A BIFURCATION RESULT FOR EQUATIONS WITH ANISOTROPIC p-LAPLACE-LIKE OPERATORS

where X is an appropriate function space and φi , F are given functions. The coefficients φi’s are different in general. In the particular case where φi(|ξ |) = |ξ |p−2, for all ξ ∈ R , for all i ∈ {1, . . . ,N}, (1.1) reduces to the p-Laplacian equation and there are several bifurcation results available (cf. [3, 4]). It seems that bifurcation problems for anisotropic elliptic operators have not been addressed in detail. As is well known in bifurcation theory, a first step is to find a “linearization” of (1.1) such that bifurcation in (1.1) can be studied through the eigenvalues and eigenfunctions of the linearization. Different from equations with compact perturbations of linear operators, we show that (1.1) can be related to a nonlinear but homogeneous equation (called the homogenization of (1.1)). Another difficulty is that since the functions φi’s may have different growths at


Introduction
This paper is about a global bifurcation result for nonlinear equations whose principal operators are anisotropic p-Laplace-like operators (sometimes called φ-Laplacians). The equation is, in the weak form, zero and infinity, the homogenization of (1.1) is defined properly only on a function space different from X. This makes the usual continuation processes (cf. [6,12,15]) to calculate the involved topological degrees and to prove global bifurcation in this case more difficult. Moreover, since the φ i 's are different in general, (1.1) does not have a variational structure and the variational approach used recently in [8,9] (by converting the equation to a variational inequality) is not applicable in our problem here.
To study the bifurcation of (1.1), we employ a resolvent operator to convert (1.1) to a fixed point inclusion. Because the principal operator is not strictly monotone, the topological invariances are calculated in terms of the Leray-Schauder degree for multivalued compact vector fields. We show that under some conditions, a nonlinear equation in a Banach space X can be associated with a simpler, homogeneous equation (in a different space) so that the bifurcation properties of (1.2) are reflected in eigenvalues and topological degrees in that "homogenized" equation (in fact, an inclusion). The maps A and B are in general not compact perturbations of linear mappings. Also, by working on a space larger than X, we are able to bypass the assumption that A is of class (S) or (S) + , which is a useful but sometimes hard to verify assumption on nonlinear elliptic operators (cf. [2,14]). Our approach is based on the classical ideas of using resolvent (Green's) operators and topological degrees. However, due to the nature of our problem (nonvariational structure, set-valued resolvent operator, principal operator not being in class (S) or (S) + , bifurcation and degree computation being in different function spaces, etc.), new arguments and techniques are needed in the proof of the abstract bifurcation theorem in Section 2. The verification of the general assumptions in the specific case of (1.1) also requires nontrivial estimates and calculations. The paper is organized as follows. In Section 2, we prove an abstract bifurcation theorem for (1.2). We define the homogenized equation and inclusion of (1.2) and prove (in Theorem 2.9) that if there is a change of topological degrees in the homogenized inclusion, then there is a global bifurcation in (1.2) in the sense of the Rabinowitz alternative. This result extends the well-known Rabinowitz global bifurcation theorem to equations whose operators are not necessarily compact perturbations of linear mappings, and the principal operators may not belong to class (S) or (S) + . In Section 3, we use the general result of Section 2 to show global bifurcation in (1.1). In this particular case, the homogenized inclusion of (1.1) is proved to be the usual p-Laplacian equation. Using degree computations at the principal eigenvalue of the p-Laplacian in [3], we prove that there is bifurcation in (1.1) at that eigenvalue, and the corresponding bifurcation branch satisfies the Rabinowitz alternative. Vy Khoi Le 231

A global bifurcation result
We are concerned in this section with the bifurcation of certain functional equations without variational structure, that is, the principal operators are not necessarily gradients of some potential functionals. Consider the following equation: in a real, reflexive Banach space X. The map A is the principal operator and B is the lower order term, depending on a real parameter λ. The set X * is the dual space of X. We use · (or · X ) and ·, · (or ·, · X,X * ) to denote the norm of X and the dual pairing between X and X * . Assume that X is compactly embedded in another Banach space Z (with norm · Z and dual pairing ·, · Z,Z * ), that is, X ⊂ Z and the embedding i : X → Z is compact. Suppose that A : X → X * is monotone, hemicontinuous, bounded, and A(0) = 0. Also, A is coercive in the following sense: The map B is a bounded, continuous mapping from Z × R to Z * such that B(0, λ) = 0, for all λ (further assumptions on A and B will be specified later). Hence, 0 is always a solution of (2.1). For convenience, we still use B to denote the restriction of B on X × R. For each f in X * , the above assumptions on A imply that the equation has at least one solution in X (cf. [10]). We denote by P (f ) = P A (f ) the set of all solutions of (2.3). Hence, P A (f ) = ∅. Equation (2.1) is equivalent to the fixed point inclusion Because of the embedding i : X → Z, (2.4) is also equivalent to the following inclusion in Z: In fact, if u satisfies (2.4), then it clearly satisfies (2.5). Conversely, if u is a solution of (2.5), then u ∈ X (since Z * ⊂ X * and P A (X * ) ⊂ X). As i(x) = x for x ∈ X, we have (2.4).Therefore, instead of studying (2.1) or (2.4), we consider the bifurcation of the equivalent inclusion (2.5), that is, We study the bifurcation of this equation in Z. First, we extend in a natural way the definition of bifurcation points and the basic global bifurcation result from 232 A bifurcation result equations to inclusions. Assume that Z is a Banach space and F mapping from Consider the inclusion Assume that 0 ∈ F (0, λ), for all λ ∈ R. Then, for all λ, 0 is a trivial solution of (2.8). We define the bifurcation points as usual. We refer to [7] or [11] for more detailed discussions and basic properties of topological degrees for multivalued compact fields. Assume λ is not a bifurcation point of (2.8). By definition, there exists an open neighborhood U of 0 such that 0 ∈ u − F (u, λ) for all u ∈ U \ {0}. Thus, the degree d(I − F (·, λ), B r (0), 0) is defined for all r > 0 sufficiently small. We have the following global bifurcation theorem for (2.8).
Theorem 2.2. Let a, b ∈ R (a < b). Assume (0, a) and (0, b) are not bifurcation points of (2.8) and that Vy Khoi Le 233 (We use the superscript Z in (2.12) to emphasize that the operation below it is taken in Z.) As usual, the proof of this theorem is based on a separation property in metric spaces (Whyburn's lemma) together with excision and homotopy invariance properties of the topological degree. Since the topological degree of compact, multivalued vector fields has these properties (cf. [7] or [11]), the proof of Theorem 2.2 is a straightforward adaptation and generalization of the corresponding result for single-valued compact vector fields, as presented, for example, in [13]. Now, we use Theorem 2.2 to the particular case of inclusion (2.5). First, we have the following lemma.
Proof. The proof of this lemma is routine. As noted previously, P A (f ) = ∅. Assume that u, w ∈ P A (f ), that is, (2.13) and that x = tu + (1 − t)w for some t ∈ [0, 1]. Since A is monotone, one can apply Minty's lemma (cf. [5]) to get for all v ∈ X. Multiplying the first inequality by t and the second by 1 − t, and adding the inequalities thus obtained, we get Apply again Minty's lemma, we have Proof. Let U be a bounded subset of X * . Assume P A (U ) is not bounded and that there exist sequences {u n } and {f n } such that {f n } ⊂ U , u n ∈ P A (f n ), for all n, and u n X → ∞ as n → ∞. By the definition of P A , A(u n ) = f n , for all n. Thus, This, however, contradicts the coercivity of A (cf. (2.2)). Hence, P A (U ) must be bounded in X.
Hence, by passing to a subsequence, we can assume that there exists u * ∈ X such that u n u * in X, (" " denotes the weak convergence). Therefore, u n → u * in Z(-strong). It follows that u = u * ∈ X and u n u in X. From Minty's lemma (cf. [5]), we have from Using Minty's lemma once more, we have Lemmas 2.3, 2.4, and 2.5 prove, in particular, that iP A [B(u, λ)] ∈ (Z) for every u ∈ Z, λ ∈ R. Another lemma is needed for our definition of the degree of I − iP A [B(·, λ)] in Z (I is the identity mapping on Z).
is relatively compact in Z by the compactness of the embedding i.  We note from the monotonicity of A that A(u), u ≥ 0, for all u ∈ X. If A has the following nondegeneracy property: In fact, assume that {(u n , λ n )} is a sequence of solutions of (2.8) such that u n → 0 in Z, u n = 0, and λ n → λ 0 . We have B(u n , λ n ) → 0 in Z * . Thus, B u n , λ n , u n = B u n , λ n , u n Z,Z * −→ 0. (2.31)

However,
A u n , u n = B u n , λ n , u n .
(2.32) Therefore, {u n } satisfies the conditions in (2.30). By the implication in (2.30), we must have u n → 0 in X, that is, λ 0 is a bifurcation point in X. The converse is trivial.

A bifurcation result
Now, we consider a homogenization procedure for (2.8). We fix a number p > 1 and define, for σ > 0: (2.34) For each σ > 0, A σ is a monotone, coercive, bounded mapping from X to X * and B σ is a continuous, bounded mapping from Z × R to Z * . It is clear that A = A 1 and B = B 1 . We assume furthermore that X 0 is a reflexive Banach space, with norm · X 0 = · 0 , that is continuously embedded in X. We have the continuous embeddings X 0 → X → Z, and thus Z * → X * → X * 0 . Suppose that there exist two mappings A 0 : X 0 → X * 0 and B 0 : Z × R → Z * such that A 0 is monotone, bounded, hemicontinuous, coercive in X 0 (in the sense of (2.2)), A 0 (0) = 0, B 0 is bounded, continuous, and B 0 (0, λ) = 0 for all λ. We assume that A σ converges to A 0 and B σ converges to B 0 as σ → 0 in the following sense. ( We also need the following assumption on the boundedness of B σ for σ small: (H2b) If {v n } is a bounded sequence in Z, λ n → λ and σ n → 0 + , then the sequence {B σ n (u n , λ n )} is bounded in Z * .
The family {A σ : σ > 0} has the following nondegeneracy condition near 0: (H3) If σ n → 0 + , v n 0 in X, and then v n −→ 0 in X. (2.38) For σ > 0, we denote by P A σ the solution mapping of the operator A σ , that is, By the same proof as in Lemma 2.6, we see that the mapping  (u, λ)]). Note that condition (H1) can also be stated equivalently as (H1 ) Assume σ n → 0 and {v n } ⊂ X, {f n } ⊂ X * are sequences such that then v ∈ X 0 and Concerning (H3), we note that the following condition is more restrictive, yet it is easier to verify (H3 ) If v n 0 in X, σ n → 0, and A σ n (v n ), v n → 0, then v n → 0 in X. We also need the following equi-coercivity condition for the family {A σ } for small values of σ : (H4) If σ n → 0 + and u n → ∞, then Now, we associate to (2.1) the "homogenized" equation which is equivalent to the fixed point inclusion This inclusion is equivalent to the following inclusion in Z: The operators A 0 and B 0 introduced above play, in some sense, the roles of the Gâteaux derivatives A (0) and ∂ u B(0, λ) in the cases where A and B are not equivalent to linear operators at 0. In classical situations where A is a linear operator and B is a perturbation of a linear operator However, in (1.1), A and B cannot be linearized at 0 and A 0 , B 0 are nonlinear in general. Nevertheless, one can define generalized eigenvalues and eigenvectors for the homogeneous equation (2.44) and relate the bifurcation of (2.1) with topological degrees at eigenvalues of (2.44).
As usual, it is expected that A σ converges to A 0 as σ → 0 + with an additional (and natural) condition: (H5) For all v ∈ X 0 , when σ → 0 * . This condition just means that A σ converges pointwise to A 0 in X 0 . Together with (H1) and (H3), (H5) shows that A 0 and B 0 are uniquely determined and are homogeneous of degree (p − 1) in u, that is, However, (H5) is not needed in the proof of our main results later. From the homegeneity of A 0 and B 0 , it immediately follows that if (u, λ) is a solution of (2.44), then so is (σ u, λ) with every σ ≥ 0. This suggests the following definition.
The following lemma gives a property of numbers which are not eigenvalues of (2.44). This is crucial to the proof of our main bifurcation result. (2.50) Proof. First, we show that for r > 0 sufficiently small, Assume otherwise that there are sequences {u n }, {σ n } such that σ n → σ ∈ [0, 1], 0 = u n X → 0, and 0 ∈ u n − P A σn B σ n u n , a , ∀n. (2.52)

Hence, u n ∈ P A σn [B σ n (u n , a)], that is,
A σ n u n = B σ n u n , a .
(2.53) Therefore, A(σ n u n ) = B(σ n u n , a). Putting v n = u n / u n X and dividing both sides of this equation by (σ n u n X ) p−1 , we get A σ n u n X v n = B σ n u n X v n , a , ∀n. Since v n X = 1, for all n, by passing to a subsequence if necessary, we can assume that v n v in X. (2.55) It follows that v n → v in Z. Because σ n u n X → 0, (H2a) implies that in Z * and then in X * . From (H1 ) with f n = B σ n u n X (v n , a), we have v ∈ X 0 and a). We useŪ Z andŪ X to denote the closure of U in Z and X, respectively.  If a and b (a < b) are not eigenvalues of (2.44) and if for some r > 0, small, then for and Ꮿ as in Theorem 2.2, then the Rabinowitz alternative (i) or (ii) holds.
Proof. Note that if a is not an eigenvalue of (2.44), then for all R > 0, all r ∈ (0, R), we have Using the homotopy invariance property of the degree for multivalued compact fields (cf. [11]), we have ·, a) , U, 0 .
(2.70) Theorem 2.9 is now a direct consequence of Theorem 2.2.

Bifurcation in equations with anisotropic p-Laplace-like operators
In this section, we apply Theorem 2.9 to a quasilinear elliptic equation that contains an anisotropic p-Laplace-like (φ-Laplace) operator. The equation is formulated in the weak form as follows: where X is a function space to be specified later. If φ i = φ, for all i ∈ {1, . . . , N}, then (3.1) reduces to the equation which has a variational structure and was treated previously in [8,9]. If the φ i 's are different then the principal operator in (3.1) does not have any potential functional and the approach used in the quoted papers seems not applicable here. Assume that for all i ∈ {1, . . . , N}, φ i : R → R + is an even continuous function such that φ i | R + is nondecreasing, φ i (0) = 0, and φ i > 0 on (0, ∞). Also, there exist γ > 1 and a, b, c, d > 0 such that Moreover, for simplicity of calculations, we assume that the φ i 's have the same behavior at 0, that is, there exists p ∈ [γ, γ * ) (γ * is the Sobolev conjugate of γ , defined by γ * = Nγ (N −γ ) −1 if γ < N and γ * = ∞ if γ ≥ N) and β ∈ R\{0} such that for all i, Suppose that F : × R × R → R is a Carathéodory function with the growth condition for a.e. x ∈ , for all λ, u ∈ R, where C ∈ L ∞ loc (R). Moreover, F (x,0, λ) = 0, for all λ ∈ R, a.e. x ∈ , and uniformly for a.e. x ∈ and for λ in bounded sets. From (3.3), we see that the integral in the left-hand side of (3.1) is well defined for u, v in the Sobolev space W 1,γ 0 ( ). Moreover, the operator A given by is well defined from X := W 1,γ 0 ( ) into X * = W −1,γ ( ) (γ = γ (γ − 1) −1 is the Hölder conjugate of γ ). Also, A is continuous and bounded in X with A(0) = 0. From the first inequality of (3.3) and from (3.4), we have a 1 and R 0 positive such that for all i = 1, . . . , N, (3.9) We have Vy Khoi Le 243 This shows that Hence, A is coercive in X in the sense of (2.2). We assume that the φ i 's satisfy the monotonicity condition for all ξ, η ∈ R N . It follows from (3.12) that A is monotone in W 1,γ 0 ( ) (but not strictly monotone in general). We put Z = L p ( ) and let B : Z × R → Z * be defined by Since p < γ * , the embeddings X → Z and Z * → X * are compact. Equation (3.5) implies that B is well defined and B(0, λ) = 0 for all λ. Moreover, Using Hölder's inequality, one can prove that there exists C > 0 such that that is, u n → 0 in X. We have proved (2.30). This implies that the definition of bifurcation points for (2.8) given in Definition 2.1 is equivalent to the usual definition of bifurcation points. Now, we put X 0 = W 1,p 0 ( ) with the usual norm Since p ≥ γ , X 0 ⊂ X and the embedding X 0 → X is continuous. Let A 0 : X 0 → X * 0 be given by It is clear that A 0 is well defined, continuous, bounded, and coercive in X 0 . Also, A 0 (0) = 0. For σ > 0, define A σ as in (2.33). We check that A σ converges to A 0 as σ → 0 + in the sense of (H1 ) (or equivalently (H1)). Assume that we have sequences {σ n }, {u n } such that

22)
A σ n u n = f n −→ f in X * . (3.23) We prove that u ∈ X 0 and A 0 (u) = f in X * 0 . We have = G σ n ∇u n , ∇u n dx, (3.25) where  This means that u ∈ X 0 = W 1,p 0 ( ). We show that (3.33) First, note that for u ∈ X 0 , In fact, by Hölder's inequality, there is a constant C > 0 independent of u and n such that On the other hand, by (3.4), Then by the dominated convergence theorem, We have from the monotonicity of A and the definition of A σ that and thus From Minty's lemma (cf. [5]), this implies which is, in its turn, equivalent to (3.33  for u, v ∈ L p ( ), λ ∈ R. It is clear that B 0 is defined and continuous. We check (H2a). Assume σ n → 0 + , v n → v in L p ( ), and λ n → λ. Since B σ n v n , λ n , v = F x, σ n v n , λ n v σ p−1 n dx, ∀v ∈ L p ( ), (3.56) by using (3.5), (3.6), and the dominated convergence theorem, we immediately have (2.36). Note that (3.56) and (3.5) also imply (H2b). On the other hand, the arguments used in the proof of (H1) also prove (H5). We have just checked all background assumptions so that the bifurcation Theorem 2.9 is applicable. Now, we consider the homogenized equation associated to (3.1). From (3.21) and (3.55) we see that the homogenized equation (2.44) associated to (2.1) is in our example given by β |∇u| p−2 ∇u · ∇v dx−λ |u| p−2 uv dx = 0, ∀v ∈ W for all λ close to βλ 1 /p and different from βλ 1 /p. Hence, by using (3.57) and (3.62) together with the bifurcation Theorem 2.9, we have the following bifurcation for (3.1).