N ⊂ X satis/es un * u and lim sup f(un ), un − u 6 0. Then there exists a subsequence {unk } such that unk → u. Lemma 2.1. Suppose that M is a >nite-dimensional Banach space, K0 is a cone of M, operator G: K0 → M is continuous satisfying the condition (WK). Then VIP(G; K0 ) has a solution. Proof. Let {u1 ; u2 ; : : : ; um } ⊂ K0 . Denote by B the convex hull of A∪{u1 ; u2 ; : : : ; um }, where A ⊂ K0 is the set in the de/nition of the condition (WK). Clearly, B is a bounded closed convex subset of K0 . Consequently, there exists v0 ∈ B such that G(v0 ); u − v0 ¿ 0
∀u ∈ B
(2)
(cf. [1]). We claim that v0 ∈ A. In fact, if v0 ∈ B \ A, by the assumption that the operator G satis/es the condition (WK), there exists z ∈ A ⊂ B such that G(v0 ); v0 − z ¿ 0, which contradicts (2). Hence v0 ∈ A. De/ne a multivalued mapping H : K0 → 2A as ∀u ∈ K0 ;
H (u) := {w ∈ A: G(w); u − w ¿ 0}:
It follows, from (2) and the fact that v0 ∈ A, that for all u ∈ B, v0 ∈ H (u). Therefore, v0 ∈∩m i=1 H (ui ), where {u1 ; : : : ; um } is the subset of K0 which appears in the beginning of the proof of this lemma. So {H (u): u ∈ K0 } has the /nite intersection property. Consequently, by the compactness of A, ∩u∈K0 H (u) = ∅, i.e., there exists w0 ∈ A such that w0 ∈ H (u) for all u ∈ K0 . We see that there exists {wn } ⊂ H (u) satisfying wn → w0 as n → ∞. It follows by the de/nition of H (u) that G(wn ); u − wn ¿ 0 and then G(w0 ); u − w0 ¿ 0
∀u ∈ K0 ;
which completes the proof of this lemma. Theorem 2.1. Let X be a re@exive Banach space, X∗ the dual space. Suppose that K is a Galerkin cone of X, {Kn }n∈N is a Galerkin approximation of K, G; F : K → X∗ are continuous operators with G satisfying the condition (S)+ , F being compact. Assume that G − F satis>es the condition (GK) for an equibounded family {Dn }n∈N . Then (1) for each n ∈ N, VIP(G − F; Kn ) has a solution un ∈ Dn ; (2) if {G(un )} is bounded, then there exists a subsequence {unj } of {un } such that unj → u∗ ∈ K, and u∗ is a solution of VIP(G − F; K). Proof. (1) It follows by Lemma 2.1 that VIP(F − G; Kn ) has a solution for each n ∈ N, that is, for each n ∈ N there exists un ∈ Kn such that G(un ) − F(un ); z − un ¿ 0
∀z ∈ Kn :
(3)
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
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Noting that G − F satis/es the condition (GK) for {Dn }, we see that for each n ∈ N, un ∈ Dn . (2) It implies from the equiboundedness of {Dn } that there exists a subsequence {unj } of {un } such that unj * u∗ ∈ K as j → ∞ (since K is weakly closed). Set uˆ n = Proj|Kn u∗
∀j ∈ N;
then, uˆ n ∈ Kn and uˆ n → u∗ . Inequality (3) yields that G(un ) − F(un ); uˆ n − un ¿ 0; hence, G(unj ); unj − uˆ nj 6 F(unj ); unj − uˆ nj : Since unj * u∗ and F is compact, we get that {F(unj )} has a subsequence, without loss of the generality, we still denote by {F(unj )}, which converges to F(u∗ ) as j goes to in/nity. It implies from the above inequality that lim supG(unj ); unj − uˆ nj 6 0:
(4)
j→∞
By the boundedness of {G(unj )}, we may assume that {G(unj )} weakly converges to some l ∈ X∗ as j goes to in/nity. Hence, lim G(unj ); uˆ nj − u∗ = 0:
(5)
j→∞
We observe that G(unj ); unj − u∗ 6 G(unj ); unj − uˆ nj + G(unj ); uˆ nj − u∗ : It follows from inequalities (4) and (5) that lim supG(unj ); unj − u∗ 6 0: j→∞
Note that G satis/es the condition (S)+ , hence there exists a subsequence of {unj }, we may denote by {unj }, such that unj → u∗ . We complete the proof by showing that u∗ is a solution of VIP(G − F; K). In fact, let z ∈ K be an arbitrary element, we denote by zn = Proj|Kn z. It implies from (3) that G(unj ) − F(unj ); znj − unj ¿ 0
∀n ∈ N:
Letting j → ∞ in the above inequality and noting the simple fact that {znj − unj } tends to z − u∗ as j tends to ∞, we have G(u∗ ) − F(u∗ ); z − u∗ ¿ 0; that is, u∗ solves the VIP(G − F; K). Let r → ’(r) be a strictly increasing continuous function that maps R+ into R+ and satis/es limr→+ ∞ ’(r) = + ∞. If · ∗ denotes the dual norm of the given norm ·
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Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
on X, the duality mapping J between X and X∗ with respect to ’ is given by J (0) = 0; J (u) = {f ∈ X∗ : f ∗ = ’( u ) and f; x = f ∗ u ; u = 0}: If ’ is the identity mapping of R+ , then J is the normalized duality mapping. In particular, if X is a Hilbert space, the normalized duality mapping is the identity mapping of X. If X is a reOexive Banach space, by the results due to Asplund, it is true that X can be renormed so that X and X∗ are both locally uniformly convex (LUC in short); in this case, J is single valued. Hence, we may assume in what follows that X and X∗ are LUC. From [3], J has the following properties: (P1 ) J (x)−J (y); x−y ¿ (’( x )−’( y ))( x − y ); x; y ∈ X. In particular, J (x)− J (y); x − y ¿ 0, i.e., J is monotone; (P2 ) J is continuous; (P3 ) J satis/es the condition (S)+ . By Theorem 2.1 we can easily get Corollary 2.1. Let K be a Galerkin of X, {Kn}n∈N a Galerkin approximation of K, J : X → X∗ the duality mapping given above, F : K → X∗ a mapping. Suppose that (i) F is strongly continuous; (ii) J − F satis>es the condition (GK) with an equibounded family {Dn }n∈N . Then VIP(J − F; K) has a solution. Corollary 2.2. Let K; J be as in Corollary 2.1. Suppose that S; T : K → X∗ with S being bounded and T being compact. Furthermore, assume that (i) J − S satis>es the condition (S)+ ; (ii) J − S − T satis>es the condition (GK) with an equibounded family {Dn }n∈N . Then VIP(J − S − T; K) has a solution. Remark 2.2. It is clear that if X is a Hilbert space, then the duality mapping J given above is the identity mapping of X; moreover, notice that in Corollary 2.1 we do not need a condition that “the function x → x; (J − F)(x) from K into R is lower semi-continuous”, therefore, Corollary 2.1 extends Theorem 1 in [3]. If X is a Hilbert space, J; S; T satisfy all conditions of Corollary 2.2, and (S + T )(K) ⊂ K, then by Corollary 2.2 we get that S + T has a /xed point in K. Hence, Corollary 2.2 improves Theorem 5 in [4].
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
1751
3. Proof of the main theorem Proof of Theorem 1.1. Clearly, K = ∪n∈N Kn . De/ne operators L; G1 ; G2 ; F: K → E∗ as L(u); v = |∇u|p−2 ∇u∇v d x
G1 (u); v = P − (x)|u|p−2 uv d x
∀u; v ∈ K:
G2 (u); v = P + (x)|u|p−2 uv d x
F(u); v = f(x; u)v d x
Easily, we see that L; G1 ; G2 ; F are well de/ned; moreover G2 ; F are completely continuous (cf. [2,12]). First we shall prove that for each ¿ 0, L + G1 satis/es the condition (S)+ . To do this, we assume that un * u in K and lim supn→∞ L(un ) + G1 (un ); un − u 6 0. Note that limn→∞ L(u) + G1 (u); un − u = 0, hence lim sup (L + G1 )(un ) − (L + G1 )(u); un − u 6 0: n→∞
It follows by the HSolder inequality that for each ¿ 0 (L + G1 )(un ) − (L + G1 )(u); un − u = (|∇un |p−2 ∇un − |∇u|p−2 ∇u)(∇un − ∇u)
+
P − (|un |p−2 un − |u|p−2 u)(un − u)
¿
p
|∇un |
×
+
¿ 0;
p
p
P |un |
−
1=p
|∇u|
−
1=p
(p−1)=p
|∇u|
p
(p−1)=p
P |un |
−
p
−
−
1=p
|∇un |
−
p
×
(p−1)=p
−
−
p
(p−1)=p
P |u|
p
P |u|
1=p
(6)
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Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
which and (6) yield that for each ¿ 0, |∇un |p d x → |∇u|p d x; P − |un |p d x → P − |u|p d x;
and then un E → u E . Therefore, un → u in K, which shows that J + G1 satis/es the condition (S)+ for each ¿ 0. Next we claim that there exists a real number . ¿ 0 such that for each : 0 ¡ ¡ 1 − 0 L(u) − Q(u) − F(u); u ¿ 0
∀ u ∈ K \ D. ;
(7)
where the operator Q := G2 − G1 and D. := {u ∈ K: u E 6 .}. By contradiction, we may assume that there exists {un } ⊂ K with un E → ∞ such that L(un ) − Q(un ) − F(un ); un 6 0; that is,
|∇un |p d x 6
P(x)|un |p d x +
f(x; un )un d x:
Let vn := un = un E , then {vn } is bounded in K. We may assume that vn * v0 ∈ K and vn (x) → v0 (x) a.e. in . By the preceding inequality: f(x; un ) p p |∇vn | d x 6 P(x)|vn | d x + v d x: (8) p−1 n un E For each n ∈ N, we de/ne n; R := {x ∈ : |un (x)| 6 R}, where R is the positive real number given in (f1 ). We split the second integral on the right-hand side of (8) into integrals over \ n; R and n; R , then by (f1 ), for each n ∈ N we have f(x; un ) f(x; un ) v d x = p un d x p−1 n u \n; R \n; R un E n E 6 0
P(x)|vn |p d x − 0
n; R
P(x) p u d x: un pE n
Since P ∈ L1 (), n; R (P(x)= un pE )unp d x → 0 as n → ∞. Using the argument in the proof of [2, Lemma 4.2], we denote n; R (K) := n; R ∩ B(0; K), where B(0; K) is the ball centered at 0 and having radius K, i.e., B(0; K)={x ∈ RN : |x| ¡ K}. By (f2 ), the HSolder inequality and the Sobolev inequality, |f(x; un )| (x)|un | + (x)|un |p |v | d x 6 dx n p−1 un pE n; R un E n; R
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
6
6
(x)
n; R
|v | d x + p−1 n
un E
c1 L(p∗ ) () un p−1 E
n; R \n; R (K)
p
(x)|vn | d x +
+ c2 LN=p (n; R \n; R (K)) +
n; R (K)
1753
(x)|un |p dx un pE
c3 (K) un pE
for some positive constants c1 , c2 and c3 (K) (dependent of K). Now ∀2 ¿ 0 we can choose K so that the second term on the right-hand side of the above inequality is 6 2=2, and then N0 large enough such that the sum of the /rst and the last terms on the right-hand side of the above inequality is 6 2=2 if n ¿ N0 . Note that P = P + − P − . We see, by (8), the de/nition of the /rst eigenvalue 1 , the fact that the mapping u → P + (x)|u|p d x is weakly continuous (cf. [12]), the weakly lower semicontinuity of the norm, that + 0 |∇v0 |p d x 6 ( + 0 ) P(x)|v0 |p d x 6 |∇v0 |p d x ¡ |∇v0 |p d x; 1 a contradiction. Accordingly, we have shown that for each ¡ 1 − 0 , Inequality (7) holds. Therefore, L − Q − F satis/es the condition (WK) for each ¡ 1 − 0 , and so L − Q − F satis/es the condition (GK) for each ¡ 1 − 0 . Finally, by applying Theorem 2.1, we complete the proof. Acknowledgements The authors would like to thank the referee for his helpful comments. References [1] P. Hartman, G. Stampacchia, On some nonlinear elliptic diTerential functional equations, Acta Math. 115 (1966) 271–310. [2] Y.S. Huang, Positive solutions of quasilinear elliptic equations, Topol. Meth. Nonlinear Anal. 12 (1998) 91–107. [3] G. Isac, Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl. 108 (1985) 563–574. [4] G. Isac, On an Altman type /xed point theorem on convex cones, Rocky Mountain J. Math. 25 (1995) 701–714. [5] P. Jebelean, Finite-dimensional approximation and coerciveness in a problem with p-Laplacian, Nonlinear Anal. TMA 33 (1998) 253–259. [6] R.S. Kubrusly, J.T. Oden, Nonlinear eigenvalue problems characterized by variational inequalities with applications to the postbuckling analysis of unilaterally-supported plates, Nonlinear Anal. TMA 5 (1981) 1265–1284. [7] V.K. Le, On global bifurcation of variational inequalities and applications, J. DiTerential Equations 141 (1997) 254–294. [8] V.K. Le, Subsolution-supersolution method in variational inequalities, Nonlinear Anal. 45 (2001) 775–800. [9] V.K. Le, K. Schmitt, Minimization problems for noncoercive functionals subject to constraints, Tran. Am. Math. Soc. 347 (1995) 4485–4513.
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[10] R.C. Riddel, Eigenvalue problems for nonlinear elliptic variational inequalities on a cone, J. funct. Anal. 26 (1977) 333–355. [11] A. Szulkin, Positive solutions of variational inequalities: a degree theoretic approach, J. DiTerential Equations 57 (1985) 90–111. [12] A. Szulkin, M. Willem, Eigenvalue problems with inde/nite weight, Studia. Math. 135 (1999) 191–201. [13] M. ThEera, Existence results for the nonlinear complementarity problem and application to nonlinear analysis, J. Math. Anal. Appl. 154 (1991) 572–586.
www.elsevier.com/locate/na
Finite-dimensional approximation for a class of elliptic obstacle problems Yisheng Huanga;∗ , Yuying Zhoub a Department
of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, People’s Republic of China
b Department
Received 27 November 2001; accepted 13 May 2002
Abstract In this paper we study the /nite-dimensional approximation for a class of elliptic obstacle problems involving unbounded domains by using the Galerkin method. The convergence of the approximation is deduced. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Obstacle problem; Finite-dimensional approximation; Galerkin cone
1. Introduction and preliminaries p ∞ Let be an arbitrary open set in RN , D1; 0 () the completion of C0 () in the p 1=p and 1 ¡ p ¡ N . Let P(x) be a sign-changing function with norm { |∇u| d x} P = P + − P − (as usual, P ± (x) := max{± P(x); 0}). We assume that P satis/es the hypothesis:
< lim x→y |x−y|p P2 (x) = (P) P ∈ L1 (); P + =P1 +P2 = 0, P1 ∈ LN=p () for every y ∈ , 0 and lim |x|→∞ |x|p P2 (x) = 0.
x∈
x∈
Denote E the space {u ∈ D01; p (): u E ¡ ∞}, where u pE = − by p P |u| d x. Let K be the cone {u ∈ E: u(x) ¿ 0 a.e. for x ∈ }.
|∇u|p d x +
Supported by NNSF of China (10161010) and NSF of Yunnan Province (1999A0003R).
∗
Corresponding author. Tel.: +86-512-625-27063; fax: +86-512-6522-2691. E-mail address: [email protected] (Y. Huang).
0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 9 2 - 4
1746
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
In this paper we are concerned with the following obstacle problem: /nd u ∈ K such that f(x; u)(v − u) (1) |∇u|p−2 ∇u · ∇(v − u) ¿ P(x)|u|p−2 u(v − u) +
∀u ∈ K, where is the real parameter and f(x; t) is the CarathEeodory function satisfying some conditions to be speci/ed later. The solvability of variational inequalities like (1), relating to an elliptic operator, has been studied extensively in recent years by various methods, such as bifurcation, variational, super-subsolution approaches and the topological /xed point method, see for example [7–9,11] and the references therein. In this paper we study the solvability of (1) in another way, which is the Galerkin method, i.e., the /nite-dimensional approximation of problem (1). This method has been used by Isac [3,4] to consider the solvability of a class of abstract nonlinear complementarity problems and in [5] to the solvability of equations involving the p-Laplacian. However, the arguments used in [5] cannot be applied to our situation even if K = E (in this case (1) is in fact an equation) since may not be bounded. Motivated by Isac [3,4], we /rst study the /nite-dimensional approximation for an abstract variational inequality involving an operator satisfying the condition (S)+ (see the de/nition below), and deduce the convergence of the approximation, especially our results extend and improve corresponding ones in [3,4]; then we prove our main theorem by applying the preceding results. Before presenting our main result, we make some assumptions on f in problem (1). Suppose that f is a CarathEeodory function satisfying (f1 ) there exist R ¿ 0 and 0 : 0 6 0 ¡ 1 such that f(x; s)s 6 0 P(x)|s|p holds ∀x ∈ and ∀s: |s| ¿ R, where 1 will be given in the following; ∗ (f2 ) |f(x; s)| 6 (x) + (x)|s|p−1 for a.e. (x; s) ∈ × R, where 0 6 (x) ∈ L(p ) () with p∗ = Np=(N − p) (the critical Sobolev exponent), (p∗ ) = p∗ =(p∗ − 1), 0 6 (x) ∈ LN=p () ∩ L∞ loc (). It was shown in [12], under the above condition (P), that the eigenvalue problem u ∈ D01; p () has the least eigenvalue 1 := inf { |∇u|p d x: u ∈ E; P|u|p d x = 1} satisfying 1 ¿ 0, where p u := div(|∇u|p−2 ∇u) is the p-Laplacian. Let {wn } be a sequence of linear independent elements of E such that n∈N Vn =E, where Vn = span{w1 ; w2 ; : : : ; wn } for each n ∈ N. Then the main result obtained in this paper is as follows. −p u = P(x)|u|p−2 u;
Theorem 1.1. Let Kn := K ∩ Vn ∀n ∈ N. Suppose that f and P satisfy conditions (f1 ) and (f2 ) and condition (P), respectively. Then, (1) ∀: 0 ¡ ¡ 1 − 0 and ∀n ∈ N, problem (1) has a solution un ∈ Kn ; (2) there exists a subsequence {unj } of {un } such that unj → u0 ∈ K as j → ∞; moreover u0 is a solution of problem (1) for each : 0 ¡ ¡ 1 − 0 .
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2. Preliminaries In this section we will present some existence results for the abstract variational inequalities, which we shall use to prove our main theorem. Hence we give some notation /rst. Let X be a real Banach space and X∗ its dual space. Denote by · ; · the duality pairing between X and X∗ , and K is a cone of X (i.e., K is a closed convex subset of X such that K ⊂ K for all ∈ R+ and K ∩ (−K) = {0}). Given a nonlinear mapping g : K → X∗ , the variational inequality problem VIP(g; K) is /nd x ∈ K; such that g(x); y − x ¿ 0 holds for all y ∈ K: If g(x) = x − T1 (x) + T2 (x), where T1 is a linear mapping and T2 is a nonlinear mapping, then the VIP(g; K) contains as a particular case the mathematical model used in the study of the post-critical equilibrium state of a thin plate subjected to unilateral conditions, see for example [3,6,10] and references therein. Denition 2.1. A Galerkin approximation of the cone K is a countable family of cones {Kn }n∈N satisfying the following properties: (1) Kn ⊂ K∀n ∈ N; (2) dim(Kn ) ¡ + ∞ ∀n ∈ N; (3) limn→∞ [Proj|Kn (x)] = x; ∀x ∈ K. We call a cone K to be a Galerkin cone if K has a Galerkin approximation. Denition 2.2. Let K be a cone of the Banach space X. A mapping f: K → X∗ is called to satisfy the weak Karamandian condition ((WK) for short) if, there exists a bounded closed convex subset A ⊂ K, such that for each x ∈ K \ A there exists z ∈ A satisfying f(x); x − z ¿ 0: Denition 2.3. Let {Kn }n∈N be a Galerkin approximation of a cone K ⊂ X, a mapping f : K → X∗ is called to satisfy the generalized Karamardian condition ((GK) for short) if, for any n ∈ N, there exists a bounded closed convex set Dn ⊂ Kn , such that for each x ∈ Kn \ Dn there exists y ∈ Dn satisfying f(x); x − y ¿ 0: Remark 2.1. De/nitions 2.1–2.3 can be seen in [3] if X is a Hilbert space or in [13] for X being a Banach space. Let X and K be as in De/nition 2.2. Recall that a mapping f : X → X∗ is called to satisfy the Karamardian condition if there exists a compact convex set C ⊂ K such that for each x ∈ K \ C there exists z ∈ C satisfying f(x); x − z ¿ 0, see [3] for more details. The following de/nition can be seen in [4].
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Denition 2.4. A mapping f : X → X∗ is called to satisfy the condition (S)+ if, {un }n∈N ⊂ X satis/es un * u and lim sup f(un ), un − u 6 0. Then there exists a subsequence {unk } such that unk → u. Lemma 2.1. Suppose that M is a >nite-dimensional Banach space, K0 is a cone of M, operator G: K0 → M is continuous satisfying the condition (WK). Then VIP(G; K0 ) has a solution. Proof. Let {u1 ; u2 ; : : : ; um } ⊂ K0 . Denote by B the convex hull of A∪{u1 ; u2 ; : : : ; um }, where A ⊂ K0 is the set in the de/nition of the condition (WK). Clearly, B is a bounded closed convex subset of K0 . Consequently, there exists v0 ∈ B such that G(v0 ); u − v0 ¿ 0
∀u ∈ B
(2)
(cf. [1]). We claim that v0 ∈ A. In fact, if v0 ∈ B \ A, by the assumption that the operator G satis/es the condition (WK), there exists z ∈ A ⊂ B such that G(v0 ); v0 − z ¿ 0, which contradicts (2). Hence v0 ∈ A. De/ne a multivalued mapping H : K0 → 2A as ∀u ∈ K0 ;
H (u) := {w ∈ A: G(w); u − w ¿ 0}:
It follows, from (2) and the fact that v0 ∈ A, that for all u ∈ B, v0 ∈ H (u). Therefore, v0 ∈∩m i=1 H (ui ), where {u1 ; : : : ; um } is the subset of K0 which appears in the beginning of the proof of this lemma. So {H (u): u ∈ K0 } has the /nite intersection property. Consequently, by the compactness of A, ∩u∈K0 H (u) = ∅, i.e., there exists w0 ∈ A such that w0 ∈ H (u) for all u ∈ K0 . We see that there exists {wn } ⊂ H (u) satisfying wn → w0 as n → ∞. It follows by the de/nition of H (u) that G(wn ); u − wn ¿ 0 and then G(w0 ); u − w0 ¿ 0
∀u ∈ K0 ;
which completes the proof of this lemma. Theorem 2.1. Let X be a re@exive Banach space, X∗ the dual space. Suppose that K is a Galerkin cone of X, {Kn }n∈N is a Galerkin approximation of K, G; F : K → X∗ are continuous operators with G satisfying the condition (S)+ , F being compact. Assume that G − F satis>es the condition (GK) for an equibounded family {Dn }n∈N . Then (1) for each n ∈ N, VIP(G − F; Kn ) has a solution un ∈ Dn ; (2) if {G(un )} is bounded, then there exists a subsequence {unj } of {un } such that unj → u∗ ∈ K, and u∗ is a solution of VIP(G − F; K). Proof. (1) It follows by Lemma 2.1 that VIP(F − G; Kn ) has a solution for each n ∈ N, that is, for each n ∈ N there exists un ∈ Kn such that G(un ) − F(un ); z − un ¿ 0
∀z ∈ Kn :
(3)
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Noting that G − F satis/es the condition (GK) for {Dn }, we see that for each n ∈ N, un ∈ Dn . (2) It implies from the equiboundedness of {Dn } that there exists a subsequence {unj } of {un } such that unj * u∗ ∈ K as j → ∞ (since K is weakly closed). Set uˆ n = Proj|Kn u∗
∀j ∈ N;
then, uˆ n ∈ Kn and uˆ n → u∗ . Inequality (3) yields that G(un ) − F(un ); uˆ n − un ¿ 0; hence, G(unj ); unj − uˆ nj 6 F(unj ); unj − uˆ nj : Since unj * u∗ and F is compact, we get that {F(unj )} has a subsequence, without loss of the generality, we still denote by {F(unj )}, which converges to F(u∗ ) as j goes to in/nity. It implies from the above inequality that lim supG(unj ); unj − uˆ nj 6 0:
(4)
j→∞
By the boundedness of {G(unj )}, we may assume that {G(unj )} weakly converges to some l ∈ X∗ as j goes to in/nity. Hence, lim G(unj ); uˆ nj − u∗ = 0:
(5)
j→∞
We observe that G(unj ); unj − u∗ 6 G(unj ); unj − uˆ nj + G(unj ); uˆ nj − u∗ : It follows from inequalities (4) and (5) that lim supG(unj ); unj − u∗ 6 0: j→∞
Note that G satis/es the condition (S)+ , hence there exists a subsequence of {unj }, we may denote by {unj }, such that unj → u∗ . We complete the proof by showing that u∗ is a solution of VIP(G − F; K). In fact, let z ∈ K be an arbitrary element, we denote by zn = Proj|Kn z. It implies from (3) that G(unj ) − F(unj ); znj − unj ¿ 0
∀n ∈ N:
Letting j → ∞ in the above inequality and noting the simple fact that {znj − unj } tends to z − u∗ as j tends to ∞, we have G(u∗ ) − F(u∗ ); z − u∗ ¿ 0; that is, u∗ solves the VIP(G − F; K). Let r → ’(r) be a strictly increasing continuous function that maps R+ into R+ and satis/es limr→+ ∞ ’(r) = + ∞. If · ∗ denotes the dual norm of the given norm ·
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on X, the duality mapping J between X and X∗ with respect to ’ is given by J (0) = 0; J (u) = {f ∈ X∗ : f ∗ = ’( u ) and f; x = f ∗ u ; u = 0}: If ’ is the identity mapping of R+ , then J is the normalized duality mapping. In particular, if X is a Hilbert space, the normalized duality mapping is the identity mapping of X. If X is a reOexive Banach space, by the results due to Asplund, it is true that X can be renormed so that X and X∗ are both locally uniformly convex (LUC in short); in this case, J is single valued. Hence, we may assume in what follows that X and X∗ are LUC. From [3], J has the following properties: (P1 ) J (x)−J (y); x−y ¿ (’( x )−’( y ))( x − y ); x; y ∈ X. In particular, J (x)− J (y); x − y ¿ 0, i.e., J is monotone; (P2 ) J is continuous; (P3 ) J satis/es the condition (S)+ . By Theorem 2.1 we can easily get Corollary 2.1. Let K be a Galerkin of X, {Kn}n∈N a Galerkin approximation of K, J : X → X∗ the duality mapping given above, F : K → X∗ a mapping. Suppose that (i) F is strongly continuous; (ii) J − F satis>es the condition (GK) with an equibounded family {Dn }n∈N . Then VIP(J − F; K) has a solution. Corollary 2.2. Let K; J be as in Corollary 2.1. Suppose that S; T : K → X∗ with S being bounded and T being compact. Furthermore, assume that (i) J − S satis>es the condition (S)+ ; (ii) J − S − T satis>es the condition (GK) with an equibounded family {Dn }n∈N . Then VIP(J − S − T; K) has a solution. Remark 2.2. It is clear that if X is a Hilbert space, then the duality mapping J given above is the identity mapping of X; moreover, notice that in Corollary 2.1 we do not need a condition that “the function x → x; (J − F)(x) from K into R is lower semi-continuous”, therefore, Corollary 2.1 extends Theorem 1 in [3]. If X is a Hilbert space, J; S; T satisfy all conditions of Corollary 2.2, and (S + T )(K) ⊂ K, then by Corollary 2.2 we get that S + T has a /xed point in K. Hence, Corollary 2.2 improves Theorem 5 in [4].
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3. Proof of the main theorem Proof of Theorem 1.1. Clearly, K = ∪n∈N Kn . De/ne operators L; G1 ; G2 ; F: K → E∗ as L(u); v = |∇u|p−2 ∇u∇v d x
G1 (u); v = P − (x)|u|p−2 uv d x
∀u; v ∈ K:
G2 (u); v = P + (x)|u|p−2 uv d x
F(u); v = f(x; u)v d x
Easily, we see that L; G1 ; G2 ; F are well de/ned; moreover G2 ; F are completely continuous (cf. [2,12]). First we shall prove that for each ¿ 0, L + G1 satis/es the condition (S)+ . To do this, we assume that un * u in K and lim supn→∞ L(un ) + G1 (un ); un − u 6 0. Note that limn→∞ L(u) + G1 (u); un − u = 0, hence lim sup (L + G1 )(un ) − (L + G1 )(u); un − u 6 0: n→∞
It follows by the HSolder inequality that for each ¿ 0 (L + G1 )(un ) − (L + G1 )(u); un − u = (|∇un |p−2 ∇un − |∇u|p−2 ∇u)(∇un − ∇u)
+
P − (|un |p−2 un − |u|p−2 u)(un − u)
¿
p
|∇un |
×
+
¿ 0;
p
p
P |un |
−
1=p
|∇u|
−
1=p
(p−1)=p
|∇u|
p
(p−1)=p
P |un |
−
p
−
−
1=p
|∇un |
−
p
×
(p−1)=p
−
−
p
(p−1)=p
P |u|
p
P |u|
1=p
(6)
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which and (6) yield that for each ¿ 0, |∇un |p d x → |∇u|p d x; P − |un |p d x → P − |u|p d x;
and then un E → u E . Therefore, un → u in K, which shows that J + G1 satis/es the condition (S)+ for each ¿ 0. Next we claim that there exists a real number . ¿ 0 such that for each : 0 ¡ ¡ 1 − 0 L(u) − Q(u) − F(u); u ¿ 0
∀ u ∈ K \ D. ;
(7)
where the operator Q := G2 − G1 and D. := {u ∈ K: u E 6 .}. By contradiction, we may assume that there exists {un } ⊂ K with un E → ∞ such that L(un ) − Q(un ) − F(un ); un 6 0; that is,
|∇un |p d x 6
P(x)|un |p d x +
f(x; un )un d x:
Let vn := un = un E , then {vn } is bounded in K. We may assume that vn * v0 ∈ K and vn (x) → v0 (x) a.e. in . By the preceding inequality: f(x; un ) p p |∇vn | d x 6 P(x)|vn | d x + v d x: (8) p−1 n un E For each n ∈ N, we de/ne n; R := {x ∈ : |un (x)| 6 R}, where R is the positive real number given in (f1 ). We split the second integral on the right-hand side of (8) into integrals over \ n; R and n; R , then by (f1 ), for each n ∈ N we have f(x; un ) f(x; un ) v d x = p un d x p−1 n u \n; R \n; R un E n E 6 0
P(x)|vn |p d x − 0
n; R
P(x) p u d x: un pE n
Since P ∈ L1 (), n; R (P(x)= un pE )unp d x → 0 as n → ∞. Using the argument in the proof of [2, Lemma 4.2], we denote n; R (K) := n; R ∩ B(0; K), where B(0; K) is the ball centered at 0 and having radius K, i.e., B(0; K)={x ∈ RN : |x| ¡ K}. By (f2 ), the HSolder inequality and the Sobolev inequality, |f(x; un )| (x)|un | + (x)|un |p |v | d x 6 dx n p−1 un pE n; R un E n; R
Y. Huang, Y. Zhou / Nonlinear Analysis 52 (2003) 1745 – 1754
6
6
(x)
n; R
|v | d x + p−1 n
un E
c1 L(p∗ ) () un p−1 E
n; R \n; R (K)
p
(x)|vn | d x +
+ c2 LN=p (n; R \n; R (K)) +
n; R (K)
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(x)|un |p dx un pE
c3 (K) un pE
for some positive constants c1 , c2 and c3 (K) (dependent of K). Now ∀2 ¿ 0 we can choose K so that the second term on the right-hand side of the above inequality is 6 2=2, and then N0 large enough such that the sum of the /rst and the last terms on the right-hand side of the above inequality is 6 2=2 if n ¿ N0 . Note that P = P + − P − . We see, by (8), the de/nition of the /rst eigenvalue 1 , the fact that the mapping u → P + (x)|u|p d x is weakly continuous (cf. [12]), the weakly lower semicontinuity of the norm, that + 0 |∇v0 |p d x 6 ( + 0 ) P(x)|v0 |p d x 6 |∇v0 |p d x ¡ |∇v0 |p d x; 1 a contradiction. Accordingly, we have shown that for each ¡ 1 − 0 , Inequality (7) holds. Therefore, L − Q − F satis/es the condition (WK) for each ¡ 1 − 0 , and so L − Q − F satis/es the condition (GK) for each ¡ 1 − 0 . Finally, by applying Theorem 2.1, we complete the proof. Acknowledgements The authors would like to thank the referee for his helpful comments. References [1] P. Hartman, G. Stampacchia, On some nonlinear elliptic diTerential functional equations, Acta Math. 115 (1966) 271–310. [2] Y.S. Huang, Positive solutions of quasilinear elliptic equations, Topol. Meth. Nonlinear Anal. 12 (1998) 91–107. [3] G. Isac, Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl. 108 (1985) 563–574. [4] G. Isac, On an Altman type /xed point theorem on convex cones, Rocky Mountain J. Math. 25 (1995) 701–714. [5] P. Jebelean, Finite-dimensional approximation and coerciveness in a problem with p-Laplacian, Nonlinear Anal. TMA 33 (1998) 253–259. [6] R.S. Kubrusly, J.T. Oden, Nonlinear eigenvalue problems characterized by variational inequalities with applications to the postbuckling analysis of unilaterally-supported plates, Nonlinear Anal. TMA 5 (1981) 1265–1284. [7] V.K. Le, On global bifurcation of variational inequalities and applications, J. DiTerential Equations 141 (1997) 254–294. [8] V.K. Le, Subsolution-supersolution method in variational inequalities, Nonlinear Anal. 45 (2001) 775–800. [9] V.K. Le, K. Schmitt, Minimization problems for noncoercive functionals subject to constraints, Tran. Am. Math. Soc. 347 (1995) 4485–4513.
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[10] R.C. Riddel, Eigenvalue problems for nonlinear elliptic variational inequalities on a cone, J. funct. Anal. 26 (1977) 333–355. [11] A. Szulkin, Positive solutions of variational inequalities: a degree theoretic approach, J. DiTerential Equations 57 (1985) 90–111. [12] A. Szulkin, M. Willem, Eigenvalue problems with inde/nite weight, Studia. Math. 135 (1999) 191–201. [13] M. ThEera, Existence results for the nonlinear complementarity problem and application to nonlinear analysis, J. Math. Anal. Appl. 154 (1991) 572–586.