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Perturbation for a p(x) -Laplacian equation involving oscillating nonlinearities in RN
Nonlinear Analysis 69 (2008) 2393–2402 www.elsevier.com/locate/na
Perturbation for a p(x)-Laplacian equation involving oscillating nonlinearities in R N I Chao Ji Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China Received 10 June 2007; accepted 8 August 2007
Abstract In this paper we study multiple solutions of the following problem in R N : ( −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = f (x, u) + λg(x, u), u ∈ W 1, p(x) (R N ),
(0.1)
R where the potential F(x, t) = 0t f (x, s)ds has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), and g is a perturbation term. Our results are a generalization of the case of the constant exponent and bounded domain from [G. Anello, G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations 234 (2007) 80–90] to the case of variable exponent and R N . c 2007 Elsevier Ltd. All rights reserved.
MSC: 35J35; 35J60; 35J70 Keywords: p(x)-Laplacian equation; Oscillating nonlinearities; Variational principle
1. Introduction In recent years the study of the various variational problems and elliptic problems with nonstandard growth conditions has seen considerable progress (e.g., see [1,2,4–12,14–16]). We refer the reader to the overview papers [4, 16] for the advances and references in this area, to [6,9] for the problem of the bounded domain involving the p(x)Laplacian, and to [2,7] for the problem in R N involving the p(x)-Laplacian. The latter is also the main study object of this paper. In [3] the authors consider a problem of the type −∆u = f (x, u) + λg(x, u) in Ω , (1.1) u = 0 on ∂Ω ,
I Research supported by the NNSF of China (10671084).
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.08.018
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Rt where Ω is a bounded open set in R N with smooth boundary ∂Ω , and the potential F(x, t) = 0 f (x, s)ds has oscillating behavior in any neighborhood of the origin or +∞. If Ω is bounded, we know that W01,2 (Ω ) ,→ L q (Ω )(1 ≤ N q < 2∗ = N2N −2 , N > 2) is compact, but the conclusion fails if Ω = R ; this is the first difficulty that we faced with. To overcome it, we refer to [7] that applies the method of the weight function. On the other hand, because the p(x)-Laplacian possesses more complex nonlinearities – for example, it is inhomogeneous – is the second difficulty. Moreover, in [3], the hypothesis of f sign-changing is very much crucial; we need it in this article as well. The paper is divided into three sections. In Section 2 we present some preliminary knowledge on the variable exponent spaces and related conclusions. In Section 3, we give the main results and their proofs. 2. Preliminary Let Ω ⊂ R N be an open set; set ∞ ∞ L + (Ω ) = p ∈ L (Ω ) : ess inf p(x) ≥ 1 . Ω
For p ∈ L ∞ + (Ω ), let p − (Ω ) = ess inf p(x),
p + (Ω ) = ess sup p(x).
x∈Ω
x∈Ω
Denote by S(Ω ) the set of all measurable real functions defined on Ω ; two measurable functions are considered as the same element of S(Ω ) when they are equal almost everywhere. For p ∈ L ∞ + (Ω ), define Z L p(x) (Ω ) = u ∈ S(Ω ) : |u| p(x) dx < ∞ , Ω
with the norm
|u| L p(x) (Ω ) = |u| p(x)
Z p(x) u = inf λ > 0 : dx ≤ 1 , Ω λ
and W 1, p(x) (Ω ) = {u ∈ L p(x) (Ω ) : |∇u| ∈ L p(x) (Ω )}, with the norm kukW 1, p(x) (Ω ) = |u| L p(x) (Ω ) + |∇u| L p(x) (Ω ) . In this paper we use the following equivalent norm on W 1, p(x) (Ω ): ! ) ( Z ∇u p(x) u p(x) dx ≤ 1 . kuk := inf λ > 0 : + λ λ Ω For some basic properties of the space W 1, p(x) (Ω ), we may refer the reader to [10,11]. Throughout the paper, we always assume that p − > 1. Proposition 2.1 ([10,11]). The spaces L p(x) (Ω ), W 1, p(x) (Ω ) are all separable and reflexive Banach spaces. R Proposition 2.2 ([7]). Set I (u) = Ω (|∇u(x)| p(x) + |u(x)| p(x) )dx. If u, u k ∈ W 1, p(x) (Ω ), then: (1) (2) (3) (4)
For u 6= 0, kuk = λ ⇔ I ( uλ ) = 1. kuk < 1(= 1; > 1) ⇔ I (u) < 1(= 1; > 1). − + If kuk > 1, then kuk p ≤ I (u) ≤ kuk p . + − If kuk < 1, then kuk p ≤ I (u) ≤ kuk p .
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(5) limk→∞ ku k k = 0 ⇔ limk→∞ I (u k ) = 0. (6) ku k k → ∞ ⇔ I (u k ) → ∞. Proposition 2.3 ([10,11]). The conjugate space of L p(x) (Ω ) is L q(x) (Ω ), where L p(x) (Ω ) and v ∈ L q(x) (Ω ), Z 1 1 + − |u| p(x) |v|q(x) ≤ 2|u| p(x) |v|q(x) . |uv|dx ≤ p− q Ω
1 p(x)
+
1 q(x)
= 1. For any u ∈
Like Proposition 2.3, we also have 1 1 1 + q(x) + r (x) = 1, then for any u ∈ L p(x) (Ω ), v ∈ L q(x) (Ω ) and ω ∈ L r (x) (Ω ), Proposition 2.4 ([7]). If p(x) Z 1 1 1 + − + − |u| p(x) |v|q(x) |ω|r (x) ≤ 3|u| p(x) |v|q(x) |ω|r (x) . |uvω|dx ≤ p− q r Ω
Proposition 2.5 ([10,11]). Let F : Ω × R → R satisfy the Carath´eodory condition, and p1 (x)
|F(x, t)| ≤ a(x) + b|t| p2 (x) ,
∀(x, t) ∈ Ω × R,
where a ∈ L p2 (x) (Ω ), b is a positive constant, p1 , p2 ∈ L ∞ + (Ω ). Denote by N F the Nemytsky operator defined by F, i.e. (N F (u))(x) = F(x, u(x)); then N F : L p1 (x) (Ω ) −→ L p2 (x) (Ω ) is a continuous and bounded map. If p(x) < N , let p ∗ :=
N p(x) . N − p(x)
Proposition 2.6 ([8]). If p : Ω × R → R is Lipschitz continuous and p + < N , then for q ∈ L ∞ + (Ω ) with p(x) ≤ q(x) ≤ p ∗ (x) there is a continuous embedding W 1, p(x) (Ω ) ,→ L q(x) (Ω ). For α, β ∈ S(Ω ), we use the symbol α β to denote ess inf (β(x) − α(x)) > 0. x∈Ω
Proposition 2.7 ([8,10,11]). Let Ω be a bounded domain in R N , p ∈ C(Ω ), p + < N . Then for any q ∈ L ∞ + (Ω ) with q p ∗ , there is a compact embedding W 1, p(x) (Ω ) ,→ L q(x) (Ω ). 3. Main results Throughout this paper, we always assume N ≥ 2, p : R N → R is Lipschitz continuous, 1 < p − ≤ p + < N and X = W 1, p(x) (R N ). We call u ∈ X a weak solution of the problem (0.1) if Z Z Z (|∇u| p(x)−2 ∇u · ∇v + |u| p(x)−2 uv)dx = f (x, u)vdx + λ g(x, u)vdx RN
for ∀v ∈ X .
RN
RN
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Set t
Z F(x, t) =
f (x, s)ds,
(x, t) ∈ R N × R,
0 t
Z
G(x, t) = g(x, s)ds, (x, t) ∈ R N × R, Z 0 1 J (u) = (|∇u(x)| p(x) + |u(x)| p(x) )dx, R N p(x) Z F(x, u)dx, Υ (u) = RN
Ψ (u) = J (u) − Υ (u), Z Φ(u) = − G(x, u)dx. RN
The following theorem is our main result when the potential F(x, ·) has an oscillating behavior in any neighborhood of the origin. Theorem 3.1. Let f, g : R N × R → R be two Carath´eodory functions. Suppose that the following conditions are satisfied: (1) There exists s0 > 0 such that sup | f (x, t)| ≤ |t|≤s0
sup |g(x, t)| ≤ |t|≤s0
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
d j (x)|t|q j (x)−1 ,
∀(x, t) ∈ R N × R,
i=1 n X
0
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , + N q 0j p ∗ , qi < p − , and there are si , s 0j ∈ L ∞ + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
qi (x) 1 + = 1. ri (x) si (x) q 0j (x) 1 + = 1. r 0j (x) s 0j (x)
(2) There exist three sequences {bn }, {cn } ⊂ (0, +∞) and {dn } ⊂ (−∞, 0) with limn→∞ cn = limn→∞ dn = 0 such that ess sup ( f (x, cn )) + λg(x, cn ) < 0, x∈R N
ess inf ( f (x, dn )) + λg(x, dn ) > 0, x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N . (3) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω lim inf > −∞, + t→0 |t| p Rt inf 0 f (x, s)ds x∈Ω lim sup = ∞. − |t| p t→0 Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than σk .
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For the proof of Theorem 3.1 we need some preliminary lemmas. Lemma 3.1 ([9]). J ∈ C 1 (X, R) and Z (|∇u(x)| p(x)−2 ∇u · ∇v + |u(x)| p(x)−2 uv)dx, J 0 (u)v =
∀u, v ∈ W 1, p(x) (R N ).
RN
J is a convex functional; J 0 : W 1, p(x) (R N ) → (W 1, p(x) (R N ))∗ is a strictly monotone, bounded homeomorphism, and is of (S+ ) type, that is un * u
limn→∞ J 0 (u n )(u n − u) ≤ 0
and
implies u n → u.
Lemma 3.2 ([7]). Suppose | f (x, t)| ≤
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
i=1
where bi (x) ≥ 0, bi ∈ L ri (x) (R N ) such that p(x) ≤ si (x) ≤ p ∗ (x),
T
+ N ∗ − ∞ N L ∞ (R N ), ri , qi ∈ L ∞ + (R ), qi p , qi < p , and there are si ∈ L + (R )
qi (x) 1 + = 1. ri (x) si (x)
Then Υ ∈ C 1 (X, R) and Υ , Υ 0 are weakly–strongly continuous, i.e., u n * u implies Υ (u n ) → Υ (u), Υ 0 (u n ) → Υ 0 (u) and Ψ is coercive. Lemma 3.3. Let f : R N × R → R be a Carath´eodory function such that f (x, d) > 0,
f (x, c) < 0,
a.e. x ∈ R N ,
for some d < 0 and c > 0. Define f (x, c) if t > c, f (x, t) = f (x, t) if d ≤ t ≤ c, f (x, d) if t < d. Then every weak solution u ∈ X of the following problem: −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = f (x, u) u∈X
(3.1)
satisfies d ≤ u(x) ≤ c for a.e. x ∈ R N . Proof. Assume that u ∈ X is a weak solution of the problem (3.1). We define c if u(x) > c, u(x) = u(x) if d ≤ u(x) ≤ c, d if u(x) < d; then u ∈ X . By the definition of a weak solution, we have Z Z p(x)−2 p(x)−2 (|∇u| ∇u · ∇(u − u) + |u| u(u − u))dx = RN
RN
f (x, u)(u − u)dx
or Z {x∈R N :u(x)
(|∇u| p(x)−2 ∇u · ∇u + |u| p(x)−2 u(u − d))dx
Z + Z
(|∇u| p(x)−2 ∇u · ∇udx + |u| p(x)−2 u(u − c))dx Z f (x, d)(u − d)dx + f (x, c)(u − c)dx
{x∈R N :u(x)>c}
= {x∈R N :u(x)
{x∈R N :u(x)>c}
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Clearly the left of the above equality is greater than 0, but, due to the hypothesis of f sign-changing, we know the right of the above equality is less than 0. So, u(x) = u(x) for a.e. x ∈ R N . Now we prove Theorem 3.1. Proof. Put n 0 ∈ N such that cn , dn ∈ [−s0 , s0 ] for all n ≥ n 0 . Without loss of generality, we assume {cn }n≥n 0 , {dn }n≥n 0 respectively decreasing and increasing. Moreover, we set N f (x, cn ) if t > cn x ∈ R , (3.2) f n (x, t) = f (x, t) if dn ≤ t ≤ cn x ∈ R N , N f (x, dn ) if t < dn x ∈ R . N g(x, cn ) if t > cn x ∈ R , (3.3) gn (x, t) = g(x, t) if dn ≤ t ≤ cn x ∈ R N , N g(x, dn ) if t < dn x ∈ R and Υn (u) =
Z
u(x)
Z RN
f n (x, t)dt dx,
0 u(x)
Z
Z
Φn (u) = −
!
RN
! gn (x, t)dt dx,
0
Ψn (u) = J (u) − Υn (u), for all u ∈ X . By condition (1), (3.2) and (3.3), we have | f n (x, t)| ≤
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
d j (x)|t|q j (x)−1 ,
∀(x, t) ∈ R N × R,
i=1
|gn (x, t)| ≤
n X
0
j=1
where bi (x), d j (x), qi (x), q 0j (x) are as in condition (1). By Lemma 3.2 we know that Ψn (u), Φn (u) ∈ C 1 (X, R) are sequentially weakly lower semicontinuous, Φn0 (u) is weakly–strongly continuous and Ψn (u) is coercive. And by condition (3), there exist M > 0, δ > 0 with δ < min{−dn , cn , 1} such that Z t + − ( f (x, s))ds ≥ −M|t| p ≥ −M|t| p , ess inf (3.4) x∈Ω
0
for all t ∈ [−δ, δ]. We choose a compact subset K ⊂ Ω , with m(K ) = (M + 1)m(Ω \ K ) and v ∈ X such that v(x) = 1
if x ∈ K ,
0 ≤ v(x) ≤ 1 v(x) = 0
(3.5)
if x ∈ Ω \ K ,
(3.6)
if x ∈ R \ Ω . N
(3.7)
From condition (3) we know there exists t ∈ R with |t| ≤ δ such that ( R ) Z t c R N (|∇v| p(x) + |v| p(x) )dx − ess inf ( f (x, s))ds ≥ max , 1 |t| p , p + m(Ω \ K ) x∈Ω 0 where c > 1. By (3.4) and (3.8), we have Z tv(x) Z t f (x, s)ds ≥ −M ess inf f (x, s)ds. 0
x∈Ω
0
(3.8)
(3.9)
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for a.e. x ∈ Ω . Combining (3.5)–(3.9), we have ! Z tv Z Z 1 p(x) p(x) f n (x, t)dt dx Ψn (tv) = (|∇tv| + |tv| )dx − R N p(x) RN 0 ! Z Z Z t − Z |t| p p(x) p(x) ≤ + f (x, t)dt dx − (|∇v| + |v| )dx − p Ω \K K 0 RN Z t − Z p |t| (|∇v| p(x) + |v| p(x) )dx − m(K ) ess inf f (x, t)dt ≤ + p x∈Ω 0 RN Z t + Mm(Ω \ K ) ess inf f n (x, t)dt x∈Ω
− |t| p
≤
Z
p+
RN
−
(|∇v|
|t| p (1 − c) p+
<
p(x)
Z RN
Z
tv
! f (x, t)dt dx
0
0
+ |v|
p(x)
)dx − m(Ω \ K ) ess inf
x∈Ω
Z
t
f (x, t)dt
0
(|∇v| p(x) + |v| p(x) )dx < 0.
So, inf Ψn < 0, X
and we know Ψn is coercive and consequently reaches its global minimum at a point u n ∈ X . Moreover by Lemma 3.3, we have dn ≤ u n (x) ≤ cn . We also know Z Z − RN
u n (x)
! f n (x, t)dt dx ≤ Ψn (u n ) < 0.
0
Hence, lim inf Ψn = lim Ψn (u n ) = 0.
n→∞ X
n→∞
(3.10)
According to (3.10), we may assume that the sequence {inf X Ψn } is strictly increasing. Fix k ∈ N ; choose ri ∈ inf Ψi , inf Ψi+1 X
X
and apply Theorem 2.1 in [13] for the functionals Ψi , Φi . So, there exists bi0 > 0 such that for each λ ∈ [0, bi0 ], the functional Ψi + λΦi has a critical point u i,λ satisfying Ψi (u i,λ ) < ri < 0 for all i ∈ N . Fix i ≥ n 0 ; by condition (2), we put 0 0 bk∗ = min{bi , bi+1 , . . . , bi+k−1 , bi0 , bi+1 , . . . , bi+k−1 }.
We have ess sup ( f (x, c j )) + λg(x, c j ) < 0, x∈R N
ess inf ( f (x, d j )) + λg(x, d j ) > 0, x∈R N
for all λ ∈ [0, bk∗ ] and all j = i, i + 1, . . . , i + k − 1. So, by Lemma 3.3, we may get k weak solutions u i,λ , u i+1,λ , . . . , u i+k−1,λ of the problem (0.1) satisfying d j ≤ u j,λ (x) ≤ c j
a.e. x ∈ R N .
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Moreover, we put ( δk = max 1,
max
p
+
Z
i≤ j≤i+k−1
RN
F(x, u j,λ )dx
1 p−
) .
By Ψ j,λ (u j,λ ) < 0, we have ku j,λ k ≤ δk for j = i, i + 1, . . . , i + k − 1. Finally we show that the k weak solutions are pairwise distinct. Fix m, n ∈ {i, i + 1, . . . , i + k − 1} and m < n. Then, by (3.2), one has Z Z Z u n,λ 1 f m (x, t)dt dx Ψm (u n,λ ) = (|∇u n,λ | p(x) + |u n,λ | p(x) )dx − R N p(x) RN 0 Z Z Z u n,λ 1 = (|∇u n,λ | p(x) + |u n,λ | p(x) )dx − f n (x, t)dt dx R N p(x) RN 0 = Ψn (u n,λ ) > rm > Ψm (u m,λ ). Hence u m,λ 6= u n,λ if m 6= n. If we replace function g with function −g, then we may get a real number bk∗∗ > 0 such that problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than δk if λ ∈ [−bk∗∗ , 0]. And we choose bk? = min{bk∗ , bk∗∗ } and immediately get the conclusions. Remark 3.1. From the proof of Theorem 3.1 we know that the smaller λ becomes, the greater the number of solutions becomes. Moreover if λ = 0, the problem (0.1) has infinitely many solutions, each one corresponding to a global minimum of Ψn . Remark 3.2. Moreover if bi ∈ L 1 , for fixed k ∈ N , the W 1, p(x) (R N )-norms of the k distinct weak solutions of the problem (0.1) may be controlled by arbitrary positive constant number δ > 0 if n is big enough. Now we state and prove the second main result for when the potential F(x, ·) has an oscillating behavior in any neighborhood of ∞. Theorem 3.2. Let f, g : R N × R → R be two Carath´eodory functions. Suppose that the following conditions are satisfied: (4) There exists d0 > 0 for every s ≥ 0, m X sup | f (·, t)| ≤ bi (x)|t|qi (x)−1 , ∀(x, t) ∈ R N × R, d0 ≤t≤s
sup |g(·, t)| ≤ d0 ≤t≤s
i=1 n X
d j (x)|t|q j (x)−1 , 0
∀(x, t) ∈ R N × R,
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , + 0 0 ∗ − ∞ N q j p , qi < p , and there are si , s j ∈ L + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
1 qi (x) + = 1. ri (x) si (x) q 0j (x) 1 + 0 = 1. r 0j (x) s j (x)
(5) There exist two sequences {bn }, {cn } ⊂ (0, +∞) with limn→∞ cn = +∞, and d0 < 0, ess sup ( f (x, cn )) + λg(x, cn ) < 0, x∈R N
ess inf ( f (x, dn )) + λg(x, dn ) > 0, x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N .
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(6) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω > −∞, lim inf − t→+∞ tp Rt inf 0 f (x, s)ds x∈Ω = ∞. lim sup + tp t→+∞ Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than σk . Proof. The proof is similar to that of Theorem 3.1; here we sketch it for the readers’ convenience. Define N f (x, cn ) if t > cn x ∈ R , f n (x, t) = f (x, t) if d0 ≤ t ≤ cn x ∈ R N , f (x, d0 ) if t < d0 x ∈ R N . N g(x, cn ) if t > cn x ∈ R , gn (x, t) = g(x, t) if d0 ≤ t ≤ cn x ∈ R N , g(x, d0 ) if t < d0 x ∈ R N . Other definitions and notation are as in the proof of Theorem 3.1. By condition (6), we know lim inf Ψn = −∞
n→∞ X
and, without loss of generality, we assume {cn } strictly increasing and {inf X Ψn } strictly decreasing. The rest of the proof is the same as that for Theorem 3.1, so we omit it. Remark 3.3. Remarks 3.1 and 3.2 made regarding Theorem 3.1 are valid for Theorem 3.2. Moreover, we may get multiple nonnegative weak solutions when g(x, 0) ≡ 0 in R N . Because of the similarity, we only work out the conclusion for when the potential F(x, ·) has an oscillating behavior in any appropriate neighborhood of the origin. Theorem 3.3. Let f, g : R N × R → R be two Carath´eodory functions with g(x, 0) ≡ 0 in R N . Suppose that the following conditions are satisfied: (7) There exists s0 > 0 such that m X bi (x)|t|qi (x)−1 , ∀(x, t) ∈ R N × R, sup | f (·, t)| ≤ 0≤t≤s0
sup |g(·, t)| ≤ 0≤t≤s0
i=1 n X
d j (x)|t|q j (x)−1 , 0
∀(x, t) ∈ R N × R,
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , N q 0j p ∗ , qi+ < p − , and there are si , s 0j ∈ L ∞ + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
1 qi (x) + = 1. ri (x) si (x) q 0j (x) 1 + 0 = 1. r 0j (x) s j (x)
(8) There exist two sequences {bn }, {cn } ⊂ (0, +∞) with limn→∞ cn = 0 such that ess sup ( f (x, cn )) + λg(x, cn ) < 0. x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N .
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(9) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω > −∞, lim inf + t→0+ tp Rt inf 0 f (x, s)ds x∈Ω lim sup = ∞. − tp t→0+ Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct nonnegative weak solutions whose W 1, p(x) (R N )-norms are less than σk . Remark 3.4. Remarks 3.1 and 3.2 made regarding Theorem 3.1 are valid for Theorem 3.3 as well. Acknowledgement The author would like to thank Professor Xianling Fan for clear valuable suggestions, and the reviewer. References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001) 121–140. [2] C.O. Alves, M.A.S. Souto, Existence of solutions for a class of problems in R N involving the p(x)-Laplacian, Progr. Nonlinear Differential Equations Appl. 66 (2005) 17–32. [3] G. Anello, G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations 234 (2007) 80–90. [4] L. Diening, P. H¨ast¨o, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Dr´abek, R´akosn´ık (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38–58. [5] D.E. Edmunds, J. R´akosnik, Sobolev embeddings with variable exponent, II, Math. Nachr. 246–247 (2002) 53–67. [6] X.L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problems involving the p(x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248–260. [7] X.L. Fan, X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in R N , Nonlinear Anal. 59 (2004) 173–188. [8] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W k, p(x)(Ω ) , J. Math. Anal. Appl. 262 (2001) 249–260. [9] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003) 1843–1852. [10] 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. [11] 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. [12] P. Marcellini, Regularity and existence of solutions of elliptic equations with ( p, q)-growth conditions, J. Differential Equations 90 (1991) 1–30. [13] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [14] M. R˚uz˘ i˘cka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [15] S. Samko, Denseness of C0∞ (R N ) in the generalized Sobolev spaces W m, p(x) (R N ), Dokl. Ross. Akad. Nauk 369 (4) (1999) 142–160. [16] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005) 461–482.
Perturbation for a p(x)-Laplacian equation involving oscillating nonlinearities in R N I Chao Ji Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China Received 10 June 2007; accepted 8 August 2007
Abstract In this paper we study multiple solutions of the following problem in R N : ( −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = f (x, u) + λg(x, u), u ∈ W 1, p(x) (R N ),
(0.1)
R where the potential F(x, t) = 0t f (x, s)ds has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), and g is a perturbation term. Our results are a generalization of the case of the constant exponent and bounded domain from [G. Anello, G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations 234 (2007) 80–90] to the case of variable exponent and R N . c 2007 Elsevier Ltd. All rights reserved.
MSC: 35J35; 35J60; 35J70 Keywords: p(x)-Laplacian equation; Oscillating nonlinearities; Variational principle
1. Introduction In recent years the study of the various variational problems and elliptic problems with nonstandard growth conditions has seen considerable progress (e.g., see [1,2,4–12,14–16]). We refer the reader to the overview papers [4, 16] for the advances and references in this area, to [6,9] for the problem of the bounded domain involving the p(x)Laplacian, and to [2,7] for the problem in R N involving the p(x)-Laplacian. The latter is also the main study object of this paper. In [3] the authors consider a problem of the type −∆u = f (x, u) + λg(x, u) in Ω , (1.1) u = 0 on ∂Ω ,
I Research supported by the NNSF of China (10671084).
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.08.018
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Rt where Ω is a bounded open set in R N with smooth boundary ∂Ω , and the potential F(x, t) = 0 f (x, s)ds has oscillating behavior in any neighborhood of the origin or +∞. If Ω is bounded, we know that W01,2 (Ω ) ,→ L q (Ω )(1 ≤ N q < 2∗ = N2N −2 , N > 2) is compact, but the conclusion fails if Ω = R ; this is the first difficulty that we faced with. To overcome it, we refer to [7] that applies the method of the weight function. On the other hand, because the p(x)-Laplacian possesses more complex nonlinearities – for example, it is inhomogeneous – is the second difficulty. Moreover, in [3], the hypothesis of f sign-changing is very much crucial; we need it in this article as well. The paper is divided into three sections. In Section 2 we present some preliminary knowledge on the variable exponent spaces and related conclusions. In Section 3, we give the main results and their proofs. 2. Preliminary Let Ω ⊂ R N be an open set; set ∞ ∞ L + (Ω ) = p ∈ L (Ω ) : ess inf p(x) ≥ 1 . Ω
For p ∈ L ∞ + (Ω ), let p − (Ω ) = ess inf p(x),
p + (Ω ) = ess sup p(x).
x∈Ω
x∈Ω
Denote by S(Ω ) the set of all measurable real functions defined on Ω ; two measurable functions are considered as the same element of S(Ω ) when they are equal almost everywhere. For p ∈ L ∞ + (Ω ), define Z L p(x) (Ω ) = u ∈ S(Ω ) : |u| p(x) dx < ∞ , Ω
with the norm
|u| L p(x) (Ω ) = |u| p(x)
Z p(x) u = inf λ > 0 : dx ≤ 1 , Ω λ
and W 1, p(x) (Ω ) = {u ∈ L p(x) (Ω ) : |∇u| ∈ L p(x) (Ω )}, with the norm kukW 1, p(x) (Ω ) = |u| L p(x) (Ω ) + |∇u| L p(x) (Ω ) . In this paper we use the following equivalent norm on W 1, p(x) (Ω ): ! ) ( Z ∇u p(x) u p(x) dx ≤ 1 . kuk := inf λ > 0 : + λ λ Ω For some basic properties of the space W 1, p(x) (Ω ), we may refer the reader to [10,11]. Throughout the paper, we always assume that p − > 1. Proposition 2.1 ([10,11]). The spaces L p(x) (Ω ), W 1, p(x) (Ω ) are all separable and reflexive Banach spaces. R Proposition 2.2 ([7]). Set I (u) = Ω (|∇u(x)| p(x) + |u(x)| p(x) )dx. If u, u k ∈ W 1, p(x) (Ω ), then: (1) (2) (3) (4)
For u 6= 0, kuk = λ ⇔ I ( uλ ) = 1. kuk < 1(= 1; > 1) ⇔ I (u) < 1(= 1; > 1). − + If kuk > 1, then kuk p ≤ I (u) ≤ kuk p . + − If kuk < 1, then kuk p ≤ I (u) ≤ kuk p .
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(5) limk→∞ ku k k = 0 ⇔ limk→∞ I (u k ) = 0. (6) ku k k → ∞ ⇔ I (u k ) → ∞. Proposition 2.3 ([10,11]). The conjugate space of L p(x) (Ω ) is L q(x) (Ω ), where L p(x) (Ω ) and v ∈ L q(x) (Ω ), Z 1 1 + − |u| p(x) |v|q(x) ≤ 2|u| p(x) |v|q(x) . |uv|dx ≤ p− q Ω
1 p(x)
+
1 q(x)
= 1. For any u ∈
Like Proposition 2.3, we also have 1 1 1 + q(x) + r (x) = 1, then for any u ∈ L p(x) (Ω ), v ∈ L q(x) (Ω ) and ω ∈ L r (x) (Ω ), Proposition 2.4 ([7]). If p(x) Z 1 1 1 + − + − |u| p(x) |v|q(x) |ω|r (x) ≤ 3|u| p(x) |v|q(x) |ω|r (x) . |uvω|dx ≤ p− q r Ω
Proposition 2.5 ([10,11]). Let F : Ω × R → R satisfy the Carath´eodory condition, and p1 (x)
|F(x, t)| ≤ a(x) + b|t| p2 (x) ,
∀(x, t) ∈ Ω × R,
where a ∈ L p2 (x) (Ω ), b is a positive constant, p1 , p2 ∈ L ∞ + (Ω ). Denote by N F the Nemytsky operator defined by F, i.e. (N F (u))(x) = F(x, u(x)); then N F : L p1 (x) (Ω ) −→ L p2 (x) (Ω ) is a continuous and bounded map. If p(x) < N , let p ∗ :=
N p(x) . N − p(x)
Proposition 2.6 ([8]). If p : Ω × R → R is Lipschitz continuous and p + < N , then for q ∈ L ∞ + (Ω ) with p(x) ≤ q(x) ≤ p ∗ (x) there is a continuous embedding W 1, p(x) (Ω ) ,→ L q(x) (Ω ). For α, β ∈ S(Ω ), we use the symbol α β to denote ess inf (β(x) − α(x)) > 0. x∈Ω
Proposition 2.7 ([8,10,11]). Let Ω be a bounded domain in R N , p ∈ C(Ω ), p + < N . Then for any q ∈ L ∞ + (Ω ) with q p ∗ , there is a compact embedding W 1, p(x) (Ω ) ,→ L q(x) (Ω ). 3. Main results Throughout this paper, we always assume N ≥ 2, p : R N → R is Lipschitz continuous, 1 < p − ≤ p + < N and X = W 1, p(x) (R N ). We call u ∈ X a weak solution of the problem (0.1) if Z Z Z (|∇u| p(x)−2 ∇u · ∇v + |u| p(x)−2 uv)dx = f (x, u)vdx + λ g(x, u)vdx RN
for ∀v ∈ X .
RN
RN
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Set t
Z F(x, t) =
f (x, s)ds,
(x, t) ∈ R N × R,
0 t
Z
G(x, t) = g(x, s)ds, (x, t) ∈ R N × R, Z 0 1 J (u) = (|∇u(x)| p(x) + |u(x)| p(x) )dx, R N p(x) Z F(x, u)dx, Υ (u) = RN
Ψ (u) = J (u) − Υ (u), Z Φ(u) = − G(x, u)dx. RN
The following theorem is our main result when the potential F(x, ·) has an oscillating behavior in any neighborhood of the origin. Theorem 3.1. Let f, g : R N × R → R be two Carath´eodory functions. Suppose that the following conditions are satisfied: (1) There exists s0 > 0 such that sup | f (x, t)| ≤ |t|≤s0
sup |g(x, t)| ≤ |t|≤s0
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
d j (x)|t|q j (x)−1 ,
∀(x, t) ∈ R N × R,
i=1 n X
0
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , + N q 0j p ∗ , qi < p − , and there are si , s 0j ∈ L ∞ + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
qi (x) 1 + = 1. ri (x) si (x) q 0j (x) 1 + = 1. r 0j (x) s 0j (x)
(2) There exist three sequences {bn }, {cn } ⊂ (0, +∞) and {dn } ⊂ (−∞, 0) with limn→∞ cn = limn→∞ dn = 0 such that ess sup ( f (x, cn )) + λg(x, cn ) < 0, x∈R N
ess inf ( f (x, dn )) + λg(x, dn ) > 0, x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N . (3) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω lim inf > −∞, + t→0 |t| p Rt inf 0 f (x, s)ds x∈Ω lim sup = ∞. − |t| p t→0 Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than σk .
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For the proof of Theorem 3.1 we need some preliminary lemmas. Lemma 3.1 ([9]). J ∈ C 1 (X, R) and Z (|∇u(x)| p(x)−2 ∇u · ∇v + |u(x)| p(x)−2 uv)dx, J 0 (u)v =
∀u, v ∈ W 1, p(x) (R N ).
RN
J is a convex functional; J 0 : W 1, p(x) (R N ) → (W 1, p(x) (R N ))∗ is a strictly monotone, bounded homeomorphism, and is of (S+ ) type, that is un * u
limn→∞ J 0 (u n )(u n − u) ≤ 0
and
implies u n → u.
Lemma 3.2 ([7]). Suppose | f (x, t)| ≤
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
i=1
where bi (x) ≥ 0, bi ∈ L ri (x) (R N ) such that p(x) ≤ si (x) ≤ p ∗ (x),
T
+ N ∗ − ∞ N L ∞ (R N ), ri , qi ∈ L ∞ + (R ), qi p , qi < p , and there are si ∈ L + (R )
qi (x) 1 + = 1. ri (x) si (x)
Then Υ ∈ C 1 (X, R) and Υ , Υ 0 are weakly–strongly continuous, i.e., u n * u implies Υ (u n ) → Υ (u), Υ 0 (u n ) → Υ 0 (u) and Ψ is coercive. Lemma 3.3. Let f : R N × R → R be a Carath´eodory function such that f (x, d) > 0,
f (x, c) < 0,
a.e. x ∈ R N ,
for some d < 0 and c > 0. Define f (x, c) if t > c, f (x, t) = f (x, t) if d ≤ t ≤ c, f (x, d) if t < d. Then every weak solution u ∈ X of the following problem: −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = f (x, u) u∈X
(3.1)
satisfies d ≤ u(x) ≤ c for a.e. x ∈ R N . Proof. Assume that u ∈ X is a weak solution of the problem (3.1). We define c if u(x) > c, u(x) = u(x) if d ≤ u(x) ≤ c, d if u(x) < d; then u ∈ X . By the definition of a weak solution, we have Z Z p(x)−2 p(x)−2 (|∇u| ∇u · ∇(u − u) + |u| u(u − u))dx = RN
RN
f (x, u)(u − u)dx
or Z {x∈R N :u(x)
(|∇u| p(x)−2 ∇u · ∇u + |u| p(x)−2 u(u − d))dx
Z + Z
(|∇u| p(x)−2 ∇u · ∇udx + |u| p(x)−2 u(u − c))dx Z f (x, d)(u − d)dx + f (x, c)(u − c)dx
{x∈R N :u(x)>c}
= {x∈R N :u(x)
{x∈R N :u(x)>c}
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Clearly the left of the above equality is greater than 0, but, due to the hypothesis of f sign-changing, we know the right of the above equality is less than 0. So, u(x) = u(x) for a.e. x ∈ R N . Now we prove Theorem 3.1. Proof. Put n 0 ∈ N such that cn , dn ∈ [−s0 , s0 ] for all n ≥ n 0 . Without loss of generality, we assume {cn }n≥n 0 , {dn }n≥n 0 respectively decreasing and increasing. Moreover, we set N f (x, cn ) if t > cn x ∈ R , (3.2) f n (x, t) = f (x, t) if dn ≤ t ≤ cn x ∈ R N , N f (x, dn ) if t < dn x ∈ R . N g(x, cn ) if t > cn x ∈ R , (3.3) gn (x, t) = g(x, t) if dn ≤ t ≤ cn x ∈ R N , N g(x, dn ) if t < dn x ∈ R and Υn (u) =
Z
u(x)
Z RN
f n (x, t)dt dx,
0 u(x)
Z
Z
Φn (u) = −
!
RN
! gn (x, t)dt dx,
0
Ψn (u) = J (u) − Υn (u), for all u ∈ X . By condition (1), (3.2) and (3.3), we have | f n (x, t)| ≤
m X
bi (x)|t|qi (x)−1 ,
∀(x, t) ∈ R N × R,
d j (x)|t|q j (x)−1 ,
∀(x, t) ∈ R N × R,
i=1
|gn (x, t)| ≤
n X
0
j=1
where bi (x), d j (x), qi (x), q 0j (x) are as in condition (1). By Lemma 3.2 we know that Ψn (u), Φn (u) ∈ C 1 (X, R) are sequentially weakly lower semicontinuous, Φn0 (u) is weakly–strongly continuous and Ψn (u) is coercive. And by condition (3), there exist M > 0, δ > 0 with δ < min{−dn , cn , 1} such that Z t + − ( f (x, s))ds ≥ −M|t| p ≥ −M|t| p , ess inf (3.4) x∈Ω
0
for all t ∈ [−δ, δ]. We choose a compact subset K ⊂ Ω , with m(K ) = (M + 1)m(Ω \ K ) and v ∈ X such that v(x) = 1
if x ∈ K ,
0 ≤ v(x) ≤ 1 v(x) = 0
(3.5)
if x ∈ Ω \ K ,
(3.6)
if x ∈ R \ Ω . N
(3.7)
From condition (3) we know there exists t ∈ R with |t| ≤ δ such that ( R ) Z t c R N (|∇v| p(x) + |v| p(x) )dx − ess inf ( f (x, s))ds ≥ max , 1 |t| p , p + m(Ω \ K ) x∈Ω 0 where c > 1. By (3.4) and (3.8), we have Z tv(x) Z t f (x, s)ds ≥ −M ess inf f (x, s)ds. 0
x∈Ω
0
(3.8)
(3.9)
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for a.e. x ∈ Ω . Combining (3.5)–(3.9), we have ! Z tv Z Z 1 p(x) p(x) f n (x, t)dt dx Ψn (tv) = (|∇tv| + |tv| )dx − R N p(x) RN 0 ! Z Z Z t − Z |t| p p(x) p(x) ≤ + f (x, t)dt dx − (|∇v| + |v| )dx − p Ω \K K 0 RN Z t − Z p |t| (|∇v| p(x) + |v| p(x) )dx − m(K ) ess inf f (x, t)dt ≤ + p x∈Ω 0 RN Z t + Mm(Ω \ K ) ess inf f n (x, t)dt x∈Ω
− |t| p
≤
Z
p+
RN
−
(|∇v|
|t| p (1 − c) p+
<
p(x)
Z RN
Z
tv
! f (x, t)dt dx
0
0
+ |v|
p(x)
)dx − m(Ω \ K ) ess inf
x∈Ω
Z
t
f (x, t)dt
0
(|∇v| p(x) + |v| p(x) )dx < 0.
So, inf Ψn < 0, X
and we know Ψn is coercive and consequently reaches its global minimum at a point u n ∈ X . Moreover by Lemma 3.3, we have dn ≤ u n (x) ≤ cn . We also know Z Z − RN
u n (x)
! f n (x, t)dt dx ≤ Ψn (u n ) < 0.
0
Hence, lim inf Ψn = lim Ψn (u n ) = 0.
n→∞ X
n→∞
(3.10)
According to (3.10), we may assume that the sequence {inf X Ψn } is strictly increasing. Fix k ∈ N ; choose ri ∈ inf Ψi , inf Ψi+1 X
X
and apply Theorem 2.1 in [13] for the functionals Ψi , Φi . So, there exists bi0 > 0 such that for each λ ∈ [0, bi0 ], the functional Ψi + λΦi has a critical point u i,λ satisfying Ψi (u i,λ ) < ri < 0 for all i ∈ N . Fix i ≥ n 0 ; by condition (2), we put 0 0 bk∗ = min{bi , bi+1 , . . . , bi+k−1 , bi0 , bi+1 , . . . , bi+k−1 }.
We have ess sup ( f (x, c j )) + λg(x, c j ) < 0, x∈R N
ess inf ( f (x, d j )) + λg(x, d j ) > 0, x∈R N
for all λ ∈ [0, bk∗ ] and all j = i, i + 1, . . . , i + k − 1. So, by Lemma 3.3, we may get k weak solutions u i,λ , u i+1,λ , . . . , u i+k−1,λ of the problem (0.1) satisfying d j ≤ u j,λ (x) ≤ c j
a.e. x ∈ R N .
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Moreover, we put ( δk = max 1,
max
p
+
Z
i≤ j≤i+k−1
RN
F(x, u j,λ )dx
1 p−
) .
By Ψ j,λ (u j,λ ) < 0, we have ku j,λ k ≤ δk for j = i, i + 1, . . . , i + k − 1. Finally we show that the k weak solutions are pairwise distinct. Fix m, n ∈ {i, i + 1, . . . , i + k − 1} and m < n. Then, by (3.2), one has Z Z Z u n,λ 1 f m (x, t)dt dx Ψm (u n,λ ) = (|∇u n,λ | p(x) + |u n,λ | p(x) )dx − R N p(x) RN 0 Z Z Z u n,λ 1 = (|∇u n,λ | p(x) + |u n,λ | p(x) )dx − f n (x, t)dt dx R N p(x) RN 0 = Ψn (u n,λ ) > rm > Ψm (u m,λ ). Hence u m,λ 6= u n,λ if m 6= n. If we replace function g with function −g, then we may get a real number bk∗∗ > 0 such that problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than δk if λ ∈ [−bk∗∗ , 0]. And we choose bk? = min{bk∗ , bk∗∗ } and immediately get the conclusions. Remark 3.1. From the proof of Theorem 3.1 we know that the smaller λ becomes, the greater the number of solutions becomes. Moreover if λ = 0, the problem (0.1) has infinitely many solutions, each one corresponding to a global minimum of Ψn . Remark 3.2. Moreover if bi ∈ L 1 , for fixed k ∈ N , the W 1, p(x) (R N )-norms of the k distinct weak solutions of the problem (0.1) may be controlled by arbitrary positive constant number δ > 0 if n is big enough. Now we state and prove the second main result for when the potential F(x, ·) has an oscillating behavior in any neighborhood of ∞. Theorem 3.2. Let f, g : R N × R → R be two Carath´eodory functions. Suppose that the following conditions are satisfied: (4) There exists d0 > 0 for every s ≥ 0, m X sup | f (·, t)| ≤ bi (x)|t|qi (x)−1 , ∀(x, t) ∈ R N × R, d0 ≤t≤s
sup |g(·, t)| ≤ d0 ≤t≤s
i=1 n X
d j (x)|t|q j (x)−1 , 0
∀(x, t) ∈ R N × R,
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , + 0 0 ∗ − ∞ N q j p , qi < p , and there are si , s j ∈ L + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
1 qi (x) + = 1. ri (x) si (x) q 0j (x) 1 + 0 = 1. r 0j (x) s j (x)
(5) There exist two sequences {bn }, {cn } ⊂ (0, +∞) with limn→∞ cn = +∞, and d0 < 0, ess sup ( f (x, cn )) + λg(x, cn ) < 0, x∈R N
ess inf ( f (x, dn )) + λg(x, dn ) > 0, x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N .
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(6) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω > −∞, lim inf − t→+∞ tp Rt inf 0 f (x, s)ds x∈Ω = ∞. lim sup + tp t→+∞ Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct weak solutions whose W 1, p(x) (R N )-norms are less than σk . Proof. The proof is similar to that of Theorem 3.1; here we sketch it for the readers’ convenience. Define N f (x, cn ) if t > cn x ∈ R , f n (x, t) = f (x, t) if d0 ≤ t ≤ cn x ∈ R N , f (x, d0 ) if t < d0 x ∈ R N . N g(x, cn ) if t > cn x ∈ R , gn (x, t) = g(x, t) if d0 ≤ t ≤ cn x ∈ R N , g(x, d0 ) if t < d0 x ∈ R N . Other definitions and notation are as in the proof of Theorem 3.1. By condition (6), we know lim inf Ψn = −∞
n→∞ X
and, without loss of generality, we assume {cn } strictly increasing and {inf X Ψn } strictly decreasing. The rest of the proof is the same as that for Theorem 3.1, so we omit it. Remark 3.3. Remarks 3.1 and 3.2 made regarding Theorem 3.1 are valid for Theorem 3.2. Moreover, we may get multiple nonnegative weak solutions when g(x, 0) ≡ 0 in R N . Because of the similarity, we only work out the conclusion for when the potential F(x, ·) has an oscillating behavior in any appropriate neighborhood of the origin. Theorem 3.3. Let f, g : R N × R → R be two Carath´eodory functions with g(x, 0) ≡ 0 in R N . Suppose that the following conditions are satisfied: (7) There exists s0 > 0 such that m X bi (x)|t|qi (x)−1 , ∀(x, t) ∈ R N × R, sup | f (·, t)| ≤ 0≤t≤s0
sup |g(·, t)| ≤ 0≤t≤s0
i=1 n X
d j (x)|t|q j (x)−1 , 0
∀(x, t) ∈ R N × R,
j=1
0 T T N where bi (x), d j (x) ≥ 0, bi ∈ L ri (x) (R N ) L ∞ (R N ), d j ∈ L r j (x) (R N ) L ∞ (R N ), ri , r 0j , qi , q 0j ∈ L ∞ + (R ), qi , N q 0j p ∗ , qi+ < p − , and there are si , s 0j ∈ L ∞ + (R ) such that
p(x) ≤ si (x) ≤ p ∗ (x), p(x) ≤ s 0j (x) ≤ p ∗ (x),
1 qi (x) + = 1. ri (x) si (x) q 0j (x) 1 + 0 = 1. r 0j (x) s j (x)
(8) There exist two sequences {bn }, {cn } ⊂ (0, +∞) with limn→∞ cn = 0 such that ess sup ( f (x, cn )) + λg(x, cn ) < 0. x∈R N
for all λ ∈ [−bn , bn ] and all n ∈ N .
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(9) There exists a nonempty bounded open subset Ω ⊂ R N such that Rt inf 0 f (x, s)ds x∈Ω > −∞, lim inf + t→0+ tp Rt inf 0 f (x, s)ds x∈Ω lim sup = ∞. − tp t→0+ Then, for each k ∈ N , there exist bk? > 0 and σk > 0 such that for every λ ∈ [−bk? , bk? ], the problem (0.1) has at least k distinct nonnegative weak solutions whose W 1, p(x) (R N )-norms are less than σk . Remark 3.4. Remarks 3.1 and 3.2 made regarding Theorem 3.1 are valid for Theorem 3.3 as well. Acknowledgement The author would like to thank Professor Xianling Fan for clear valuable suggestions, and the reviewer. References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001) 121–140. [2] C.O. Alves, M.A.S. Souto, Existence of solutions for a class of problems in R N involving the p(x)-Laplacian, Progr. Nonlinear Differential Equations Appl. 66 (2005) 17–32. [3] G. Anello, G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations 234 (2007) 80–90. [4] L. Diening, P. H¨ast¨o, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Dr´abek, R´akosn´ık (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38–58. [5] D.E. Edmunds, J. R´akosnik, Sobolev embeddings with variable exponent, II, Math. Nachr. 246–247 (2002) 53–67. [6] X.L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problems involving the p(x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248–260. [7] X.L. Fan, X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in R N , Nonlinear Anal. 59 (2004) 173–188. [8] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W k, p(x)(Ω ) , J. Math. Anal. Appl. 262 (2001) 249–260. [9] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003) 1843–1852. [10] 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. [11] 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. [12] P. Marcellini, Regularity and existence of solutions of elliptic equations with ( p, q)-growth conditions, J. Differential Equations 90 (1991) 1–30. [13] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [14] M. R˚uz˘ i˘cka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [15] S. Samko, Denseness of C0∞ (R N ) in the generalized Sobolev spaces W m, p(x) (R N ), Dokl. Ross. Akad. Nauk 369 (4) (1999) 142–160. [16] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005) 461–482.