Eigenvalues of the p(x) -Laplacian Neumann problems

Nonlinear Analysis 67 (2007) 2982–2992 www.elsevier.com/locate/na

Eigenvalues of the p(x)-Laplacian Neumann problemsI Xianling Fan ∗ Department of Mathematics, Lanzhou University, Lanzhou, 730000, PR China Received 20 April 2006; accepted 25 September 2006

Abstract We study the eigenvalues of the p(x)-Laplacian operator with zero Neumann boundary condition on a bounded domain, where p(x) is a continuous function defined on the domain with p(x) > 1. We show that, similarly to the p-Laplacian case, the smallest eigenvalue of the problem is 0 and it is simple, and the supremum of all the eigenvalues is infinity, however, unlike the p-Laplacian case, for very general variable exponent p(x), the first eigenvalue is not isolated, that is, the infimum of all positive eigenvalues of the problem is 0. We also study some properties of the set of functions having p(x)-average value zero. c 2006 Elsevier Ltd. All rights reserved.

MSC: 35B45; 35J60; 35J70 Keywords: p(x)-Laplacian; Neumann problem; Eigenvalue; Variable exponent Sobolev space; Ljusternik–Schnirelman principle

1. Introduction In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent. The main references in this field can be found in an overview paper [6]. For the application backgrounds of the p(x)-Laplacian equations we refer to the monographs [13,18]. The existence of solutions of p(x)-Laplacian Dirichlet problems has been studied by several authors (see e.g. [9,10,16]), in particular for the eigenvalues of the p(x)-Laplacian Dirichlet problem see [10]. The purpose of the present paper is to study the eigenvalues of the p(x)Laplacian Neumann problem. This is a new topic. Throughout the paper, Ω will be a bounded domain in R N with smooth boundary ∂Ω , p ∈ C(Ω ) and 1 < p− = p− (Ω ) := inf p(x) ≤ p+ = p+ (Ω ) := sup p(x) < ∞. x∈Ω

(1.1)

x∈Ω

Consider the eigenvalue problem of p(x)-Laplacian with Neumann boundary condition  −1 p(x) u := −div(|∇u| p(x)−2 ∇u) = λ|u| p(x)−2 u in Ω ∂u  = 0 on ∂Ω , ∂ν where ν is the outward unit normal to ∂Ω . I Research supported by NNSF of China (10371052, 10671084). ∗ Tel.: +86 931 8911173; fax: +86 931 8911100.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.052

(1.2)

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In the case when p(x) ≡ p ∈ (1, ∞), the eigenvalues of the p-Laplacian operator have been extensively investigated, see e.g. [1,2,4,5,7,8,12,15,17,19] and references therein. In particular in [15] the eigenvalue problems of p-Laplacians subject to different kinds of boundary conditions are studied by using a unified treatment. It is well known that (see e.g. [12,15]), there is a nondecreasing sequence {λn } of the eigenvalues of the p-Laplacian Neumann problem such that λ1 = 0, λn → +∞ and λ2 = inf{λ > 0 : λ is an eigenvalue of the p-Laplacian Neumann problem} > 0. Set Λ = {λ : λ is an eigenvalue of (1.2)}. It is easy to see that, similarly to the constant exponent case, λ1 = min Λ = 0, λ1 is simple, all eigenfunctions associated with λ1 = 0 are nonzero constant functions, and sup Λ = +∞. However, we will show that (see Theorem 3.3 below), unlike the constant exponent case, for very general variable exponent p(x), inf{λ > 0 : λ ∈ Λ} = 0. The main cause for this occurrence is the inhomogeneity of the p(x)-Laplacian operator. It is easy to see that all eigenfunctions of (1.2) associated with λ > 0 are in the set   Z V p(x) := v ∈ W 1, p(x) (Ω ) : |v| p(x)−2 vdx = 0 .

(1.3)

Ω

In Section 2 we present some elementary properties of the set V p(x) , which are needed in Section 3 and are also of independent interest. In Section 3 we prove the main results on the eigenvalues of (1.2). 2. Space W 1, p(x) (Ω ) and set V p(x) We first recall some basic facts about the variable exponent Lebesgue–Sobolev spaces. Let Ω ⊂ R N and p ∈ C(Ω ) be as in Section 1. The variable exponent Lebesgue space L p(x) (Ω ) is defined by   Z p(x) p(x) L (Ω ) = u | u : Ω → R is measurable, |u| dx < ∞ Ω

with the norm  Z p(x) u dx ≤ 1 . = inf σ > 0 : Ω σ 

|u| L p(x) (Ω ) = |u| p(x)

The variable exponent Sobolev space W 1, p(x) (Ω ) is defined by W 1, p(x) (Ω ) = {u ∈ L p(x) (Ω ) : |∇u| ∈ L p(x) (Ω )} with the norm kukW 1, p(x) (Ω ) = kuk1, p(x) = |u| p(x) + |∇u| p(x) . 1, p(x)

1, p(x)

Define W0 (Ω ) as the closure of C0∞ (Ω ) in W 1, p(x) (Ω ). The spaces L p(x) (Ω ), W 1, p(x) (Ω ) and W0 (Ω ) all 1, p(x) are separable and reflexive Banach spaces. We refer to [6,11,14] for the elementary properties of the space W (Ω ). Define   N p(x) , if p(x) < N p ∗ (x) = N − p(x) ∞, if p(x) ≥ N . Proposition 2.1 (See [11]). Let q ∈ C(Ω ) and 1 ≤ q(x) < p ∗ (x) for all x ∈ Ω . Then there is a compact imbedding W 1, p(x) (Ω ) ,→ L q(x) (Ω ).

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Let us turn to the study of the set V p(x) . For a function u(x), define u + (x) = max{u(x), 0}, u − (x) = max{−u(x), 0}. R R R Note that Ω |u| p(x)−2 udx = 0 ⇔ Ω |u + | p(x)−1 dx = Ω |u − | p(x)−1 dx. Proposition 2.2. For every u ∈ L p(x) (Ω ), there exists a unique real number r = r (u) such that Z |u − r | p(x)−2 (u − r )dx = 0.

(2.1)

Ω

Proof. Let u ∈ L p(x) (Ω ). For t ∈ R define Z Z + p(x)−1 |(u − t)− | p(x)−1 dx. |(u − t) | dx, g(t) = f (t) = Ω

Ω

It is easy to see that the functions f, g : R → R are continuous, f is nonincreasing, g is nondecreasing, ( f − g) is strictly decreasing, f (t) − g(t) → +∞ as t → −∞, and f (t) − g(t) → −∞ as t → +∞. Hence there exists a unique r = r (u) such that f (r ) − g(r ) = 0, i.e. (2.1) holds.  We call the real number r = r (u) satisfying (2.1) the p(x)-average value of u. By Proposition 2.2, there is a real function r : L p(x) (Ω ) → R satisfying (2.1). Proposition 2.3. The p(x)-average value function r : L p(x) (Ω ) → R has the following properties: (1) Z Z Z r (u) = 0 ⇐⇒ |u| p(x)−2 udx = 0 ⇐⇒ |u + | p(x)−1 dx = |u − | p(x)−1 dx; Ω Ω Ω Z Z Z p(x)−2 + p(x)−1 r (u) > 0 ⇐⇒ |u| udx > 0 ⇐⇒ |u | dx > |u − | p(x)−1 dx; Ω Ω Ω Z Z Z r (u) < 0 ⇐⇒ |u| p(x)−2 udx < 0 ⇐⇒ |u + | p(x)−1 dx < |u − | p(x)−1 dx. Ω

Ω

Ω

(2) r : L p(x) (Ω ) → R is bounded on every bounded set in L p(x) (Ω ). (3) r : L p(x) (Ω ) → R is continuous. Proof. Assertion (1) is obvious. Here we only prove assertions (2) and (3). To prove the boundedness of r , let u ∈ L p(x) (Ω ) with |u| p(x) ≤ K , where K is a positive constant. We will prove that |r (u)| ≤ C, where C isR a positive constantRdepending on K but independent of u. Without loss of generality we may assume r (u) > 0, i.e. Ω |u + | p(x)−1 dx > Ω |u − | p(x)−1 dx. Then we have that Z Z + p(x)−1 |(u − r (u)) | dx = |(u − r (u))− | p(x)−1 dx, Ω Ω Z Z Z |(u − r (u))+ | p(x)−1 dx < |u + | p(x)−1 dx ≤ (1 + |u| p(x) )dx ≤ C1 , Ω

Ω

Ω

where C1 is a positive constant depending only on K , |Ω |, p− and p+ . Since |u| p(x) ≤ K , there is a positive constant M, depending only on K , |Ω |, p− and p+ , such that |Ω | . 2 Take d > 1 large enough such that |{x ∈ Ω : u(x) ≤ M}| ≥

(d − 1)M > 1,

((d − 1)M) p− −1 ·

Then when t > d M, Z Z |(u − t)− | p(x)−1 dx ≥ Ω

|Ω | > C1 . 2

((d − 1)M) p(x)−1 ≥ ((d − 1)M) p− −1 · {u≤M}

This shows r (u) ≤ d M. Assertion (2) is proved.

|Ω | > C1 . 2

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To prove the continuity of r , let u n → u 0 in L p(x) (Ω ), r (u n ) = rn and r (u 0 ) = r0 . We will prove rn → r0 . To this end it is sufficient to show that every subsequence of {rn } has a subsequence converging to r0 . Given a subsequence of {rn }, which is denoted still by {rn }, by assertion (2), there is a convergent subsequence of {rn } : rn i → r∗ . Now Z Z |(u n i − rn i )+ | p(x)−1 dx = |(u ni − rn i )− | p(x)−1 dx. Ω

Ω

Going to the limit for i → ∞ in the above equality, yields Z Z + p(x)−1 |(u 0 − r∗ )− | p(x)−1 dx. |(u 0 − r∗ ) | dx = Ω

Ω

By Proposition 2.2, r∗ = r0 . The proof is complete.



Let V p(x) be as in (1.3), that is V p(x) = {v ∈ W 1, p(x) (Ω ) : r (v) = 0}. Since W 1, p(x) (Ω ) ,→ L p(x) (Ω ), we may think of r (u) as a function defined on W 1, p(x) (Ω ). Because the imbedding W 1, p(x) (Ω ) ,→ L p(x) (Ω ) is compact, from Propositions 2.2 and 2.3 we obtain the following Proposition 2.4. (1) Given any u ∈ W 1, p(x) (Ω ), there exists a unique r (u) ∈ R such that u − r (u) ∈ V p(x) . (2) The function r : W 1, p(x) (Ω ) → R is sequentially weakly continuous.  Define   Z V2 = u ∈ W 1, p(x) (Ω ) : udx = 0 . Ω

R For u ∈ W 1, p(x) (Ω ), denote u = |Ω1 | Ω udx and e u = u − u. Then u = u + e u , where u ∈ R and e u ∈ V2 . So 1, p(x) 1, p(x) (Ω ) = R ⊕ V2 . V2 is a closed linear subspace of W (Ω ) with codimension 1. W V p(x) is a sequentially weakly closed subset of W 1, p(x) (Ω ). V p(x) is symmetric, that is, u ∈ V p(x) implies −u ∈ V p(x) . When p(x) ≡ p, V p is positively homogeneous, that is, u ∈ V p implies tu ∈ V p for every t > 0. For general p(x), V p(x) is not positively homogeneous. By statement (1) in Proposition 2.4, we can write W 1, p(x) (Ω ) = R ⊕ V p(x) , which means that, given any u ∈ W 1, p(x) (Ω ), there exist a unique r (u) ∈ R and a unique v ∈ V p(x) such that u = r (u) + v. The following proposition shows that there exists a natural homeomorphism between V p(x) and V2 . Proposition 2.5. Define h(v) = v − v

for v ∈ V p(x) .

Then h : V p(x) → V2 is a homeomorphism, and the mappings h and h −1 are bounded. Proof. It is easy to see that the mapping h : V p(x) → V2 is continuous and bounded. Let v1 , v2 ∈ V p(x) and h(v1 ) = h(v2 ). Then v1 − v1 = v2 − v2 , consequently v1 = v2 + (v1 − v2 ) ∈ V p(x) , which implies that v1 − v2 = 0 and R v1 = v2 . This shows that h is injective. Given any u ∈ V2 , then Ω udx = 0. By Proposition 2.4 (1), there exists a unique r = r (u) such that u −r ∈ V p(x) . Setting v = u − r , then v ∈ V p(x) and Z 1 h(v) = v − v = (u − r ) − (u − r )dx = u − r + r = u. |Ω | Ω This shows that h is surjective. It is easy to see that h −1 (u) = u − r (u) for every u ∈ V2 , therefore, the mapping h −1 : V2 → V p(x) is continuous and bounded. The proof is complete.  Proposition 2.6. There exists a positive constant C0 such that |u| p(x) ≤ C0 |∇u| p(x) ,

∀u ∈ V2 .

(2.2)

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Proof. Arguing by contradiction, assume that there exists a sequence {u n } ⊂ V2 such that |u n | p(x) ≥ n|∇u n | p(x) . Without loss of generality we may assume |u n | p(x) = 1, then |∇u n | p(x) ≤ n1 . We may assume, taking a subsequence w

if necessary, that u n * u 0 in W 1, p(x) (Ω ), u n → u 0 in L p(x) (Ω ) and u n (x) → u 0 (x) for a.e. x ∈ Ω . So we have that |u 0 | p(x) = 1, Z Z u 0 dx = lim u n dx = 0, n→∞ Ω

Ω

and Z

|∇u 0 | p(x) dx ≤ lim

Z

n→∞ Ω

Ω

|∇u n | p(x) dx = 0.

It follows that u 0 ∈ V2 and ∇u 0 = 0, consequently u 0 = 0, which contradicts |u 0 | p(x) = 1.



Proposition 2.7. Let v ∈ V p(x) be given. Define ϕ : R → R by Z |v − t| p(x) dx for t ∈ R. ϕ(t) = p(x) Ω Then ϕ 0 (0) = 0, ϕ 0 (t) < 0 for t < 0, and ϕ 0 (t) > 0 for t > 0, so the function ϕ(t) attains its minimum at t = 0. Proof. It is easy to verify that Z ϕ 0 (t) = − |v − t| p(x)−2 (v − t)dx. Ω

Then Proposition 2.7 follows from Proposition 2.3.



For each v ∈ V p(x) , define L v := {v + t : t ∈ R} = v ⊕ R. We call the straight line L v the fibre over v ∈ V p(x) . Define Z |u| p(x) G(u) = dx for u ∈ W 1, p(x) (Ω ). p(x) Ω Proposition 2.7 shows that, for each v ∈ V p(x) , G(v) < G(v + t) for all t 6= 0, in other words, G(v) < G(u) for u ∈ L v \ {v}. Remark 2.1. For u ∈ L p(x) (Ω ), define   Z 1 u p(x) dx ≤ 1 . |u|0p(x) = inf σ > 0 : Ω p(x) σ Then | • |0p(x) is a norm on L p(x) (Ω ) equivalent to | • | p(x) . By Proposition 2.6, |u|0p(x) ≤ C00 |∇u|0p(x) ,

∀u ∈ V2 .

Set ( M1 = u ∈ W

1, p(x)

) |u| p(x) (Ω ) : dx = 1 . Ω p(x) Z

R p(x) Note that Ω |u|p(x) dx = 1 if and only if |u|0p(x) = 1. Take any v ∈ V p(x) ∩ M1 and denote u = h(v) = v − v ∈ V2 . Then |u|0p(x) ≤ C00 |∇u|0p(x) . By Proposition 2.7, G(u) ≥ G(v) = 1, which implies |u|0p(x) ≥ 1 = |v|0p(x) . Noting that ∇u = ∇v, we get |v|0p(x) ≤ C00 |∇v|0p(x) ,

∀v ∈ V p(x) ∩ M1 .

(2.3)

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When p(x) ≡ p, by the homogeneity of V p , it follows from (2.3) that |v|0p ≤ C00 |∇v|0p for all v ∈ V p . For general p(x) we can only get the following result: If A ⊂ V p(x) is such that |∇v|0p(x) ≤ K for all v ∈ A, where K is a positive constant, then there exists a positive constant C K depending on K such that |v|0p(x) ≤ C K for all v ∈ A. 3. Eigenvalues Now let us consider the eigenvalue problem (1.2). Definition 3.1. Let u ∈ W 1, p(x) (Ω ) and λ ∈ R. (1) (u, λ) is called a (weak) solution of (1.2) if Z Z |u| p(x)−2 uvdx, |∇u| p(x)−2 ∇u∇vdx = λ Ω

Ω

∀v ∈ W 1, p(x) (Ω ).

(3.1)

(2) λ is called an eigenvalue of (1.2) if there exists u ∈ W 1, p(x) (Ω ) \ {0} such that (u, λ) is a solution of (1.2). In this case u is called an eigenfunction associated with λ. Set Λ = {λ ∈ R : λ is an eigenvalue of (1.2)}. Theorem 3.1. (1) λ ≥ 0 for every λ ∈ Λ. (2) λ1 = 0 ∈ Λ is the smallest eigenvalue of (1.2) and only eigenfunctions associated with λ1 are nonzero constant functions. R (3) If (u, λ) is a solution of (1.2) and λ 6= 0, then Ω |u| p(x)−2 udx = 0, that is u ∈ V p(x) . Proof. Taking v = u and v = 1 in (3.1) respectively, we can prove assertions (1) and (3). Assertion (2) is obvious. Obviously, the problem (1.2) is equivalent to the following problem  −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = µ|u| p(x)−2 u in Ω ∂u  = 0 on ∂Ω , ∂ν with λ = µ − 1. Define F, G : W 1, p(x) (Ω ) → R by  Z  1 1 |∇u| p(x) + |u| p(x) dx, F(u) = p(x) p(x) ZΩ 1 G(u) = |u| p(x) dx, ∀u ∈ W 1, p(x) (Ω ). p(x) Ω



(3.2)

∀u ∈ W 1, p(x) (Ω ),

For brevity, we will write X instead of W 1, p(x) (Ω ) and write kuk instead of kukW 1, p(x) (Ω ) . The proof of the 1, p(x)

following proposition is standard, for the case when X = W0

(Ω ), see [9].

Proposition 3.1. (1) F, G ∈ C 1 (X, R), and Z Z p(x)−2 0 F (u)v = |∇u| ∇u∇vdx + |u| p(x)−2 uvdx, Ω ZΩ 0 p(x)−2 G (u)v = |u| uvdx, ∀v ∈ X.

∀v ∈ X,

Ω

(2) : X → X ∗ is a strictly monotone homeomorphism, the mappings F 0 and (F 0 )−1 are bounded, F 0 is of (S+ ) type, namely, F0

w

un * u0

in X

and

lim F 0 (u n )(u n − u 0 ) ≤ 0

n→∞

imply u n → u 0

in X. w

(3) G 0 : X → X ∗ is sequentially weakly–strongly continuous, namely, u n * u 0 in X implies G 0 (u n ) → G 0 (u 0 ) in ∗ X . 

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It is clear that (u, µ) is a solution of (3.2) if and only if F 0 (u) = µG 0 (u).

(3.3)

As is well known, to solve the eigenvalue problem (3.3), the constrained variational method can be applied. The usual method in most references is to take G as an objective functional and F as a constraint functional (see e.g. [2,3, 15,20,21]). Of course, the dual method, that is, by taking F as an objective functional and G as a constraint functional, can also be used equivalently (see e.g. [7,8]). In this paper we well use the latter. For each t > 0, denote Mt = {u ∈ X : G(u) = t}. Then Mt is a C 1 -submanifold of X with codimension 1 since 0 e : Mt → R G (u) 6= 0 for u ∈ Mt . We denote by Tu (Mt ) the tangent space at u ∈ Mt , i.e. Tu (Mt ) = ker G 0 (u), by F 0 e the derivative of F e at u ∈ Mt , i.e. the restriction of F (u) on Tu (Mt ). the restriction of F on Mt , and by d F(u) e : Mt → R satisfies the (P.S) condition, namely, any sequence Proposition 3.2. For each t > 0, the functional F e e {u n } ⊂ Mt such that F(u n ) → c and d F(u n ) → 0 contains a converging subsequence. Proof. Proposition 3.2 can be proved by using the standard arguments (see e.g. [20]). Here we give a proof of Proposition 3.2, in which the duality mapping J : X → X ∗ used in the standard arguments is replaced by the homeomorphism F 0 : X → X ∗ . Let u ∈ Mt . It is easy to see that G 0 (u) 6= 0 and (F 0 )−1 (G 0 (u)) 6∈ Tu (Mt ). Thus X = Tu (Mt ) ⊕ {α(F 0 )−1 (G 0 (u)) : α ∈ R}. Let P : X → Tu (Mt ) be the natural projection. Then for every v ∈ X , there is a unique α ∈ R such that v = Pv + α(F 0 )−1 (G 0 (u)). Acting on the above equation by G 0 (u) and noting that hG 0 (u), Pvi = 0, we get α=

hG 0 (u), vi , hG 0 (u), (F 0 )−1 (G 0 (u))i

consequently, e hd F(u), vi = hF 0 (u), Pvi  = hF 0 (u), vi − F 0 (u), = hF 0 (u), vi −

hG 0 (u), vi (F 0 )−1 (G 0 (u)) 0 hG (u), (F 0 )−1 (G 0 (u))i



hF 0 (u), (F 0 )−1 (G 0 (u))i 0 hG (u), vi, hG 0 (u), (F 0 )−1 (G 0 (u))i

which shows that e d F(u) = F 0 (u) −

hF 0 (u), (F 0 )−1 (G 0 (u))i 0 G (u) := F 0 (u) − µ(u)G 0 (u). hG 0 (u), (F 0 )−1 (G 0 (u))i

(3.4)

e n ) → c and d F(u e n ) → 0. Then obviously, {ku n k} is bounded, so we may Now let {u n } ⊂ Mt be such that F(u w assume that u n * u 0 in X , and consequently, u n → u 0 in L p(x) (Ω ), G(u n ) → G(u 0 ) and G 0 (u n ) → G 0 (u 0 ). e n ) = F 0 (u n ) − µn G 0 (u n ), where µn = µ(u n ). Denote It is clear that u 0 ∈ Mt . By (3.4) we have d F(u 0 −1 0 0 0 wn = (F ) (G (u n )). Then F (wn ) = G (u n ), and hence hG 0 (u n ), (F 0 )−1 (G 0 (u n ))i = hF 0 (wn ), wn i Z = (|∇wn | p(x) + |wn | p(x) )dx. Ω

 Noting that wn → w0 6= 0 in X because G 0 (u n ) → G 0 (u 0 ) 6= 0 in X ∗ , and |hF 0 (u n ), (F 0 )−1 (G 0 (u n ))i| is bounded, we can conclude that {µn } is bounded. So we may assume, taking a subsequence if necessary, that µn → µ0 . Then e n ) → 0. Hence u n → u 0 in X because the µn G 0 (u n ) → µ0 G 0 (u 0 ), consequently F 0 (u n ) → µ0 G 0 (u 0 ) because d F(u mapping F 0 is of (S+ ) type. The proof is complete. 

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e be as above. Set Let t, Mt , F, G and F Σt = {A ⊂ Mt : A is closed and − A = A}. Denote by γ (A) the genus of A ∈ Σt (see e.g. [3,21]). Define cn,t =

inf

e sup F(u),

A∈Σt ,γ (A)≥n u∈A

n = 1, 2, . . . .

(3.5)

By the theorem of Ljusternik–Schnirelman (see e.g. [3,20,21]), we know that each cn,t (n = 1, 2, . . . , ) is a critical e and cn,t ≤ cn+1,t . Using the standard arguments we can prove that cn,t → +∞ as n → ∞. It is easy to value of F e has infinitely many critical points {±u n,t : n = 1, 2, . . . , } ⊂ Mt such that see that c1,t = inf Mt F(u) = t. Thus F F(±u n,t ) = cn,t . This shows that the problem (3.2) has infinitely many solutions {(±u n,t , µn,t ) : n = 1, 2, . . . , }, and consequently the problem (1.2) has infinitely many solutions {(±u n,t , λn,t ) : n = 1, 2, . . . , } with λn,t = µn,t − 1. It follows from c1,t = t that λ1,t = 0, and from cn,t → +∞ that λn,t → +∞. Let Ct be the unique positive constant R p(x) t| satisfying Ω |Cp(x) dx = t. Then it is clear that, for u ∈ Mt , F(u) = t if and only if u = ±Ct . By the multiplicity theorem of Ljusternik–Schnirelman, we can know that c1,t < c2,t and 0 = λ1,t < λ2,t . Let us formulate these results in the form of the following theorem. e cn,t ≤ cn+1,t , t = c1,t < c2,t , Theorem 3.2. Each cn,t (n = 1, 2, . . . , ) defined by (3.5) is a critical value of F, and cn,t → +∞ as n → ∞. Moreover the problem (1.2) has infinitely many solutions {±u n,t , λn,t } such that G(±u n,t ) = t, F(±u n,t ) = cn,t , 0 = λ1,t < λ2,t , and λn,t → +∞ as n → ∞.  Define λ∗ = inf{λ > 0 : λ ∈ Λ}.

(3.6) c

It is well known that (see e.g. [12,15,17]), in the case when p(x) ≡ p, thanks to the homogeneity, the values n,t t are independent of t > 0, so are µn,t and λn,t , in particular, λ2 = λ2,t > 0, and in fact λ2 = λ∗ . However for general p(x), this is not the case due to the loss of the homogeneity. We will show that, for very general p(x), λ∗ = 0 (see Theorem 3.3 below). Now let us observe c2,t . As noted above, R also denotes the set of all constant functions in X . For u ∈ X \ R, we denote by πu the plane generated by u and R, that is πu = {tu + c : t ∈ R, c ∈ R}, and by πu+ the positive half plane: πu+ = {tu + c : t > 0, c ∈ R}. R + 1 Obviously, πu = π−u and π−u = −πu+ . We define I (u) = Ω p(x) |∇u| p(x) dx for u ∈ X . Thus F(u) = I (u) + G(u). Let A, B ⊂ X . We say that A and B are radially homeomorphic if 0 6∈ A, for each x ∈ A the ray {t x : t > 0} and B have a unique common point which is denoted by Q(x), and the mapping Q : A → B is a homeomorphism. Proposition 3.3. Let t > 0 be fixed, and let v ∈ Mt ∩ V p(x) . Then (1) L v = v + R and πv+ ∩ Mt are radially homeomorphic. (2) I (v) = max{I (w) : w ∈ πv+ ∩ Mt }. Proof. (1) Note that L v is a straight line in πv+ paralleling R and πv+ ∩ Mt is an open arc passing v and connecting Ct and −Ct . Take any u = v + r ∈ L v . Then there is a unique positive number s = s(u) such that G(su) = t, i.e. su ∈ Mt . Obviously s(v) = 1 since v ∈ Mt . When u 6= v i.e. r 6= 0, by Proposition 2.7, G(u) > G(v) = t, hence s(u) < 1. We define Q : L v → πv+ ∩ Mt by Q(u) = s(u)u for u ∈ L v . It is easy to show that the mapping Q : L v → πv+ ∩ Mt is a homeomorphism. (2) Let w ∈ πv+ ∩ Mt and w 6= v. Setting Q −1 (w) = u, then w = su with s ∈ (0, 1), which implies I (w) < I (u). Since u = v + r, ∇u = ∇v and I (u) = I (v). Thus I (w) < I (v) and the proof is complete. 

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Remark 3.1. In fact, it can be proved that the cylindrical surface (V p(x) ∩ Mt ) ⊕ R and Mt \ {±Ct } are radially homeomorphic. Proposition 3.4. c2,t = inf{F(v) : v ∈ V p(x) ∩ Mt }. Proof. Denote dt = inf{F(v) : v ∈ V p(x) ∩ Mt }. By Theorem 3.2, there is u 2,t ∈ Mt such that F(u 2,t ) = c2,t > t and −1 p(x) u 2,t = λ2,t u 2,t with λ2,t > 0. So u 2,t 6∈ R and u 2,t ∈ V p(x) , consequently dt ≤ c2,t . On the other hand, since the functional F : X → R is coercive and sequentially weakly lower semicontinuous, and V p(x) ∩ Mt is a sequentially weakly closed subset of X , there exists v∗ ∈ V p(x) ∩ Mt such that F(v∗ ) = dt . Set A = πv∗ ∩ Mt . Then γ (A) = 1. By Proposition 3.3(2), F(v∗ ) = I (v∗ ) + t = maxu∈A F(u). By the definition of c2,t , c2,t ≤ F(v∗ ) = dt . Hence c2,t = dt .  A main result in this paper is the following Theorem 3.3. Suppose that one of the following two conditions, either (H0 ) or (H∞ ), holds: (H0 ) There exists a compact subset Γ of Ω such that Ω \ Γ = Ω1 ∪ Ω2 , where Ω1 and Ω2 are two disjoint relative open subsets of Ω , and there exist x1 ∈ Ω1 and x2 ∈ Ω2 such that max{ p(x1 ), p(x2 )} < inf{ p(x) : x ∈ Γ }.

(3.7)

(H∞ ) In the condition (H0 ), (3.7) is replaced by min{ p(x1 ), p(x2 )} > sup{ p(x) : x ∈ Γ }.

(3.8)

Then λ∗ = 0. More precisely, if (H0 ) holds, then λ2,t → 0 as t → 0; and if (H∞ ) holds, then λ2,t → 0 as t → ∞. Below we write for u ∈ X \ {0}, R |∇u| p(x) dx , α(u) = RΩ p(x) dx Ω |u|

R

1 p(x) dx Ω p(x) |∇u| 1 p(x) dx Ω p(x) |u|

β(u) = R

=

I (u) . G(u)

To prove Theorem 3.3, we first prove the following Proposition 3.5. The following assertions are mutually equivalent: (1) λ∗ = 0. (2) inf{α(u) : u ∈ V p(x) } = 0. (3) inf{β(u) : u ∈ V p(x) } = 0. (4) inf{λ2,t : t > 0} = 0. Proof. (1) H⇒ (2): Let λ∗ = 0. Then there exists {λn } ⊂ Λ such that λn > 0 and λn → 0 as n → ∞. Let u n be the eigenfunction associated with λn . Then u n ∈ V p(x) and α(u n ) = λn → 0, which shows that (2) holds. It follows from (1.1) that (2) ⇐⇒ (3). It is obvious that (4) H⇒ (1). It remains to show that (3) H⇒ (4). Now let (3) hold. Then there exists {vn } ⊂ V p(x) such that β(vn ) → 0 as n → ∞. Let G(vn ) = tn and let c2,tn , u 2,tn and λ2,tn be as in Theorem 3.2. Then, by Proposition 3.4, for each n, c2,tn F(v) = inf = inf β(v) + 1 ≤ β(vn ) + 1. v∈V p(x) ∩Mtn G(v) v∈V p(x) ∩Mtn tn Hence when n → ∞, F(u 2,tn ) c2,tn = → 1, tn tn which implies that β(u 2,tn ) → 0, and consequently α(u 2,tn ) → 0, i.e. λ2,tn → 0, so (4) holds. The proof is complete.  Proof of Theorem 3.3. Here we only give the proof in the case (H0 ) because the proof in the case (H∞ ) is similar. Now assume that (H0 ) holds. We will prove that inf{α(u) : u ∈ V p(x) } = 0. Denote for A ⊂ Ω and δ > 0, B(A, δ) = {x ∈ R N : dist(x, A) < δ}

and

BΩ (A, δ) = B(A, δ) ∩ Ω .

X. Fan / Nonlinear Analysis 67 (2007) 2982–2992

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By the continuity of p(x), there exists δ > 0 sufficiently small such that BΩ (x1 , δ) ∩ BΩ (Γ , δ) = ∅, BΩ (x2 , δ) ∩ BΩ (Γ , δ) = ∅, and p1 := max{ p+ (BΩ (x1 , δ)), p+ (BΩ (x1 , δ))} < p− (BΩ (Γ , δ)) := p2 . Take u 0 ∈ C ∞ (Ω ) such that 0 ≤ u 0 ≤ 1, u 0 (x) = 1 if x ∈ Ω1 \ B(Γ , δ), and u 0 (x) = 0 if x 6∈ Ω1 . Take v0 ∈ C ∞ (Ω ) such that 0 ≤ v0 ≤ 1, v0 (x) = 1 if x ∈ Ω2 \ B(Γ , δ), and v0 (x) = 0 if x 6∈ Ω2 . Choose a positive constant r0 such that Z Z |r0 v0 | p(x)−1 dx. (3.9) |u 0 | p(x)−1 dx = Ω

Ω

Set w0 = r0 v0 . (3.9) implies that u 0 − w0 ∈ V p(x) . It is easy to see that, given any t > 0, there is s(t) > 0 such that tu 0 − s(t)w0 ∈ V p(x) and s(t) → 0 as t → 0. For sufficiently small t > 0, we have R R p(x) |∇u | p(x) dx |t∇u 0 | p(x) dx 0 c1 B(Γ ,δ)∩Ω1 t Ω R ≤ ≤ t p2 − p1 , α(tu 0 ) = R p(x) p(x) p(x) c2 dx |u 0 | dx Ω |tu 0 | BΩ (x1 ,δ) t R R where c1 = B(Γ ,δ)∩Ω1 |∇u 0 | p(x) dx and c2 = B (x1 ,δ) |u 0 | p(x) dx are positive constants. Repeating the same Ω reasoning, we can obtain that c3 α(s(t)w0 ) ≤ (s(t)) p2 − p1 . c4 Given any ε > 0, since p1 < p2 , for sufficiently small t > 0, we have α(tu 0 ) ≤ ε and α(s(t)w0 ) ≤ ε, consequently, R R p(x) dx + p(x) dx Ω1 |t∇u 0 | Ω2 |s(t)∇w0 | R R ≤ ε. α(tu 0 − s(t)w0 ) = p(x) dx + p(x) dx Ω1 |tu 0 | Ω2 |s(t)w0 | It follows from this that inf{α(u) : u ∈ V p(x) } = 0, which implies that λ∗ = 0. In addition, from the above proof we can see that, when (H0 ) holds, λ2,t → 0 as t → 0. The proof is complete.  Corollary 3.1. Let x1 ∈ Ω be a global minimizer (or maximizer) of p on Ω . If there exists a strictly local minimizer (correspondingly, maximizer) x2 ∈ Ω of p on Ω such that x1 6= x2 , then λ∗ = 0. Proof. In the “minimizer” case, taking ε > 0 small enough and setting Γ = {x ∈ Ω : dist(x, x2 ) = ε}, Ω2 = BΩ (x2 , ε) and Ω1 = Ω \ Ω2 . Then the condition (H0 ) in Theorem 3.3 is satisfied. Analogously, in the “maximizer” case, (H∞ ) holds.  Remark 3.2. Robinson [17] has shown that nonconstant eigenfunctions of the p-Laplacian Neumann problem do not necessarily have an average value of 0, i.e. are not necessarily in V2 . Acknowledgments The author is grateful to the reviewers for their valuable comments and suggestions. References [1] A. Anane, N. Tsouli, On the second eigenvalue of the p-Laplacian, in: Nonlinear Partial Differential Equations (F´es, 1994), in: Pitman Res. Notes Math. Ser., vol. 343, Longman, Harlow, 1996, pp. 1–9. [2] J.P.G. Azorero, I.P. Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987) 1389–1430. [3] F. Browder, Existence theorems for nonlinear partial differential equations, in: Global Analysis (Proceedings of the Symposium Pure Mathematics, vol. 16, Berkeley, California, 1968), American Mathematics Society, Providence, RI, 1970, pp. 1–60. [4] M. Cuesta, D.G. de Figueiredo, J.P. Gossez, The beginning of the Fuˇcik spectrum for the p-Laplacian, J. Differential Equations 159 (1999) 212–238. [5] M. Cuesta, D.G. de Figueiredo, J.P. Gossez, A nodal domain property for the p-Laplacian, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000) 669–673.

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X. Fan / Nonlinear Analysis 67 (2007) 2982–2992

[6] L. Diening, P. H¨ast¨o, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Dr´abek, J. R´akosn´ık (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38–58. [7] P. Dr´abek, S. Robinson, Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1999) 189–200. [8] P. Dr´abek, S. Robinson, On the generalization of the courant nodal domain theorem, J. Differential Equations 181 (2002) 58–71. [9] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. [10] X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005) 306–317. [11] X.L. Fan, D. Zhao, On the spaces L p(x) (Ω ) and W m, p(x) (Ω ), J. Math. Anal. Appl. 263 (2001) 424–446. [12] L. Friedlander, Asymptotic behavior of the eigenvalues of the p-Laplacian, Comm. Partial Differential Equations 14 (1989) 1059–1069. [13] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. [14] O. Kov´acˇ ik, J. R´akosn´ık, On spaces L p(x) (Ω ) and W k, p(x) (Ω ), Czechoslovak Math. J. 41 (1991) 592–618. [15] A. Lˆe, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006) 1057–1099. [16] P. Marcellini, Regularity and existence of solutions of elliptic equations with ( p, q)-growth conditions, J. Differential Equations 90 (1991) 1–30. [17] S.B. Robinson, On the average value for nonconstant eigenfunctions of p-Laplacian assuming Neumann boundary data, Electron. J. Differential Equations 10 (2003) 251–256. Conference. [18] M. R˚uzˇ iˇcka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [19] V.L. Shapiro, Superlinear quasilinearity and the second eigenvalue, Nonlinear Anal. 44 (2001) 81–96. [20] A. Szulkin, Ljusternik–Schnirelmann theory on C 1 -manifolds, Ann Inst. H. Poincar´e Anal. Non Lin´eaire 5 (1988) 119–139. [21] E. Zeidler, Variational Methods and Optimization, in: Nonlinear Functional Analysis and its Applications, vol. 3, Springer-Verlag, Berlin, 1990.