AN INDEX FORMULA FOR THE DEGREE OF ( S ) +-MAPPINGS ASSOCIATED WITH ONE-DIMENSIONAL p-LAPLACIAN

The topological degree for (S) + -mappings concerning a nonlinear eigenvalue problem associated with one-dimensional p -Laplacian is evaluated. The result is applied to a variational inequality, where the multiple existence of solutions is discussed.


Introduction
In this paper, we evaluate the topological degree for (S) + -mappings concerning the following nonlinear eigenvalue problem: p−2 u (x) = µ u(x) p−2 u(x), x ∈ (0,1), where p > 1 and µ ∈ R. It is shown by the second author [12] that all eigenvalues for (1.1) are explicitly written in terms of the beta function as follows: and that µ k (p) are all simple.Define the (S) + -mapping (see [2]) T p µ : W 1,p 0 (0,1) → W −1, p (0,1) by Then, we have the following theorem.
These kind of results are useful for the study of the (multiple) existence of the equations of the type − div |∇u| p−2 ∇u = f (x,u) in Ω, u = 0 on ∂Ω (1.5) as well as the variational inequalities of the type where f is a Carathéodory function and ϕ is a lower semicontinuous convex function.
The index formulas for mappings of class (S) + or "densely defined" mappings which satisfy a variant of (S) + condition have been investigated in [7,8,9,13] in abstract settings.However, they assume that the leading term of mappings does not degenerate in a sense, and hence their results cannot be applied directly to our problem.
To prove Theorem 1.1, we employ a technique similar to those in [3,4], a homotopic deformation along p to the case p = 2.In [3], such a deformation is applied in C[0,1], the same Banach space where the degree is considered in the formula corresponding to (1.4).On the other hand, in [4], the corresponding result to Theorem 1.1 is considered in W 1,p 0 (Ω), which varies with p, so more delicate arguments are required.One needs a lemma which provides a connection between two degrees in different Banach spaces, the degree in W 1, p 0 (Ω), and the degree in L q (Ω) for some fixed q (see [4,Lemma 2.4]).Then, a homotopic deformation is used in L q (Ω).
Our strategy is similar to that of the latter case.We employ the degree theory for subdifferential operators which is developed in our previous work [10].It is shown that the left-hand side of (1.4) coincides with the degree in L 2 (0,1) for some mapping given as the sum of a subdifferential operator and a perturbation associated with (1.1).Then, a homotopic deformation along p is applied in the fixed space L 2 (0,1).
This paper is composed of four sections.In Section 2, we recall some basic facts on the degree for (S) + -mappings as well as the degree for subdifferential operators.A proof of Theorem 1.1 is given in Section 3, where the procedure mentioned above is carried out.In Section 4, we give an example of applications to variational inequalities, where the multiple existence of solutions is discussed.

Preliminaries
If X is a Banach space, then the norm of X will be denoted by • X or | • | X .We denote by •, • X the duality pairing between X and its dual X * and by U X (u,r) the open ball of X centered at u with radius r > 0. For a subset A of X, the closure and the boundary of A with respect to the topology of X are designated by A X and ∂ X A, respectively.
Usually, deg(M,D, f ) stands for the degree for a mapping M relative to a bounded open subset D at a point f .By substituting M − f for M, we can always assume that f = 0. We simply denote deg(M,D) instead of deg(M,D,0).

Degree for (S)
+ -mappings.Let X be a real reflexive Banach space.A single-valued mapping S from D(S) ⊂ X into X * is said to be class (S) + if for any sequence (u n ) of D(S), the conditions imply that u n → u strongly.We here recall how to define the degree for S.
Let D be a bounded open subset of X and let S be a demicontinuous (S) + -mapping from D into X * .Let {X α : α ∈ A} be the family of all finite-dimensional subspaces of X and let S α be the Galerkin approximation of S with respect to X α , that is, ( Then, deg (2.5) 984 An index formula associated with p-Laplacian Then, as it is well known, ∂ϕ becomes a maximal monotone operator in H.For λ > 0, we denote by J λ and ∂ϕ λ the resolvent and the Yosida approximation of ϕ, respectively, that is, where I is the identity on H.It follows from (A.0) that 0 ∈ ∂ϕ(0) and hence J λ 0 = 0, ∂ϕ λ (0 (Concerning other type of conditions equivalent to (A.P) In [10], we introduced two classes of multivalued perturbations (and their homotopies).For our purpose here, we only recall one of them in a restricted form (especially we consider a class of single-valued perturbations).Definition 2.3.For a given ϕ ∈ Φ C (H), denote by ᏮᏰ 1 (ϕ) the collection of mappings B from D(B) ⊂ H into H which satisfy the following conditions: Bu, (A.3) there exist k ∈ (0,1), α ∈ (0,2), and a positive, monotone increasing function l such that 3) is satisfied uniformly in t, that is, the constants k, α and the function l can be chosen independently of t.
Let ϕ ∈ Φ C (H) and B ∈ ᏮᏰ 1 (ϕ).We here sketch how to define the degree for ∂ϕ + B. Since H is separable, there exists a sequence (H i ) i∈N of finite-dimensional subspaces such that (2.9) For i ∈ N and λ > 0, we put B i,λ : for sufficiently small λ > 0 and for sufficiently large i ∈ N.
In the following, for a mapping M on H, we denote by M i the Galerkin approximation of M with respect to H i : ( It then holds that for all u ∈ H i , We also note that if i ≥ j, then (P j • M) i coincides with M i .Therefore, by (2.3) (and the fact mentioned below it) and by (2.10), we have for sufficiently small λ > 0 and for sufficiently large (2.14) Then, it easily follows that ) in the sense of distribution with the domain Then, B p µ ∈ ᏮᏰ 1 (ϕ p ) holds.Indeed, the verification of (A.1) is clear.The condition (A.2) follows from the fact that B p µ is weakly-strongly continuous from W 1, p 0 (0,1) into H.

An index formula associated with p-Laplacian
As for the boundedness condition (A.3), by the Gagliardo-Nirenberg inequality, there exist (2.17) Hence, (A.3) follows.

Proof of Theorem 1.1
In this section, let V p and H denote W 1, p 0 (0,1) and L 2 (0,1), respectively.Then, where each injection is dense and compact.Let T p µ , ϕ p , and B p µ be given in (1.3), (2.14), and (2.16), respectively.It then follows from (2.15) and (2.17) that, for all u ∈ V p , Moreover, there exists a sequence (H i ) i∈N of finite-dimensional subspaces which satisfies (2.9) and and We also notice that for each L > 0, the level set {u ∈ H : Now, let p ∈ (1,∞) and µ = µ k (p) for all k ∈ N. We are going to show by using the homotopy between the Galerkin approximations In the following three lemmas, we drop p and µ for simplicity.
Lemma 3.1.Let r 1 > 0 and r 2 > 0 be such that U H (0,r 1 ) ⊃ U V (0,r 2 ) V .Then, there exists i 0 ∈ N such that for all i ≥ i 0 , J. Kobayashi and M. Ôtani 987 Proof.Suppose that the assertion of the lemma was false.Then, we could find sequences and T in (u n ) = 0. Especially, we have which implies Therefore, it follows from (2.17) that ϕ(u n ) is bounded as n→∞.Hence, u n V is bounded, and so is T(u n ) V * .Passing to subsequence if necessary, we may assume that u n u weakly in V .Then, taking a sequence (v n ) such that v n ∈ H n and v n → u strongly in V (see (3.4)), we get Since T is of class (S) + , it follows that u n → u strongly in V (and hence u V ≥ r 2 ).From the continuity of T, we easily deduce T(u) = 0, which implies u = 0. Thus, we are led to a contradiction.
Let r,ρ > 0. By choosing r 2 > 0 such that U H (0,r) ⊃ U V (0,r 2 ) V and applying the lemma above and the excision property of degree, we deduce for large i ∈ N.
Lemma 3.3.Let r > 0, and let λ be fixed to satisfy (3.12).Then, there exists i 0 = i 0 (λ) ∈ N such that for all i ≥ i 0 for some u,v * ∈ H.Moreover, by the estimate we see that ϕ(J λ u n ) is bounded.We may assume that J λ u n converges to some v weakly in V and strongly in H.Then, by the demiclosedness of ∂ϕ, we have [v,v * ] ∈ ∂ϕ.Moreover, letting n → ∞ in the equation u n = J λ u n + λ∂ϕ λ (u n ), we get u = v + λv * .Therefore, by the definition of J λ and ∂ϕ λ , we obtain v = J λ u and v * = ∂ϕ λ (u).Thus, Since T is of class (S) + , it follows that u n → u strongly in V , and hence We are going to show that P in T(u n ) T(u) weakly in V * .Let w be arbitrarily taken in V .Take a sequence (w n ) such that w n ∈ H in and w n → w strongly in V .Then, noticing that which contradicts (3.12) since |u| H = r.Now let t = 0, that is, t n → 0.Then, since Moreover, combining the same argument as in the proof of the previous lemma and the same one as above, we get which implies u = 0. We are going to show that u n → u strongly in H (and therefore |u| H = r), which leads to a contradiction.By (3.28) and (3.35), lim n→∞ Therefore, noticing that P in u → u strongly in H, we get Since ∂ϕ λ is of class (S) + in H, we conclude that u n → u strongly in H.
We have thus proved the lemma.
Let λ > 0 be so small and i ∈ N be so large that (3.21) is satisfied.Then, by (2.13) and by the homotopy invariance of the Brouwer degree, we have Combining this equality and (3.11), we deduce (3.5).
In order to employ the homotopy invariance of deg H (•,•) along p, we need the following lemma.

Application
Consider the following variational inequality: Find where 1 < p < ∞, f : (0,1) × R → R is a Carathéodory function and for some C > 0. In [11], the higher-dimensional version of (4.1) is considered, where the multiple existence of solutions is shown under the condition that lim τ→0 f (x,τ)/|τ| p−2 τ lies between the first and the second eigenvalues of p-Laplacian in a sense.As for the one-dimensional case, we have the following theorem.
Proof.Since a proof is quite similar to that of [11,Theorem 4.1], we only give an outline of it.We employ the degree for (S) + -mappings with maximal monotone perturbations.
It will be also denoted by deg (S)+ (•,•).Since K forms a bounded subset of L ∞ (0,1), we may assume that f (x,τ) ≤ ρ(x) a.e.x ∈ (0,1), ∀τ ∈ R, (4.5) for some ρ ∈ L 1 (0,1) by cutting off the function f with respect to the second variable.Set V = W where I K is the indicator function of K and ∂ V I K : V → V * is its subdifferential in V .Moreover, u is a solution of (4.7) if and only if it is a critical point of the functional Ψ + I K , where We proceed to find three solutions of (4.7).The first solution is u = 0 since f (x,0) = 0 by (4.4).The second one is the global minimizer of Ψ + I K .Indeed, Ψ + I K is bounded below by (4.5), and its infimum is achieved by some u 0 ∈ K since K is bounded and Ψ + I K is weakly lower semicontinuous.Moreover, it easily follows from (4.4) that Ψ(u 0 ) < 0 = Ψ(0), which implies u 0 = 0.
weakly and Bu n b weakly, then b =

( 3 .
46)Hence, u should satisfy ∂ϕ p ( u) = v, which implies u = u.Thus, (A.P) h is verified with u n = u n and v n ≡ v. Now, we are ready to prove Theorem 1.1.