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Infinitely many solutions for a p(x) -Laplacian equation in RN
Nonlinear Analysis 71 (2009) 1133–1139
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Infinitely many solutions for a p(x)-Laplacian equation in RN I Guowei Dai ∗ Department of Mathematics, Lanzhou University, Lanzhou, 730000, PR China
article
info
a b s t r a c t This paper deals with a p(x)-Laplacian equation in RN . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose W 1,p(x) (RN )-norms tend to zero (to infinity, respectively) under weaker hypotheses about nonlinearity at zero (at infinity, respectively). © 2008 Elsevier Ltd. All rights reserved.
Article history: Received 22 May 2008 Accepted 5 November 2008 MSC: 35D05 35J20 35J70 Keywords: p(x)-Laplacian Variational method Infinitely many solutions
1. Introduction In this paper we study the following problem:
−div(|∇ u|p(x)−2 ∇ u) + |u|p(x)−2 u = f (x, u) in RN , u ∈ W 1,p(x) (RN ),
(1.1)
where p(x) = p(|x|) ∈ C (RN ) with 2 ≤ N < p− := infRN p(x) ≤ p+ := supRN p(x) < +∞, and f : RN × R → R satisfies the Carathéodory condition and is radial respect to x, with f (x, 0) = 0 and for each r > 0 sup|t |≤r |f (x, t )| ∈ L1 (RN ).
The operator −div(|∇ u|p(x)−2 ∇ u) is said to be p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been receiving considerable attention in recent years. These problems are interesting in applications (see e.g. [14,19]) and raise many difficult mathematical problems. We refer the reader to [2,10,15,17,18] for an overview of and references on this subject, and to [1,4–9] for the study of the p(x)-Laplacian equations and the corresponding variational problems. In order to describe our results precisely, we recall some basic concepts. For simplicity we write E = W 1,p(x) (Ω ), and employ B(y, ρ) to denote the open N-dimensional balls with center y ∈ RN and radius ρ > 0 in RN and the letter C to denote any constant (the exact value of C may change from line to line). The energy functional I : E → R associated with problem (1.1) I (u) =
Z
1 RN
p(x)
(|∇ u|p(x) + |u|p(x) )dx −
Z RN
F (x, u)dx
I Research supported by the National Natural Science Foundation of China (10671084).
∗
Tel.: +86 931 8794115. E-mail address: [email protected].
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.11.037
(1.2)
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G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
Ru
is well defined, where F (x, u) = 0 f (x, t )dt. Since f (x, 0) = 0 we may assume that f is extended to the whole real line with zero on (−∞, 0] for almost all x ∈ RN . Now, we are in a position to state our main results. First, we make the following assumptions. (S0 ) There are two sequences {ak }k∈N , {bk }k∈N such that 0 < bk+1 < ak < bk , limk→+∞ bk = 0, and f (x, s) ≤ 0 for almost all x ∈ RN and s ∈ [ak , bk ], k ∈ N.
(F0 ) There exist a sequence {ηk }k∈N ⊂ [0, bk ] and a constant h0 > γ
+ − 2p (2n −1)+γ p 2n − − p γ p
p−
such that F (x, ηk ) ≥ h0 ηk for almost all
x ∈ RN , and F (x, u(x)) ≥ 0 for a.e. x ∈ B(0, γ ) \ B(0, 2 ), where γ is a positive constant. (S∞ ) There are two sequences {ak }k∈N , {bk }k∈N such that 0 < ak < bk < ak+1 , limk→+∞ bk = +∞, and f (x, s) ≤ 0 for almost all x ∈ RN and s ∈ [ak , bk ], k ∈ N.
(F∞ ) There exist a sequence {ηk }k∈N ⊂ [0, bk ] and a constant h0 > γ
+ + 2p (2n −1)+γ p 2n + − p γ p
p+
such that F (x, ηk ) ≥ h0 ηk for almost
all x ∈ RN , and F (x, u(x)) ≥ 0 for a.e. x ∈ B(0, γ ) \ B(0, 2 ), where γ is a positive constant. We are interested in the existence of infinitely many solutions for problem (1.1). Our main results are the following two theorems, which are generalizations of Theorems (1.1) and (1.2) of [11]. Theorem 1.1. Assume f satisfies (S0 ) and (F0 ). Then, there exists a radially symmetric sequence {uGk } ⊂ X of nonnegative, homoclinic weak solutions of (1.1) such that lim I (uGk ) = 0 and
k→+∞
lim kuGk kE = 0.
k→+∞
Theorem 1.2. Assume f satisfies (S∞ ) and (F∞ )). Then, there exists a radially symmetric sequence {uGk } ⊂ X of nonnegative, homoclinic weak solutions of (1.1) such that lim I (uGk ) = −∞ and
lim kuGk kE = +∞.
k→+∞
k→+∞
Example 1.1. A simple function which satisfies (F0 ) and (S0 ) is F (x, s) = α(x)sq(x) sin 1 < q(x) ≤ p− , α ∈ L∞ (RN ) and essinfRN α ≥ h0 .
1 s
Example 1.2. A simple function which satisfies (F∞ ) and (S∞ ) is F (x, s) = α(x)sq(x) sin p+ ≤ q(x), α ∈ L∞ (RN ) and essinfRN α ≥ h0 .
for s > 0, and f (x, 0) = 0, where
1 s
for s > 0, and f (x, 0) = 0, where
Remark 1.1. It is possible to handle the case where the nonlinear term f locally essentially bounded (not necessarily continuous) with respect to the second argument u in Theorems 1.1 and 1.2; in such a case, a differential inclusion problem is formulated instead of (1.1). Remark 1.2. Note that in this paper we take G = O(N ) which is a particular form of [11], where the authors work within a very general framework, considering instead of O(N ) the form GN = {G ⊆ O(N ) : G = O(N1 ) × · · · × O(Nk ), k ≥ 1, N1 + · · · + Nk = N , Nj ≥ 2, j = 1, . . . , k}. Remark 1.3. In [11], the nonlinearity oscillates at zero (at infinity, respectively), while, our condition (F0 ) ((F∞ ), respectively) is weaker than it, even in the case of constant exponent. The aim of the present paper is to improve and generalize the main results of [11]. f is independent of x in [11], while in the variable exponent case we need to consider f depending on x in our problem (1.1), which raises some essential difficulties. On the other hand, the p(x)-Laplacian possesses more complex nonlinearities, which is the second difficulty. This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and give two key embedding results. In Sections 3 and 4, we prove Theorems 1.1 and 1.2, respectively. 2. Preliminaries 2.1. Variable exponent Sobolev space 1,p(x)
In order to discuss problem (1.1), we need some theories on W0
(Ω ) which we call the variable exponent Sobolev 1,p(x)
space, where Ω is a open subset of R . Firstly we state some basic properties of spaces W0 (Ω ) which will be used later (for details, see [9]). Denote by S(Ω ) the set of all measurable real functions defined on Ω . Two functions in S(Ω ) are considered as the same element of S(Ω ) when they are equal almost everywhere. Write N
Lp(x) (Ω ) =
u ∈ S(Ω ) :
Z Ω
|u(x)|p(x) dx < +∞
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1135
with the norm
) Z u(x) p(x) = inf λ > 0 : λ dx ≤ 1 , Ω (
|u|Lp(x) (Ω ) = |u|p(x) and
W 1,p(x) (Ω ) = u ∈ Lp(x) (Ω ) : |∇ u| ∈ Lp(x) (Ω )
with the norm
kukW 1,p(x) (Ω ) = |u|Lp(x) (Ω ) + |∇ u|Lp(x) (Ω ) . 1,p(x)
Denote by W0
(Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ) . 1,p(x)
Proposition 2.1 (See [9]). The spaces Lp(x) (Ω ) , W 1,p(x) (Ω ) and W0 Proposition 2.2 (See [9]). Set ρ(u) =
R
Ω
(Ω ) are separable and reflexive Banach spaces.
|u(x)|p(x) dx. For any u ∈ Lp(x) (Ω ), then:
(1) For u 6= 0, |u|p(x) = λ ⇔ ρ( λu ) = 1 (2) |u|p(x) < 1(= 1; > 1) ⇔ ρ(u) < 1(= 1; > 1). p− p+ (3) If |u|p(x) > 1, then |u|p(x) ≤ ρ (u) ≤ |u|p(x) . (4) (5) (6)
p+
p−
If |u|p(x) < 1, then |u|p(x) ≤ ρ (u) ≤ |u|p(x) . limk→+∞ |uk |p(x) = 0 ⇐⇒ limk→+∞ ρ(uk ) = 0 |uk |p(x) → +∞ ⇐⇒ ρ(uk ) → +∞. 1,p(x)
Proposition 2.3 (See [9]). In W0
(Ω ) the Poincaré inequality holds, that is, there exists a positive constant C such that
∀u ∈ W01,p(x) (Ω ) .
|u|Lp(x) (Ω ) ≤ C |∇ u|Lp(x) (Ω ) ,
Proposition 2.4 (See [7,9]). The conjugate space of Lp(x) (Ω ) is Lp (x) (Ω ), where p(1x) + R 0 v ∈ Lp (x) (Ω ) , Ω |uv| dx ≤ 2 |u|p(x) |v|p0 (x) . 0
1 p0 (x)
= 1. For any u ∈ Lp(x) (Ω ) and
2.2. A key embedding result The action of the orthogonal group G = O(N ) on W 1,p(x) RN can be defined by (gu)(x) = u(g −1 x) for every g ∈ O(N ), x ∈ RN . It is clear that this group acts linearly and isometrically; in particular kgukE = kukE for every g ∈ O(N ) and u ∈ E. Defining the subspace of radially symmetric functions of E by
Wr1,p(x) (RN ) = {u ∈ W 1,p(x) RN : gu = u
∀g ∈ O(N )},
we can state the following crucial result, which is a generalization of Theorem 3.1 in [12]. From now on, we write X = 1,p(x) Wr (RN ) and e X = L∞ (RN ) for simplicity. Theorem 2.1. The embedding E ,→ e X is continuous whenever 2 ≤ N < p− ≤ p+ < +∞. Proof. The proof is similar to that for the case of constant exponent (see Morrey’s inequality in [3]), so we only outline the proof. We only need to prove that there exists a constant C , depending only on p+ , p− and N, such that sup |u| ≤ C kukE . RN
Firstly choose any ball B(x, r ) ⊂ RN . Using the same method as in [3], we can claim there exists a constant C , depending only on N, for any y ∈ B(x, r ), such that
Z B(x,r )
|u(x) − u(y)|dy ≤ C
Z B(x,r )
|∇ u(y)| dy. |x − y|N −1
(2.5)
Now fix x ∈ RN . We apply inequality (2.5) and Proposition 2.4 as follows:
|u(x)| ≤
1
|B(x, 1)| Z
Z B(x,1)
|u(x) − u(y)|dy +
1
|B(x, 1)|
Z B(x,1)
|u(y)|dy
|∇ u(y)| dy + C |u|Lp(x) (B(x,1)) | x − y|N −1 B(x,1) 1 ≤ C |∇ u|Lp(x) (RN ) + C |u|Lp(x) (RN ) . |x − y|(N −1) p0 (x) ≤C
L
(B(x,1))
1136
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
p(x) Since p(x) ≥ p− > N implies (N − 1) p(x)−1 < N, we have
ρ
1
Z
1
=
|x − y|N −1
B(x,1)
Using Proposition 2.2, one has |
|x − y|(N −1)p0 (x) 1
|0 |x−y|(N −1) p (x)
dy < +∞.
< +∞. Therefore, we obtain |u(x)| ≤ C kukE . This ends the proof.
Theorem 2.2. The embedding X ,→ e X is compact whenever 2 ≤ N < p− ≤ p+ < +∞. Proof. Let un be a bounded sequence in X . Up to a subsequence, un * u in X for some u ∈ X . Let ρ > 0 be an arbitrarily fixed number. Due to the radially symmetric properties of u and un , we have
kun − ukW 1,p(x) (B(g1 y,ρ)) = kun − ukW 1,p(x) (B(g2 y,ρ))
(2.1)
for every g1 , g2 ∈ O(N ) and y ∈ RN . For a fixed y ∈ RN , we can define m(y, ρ) = sup{n ∈ N : ∃gi ∈ O(N ), i ∈ N such that B(gi y, ρ) ∩ B(gj y, ρ) = ∅, ∀i 6= j}. By virtue of (2.1), for every y ∈ RN and n ∈ N, we have
k un − uk E ≤ ≤ m(y, ρ)
kun − ukW 1,p(x) (B(y,ρ))
sup kun kE + kukE n∈N
m(y, ρ)
.
The right-hand side does not depend on n, and m(y, ρ) → +∞ whenever |y| → +∞. Thus, for every ε > 0 there exists Rε > 0 such that for every y ∈ RN with |y| ≥ Rε one has
kun − ukW 1,p(x) (B(y,ρ)) < ε(2Sρ )−1 ∀n ∈ N,
(2.2)
where Sρ > 0 is the embedding constant of W 1,p(x) (BN (y, ρ)) ,→ C 0 (B[y, ρ]), where B[y, ρ] denotes the closure of B(y, ρ). Thus, in view of (2.2), one has that sup kun − ukC 0 (B[y,ρ]) ≤
|y|≥Rε
ε 2
∀n ∈ N.
(2.3)
On the other hand, since un * u in X , then by the compact embedding of X ,→ C 0 (B[0, Rε ]) (because X is a continuous 1,p(x) embedding Wr (B[0, Rε ]) and Wr1,p(x) (B[0, Rε ]) ,→ C 0 (B[0, Rε ]) is compact). It follows that there exists nε ∈ N such that
kun − ukC 0 (BN [0,Rε ]) ≤
ε 2
∀ n ≥ nε .
(2.4)
Combining (2.3) with (2.4), one concludes that kun − uke X < ε for every n ≥ nε . This ends the proof.
3. Proof of Theorem 1.1 Due to the principle of symmetric criticality of Palais (see [16]), the critical points of IG are critical points of I as well, where IG stands for the restriction of I to X (note that the fixed point space of the action G on the space E is exactly X ). Proposition 3.1. Functional IG is sequentially weakly lower semicontinuous on X . Proof. The function
1 (|∇ u|p(x) RRN p(x)
R
+ |u|p(x) )dx is clearly sequentially weakly lower semicontinuous on X . Let us prove
that the function Φ = RN F (x, u)dx is sequentially weakly continuous. Suppose the contrary, i.e. let {un } ⊂ X be a sequence which converges weakly to u ∈ X but Φ (un ) 9 Φ (u) as n → +∞. Therefore, up to a subsequence, one can find a number ε0 > 0 such that
|Φ (un ) − Φ (u)| ≥ ε0 > 0 ∀n ∈ N.
(3.1)
By Theorem 2.2, for every n ∈ N one has 0 < θn < 1 such that
|Φ (un ) − Φ (u)| ≤
Z RN
|f (x, u + θn (un − u))| · |un − u|dx
≤ | sup |f (x, t )||1 · |un − u|∞ → 0 as n → +∞, |t |≤Mn
where Mn = |u|∞ + |un |∞ < +∞, a contradiction to (3.1).
Let us fix a number r < 0 arbitrarily, and for every k ∈ N, consider the set Sk = {u ∈ X : r ≤ u(x) ≤ bk
a.e. x ∈ RN },
where bk is from (S0 ). Proposition 3.2. The functional IG is bounded from below on Sk and its infimum mk on Sk is attained at uGk ∈ Sk .
(3.2)
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1137
Proof. Firstly, for every u ∈ Sk , we have
Z
1
(|∇ u|p(x) + |u|p(x) )dx − Φ (u) Z ≥ −Φ (u) ≥ − max{bk , −r } sup
IG (u) =
RN
p(x)
RN |t |≤max{bk ,−r }
|f (x, t )|dx > −∞.
Thus, IG is bounded from below on Sk . It is clear that Sk is convex. Moreover, we have closed X due to Theorem 2.1. Thus Sk is weakly closed in X . Let mk = infSk IG , and {un } be a sequence in Sk such that mk ≤ IG (un ) ≤ mk + 1n for all n ∈ N. Then, if kun kSk ≤ 1, we are done; otherwise, we have −
kun kpSk
≤ mk + 1 + max{bk , −r }
p+
Z sup
RN |t |≤max{bk ,−r }
f (x, t )dx
(3.3)
for all n ∈ N, and thus {un } is bounded in Sk . So, up to a subsequence, {un } weakly converges in Sk to some uGk ∈ Sk . By Proposition 3.1, IG is sequentially weakly lower semicontinuous in Sk , which implies that I (uGk ) = mk = infSk IG . Proposition 3.3. 0 ≤ uGk (x) ≤ ak a.e. x ∈ RN . Proof. Let W = {x ∈ RN : uGk (x)∈[0, ak ]} and suppose that meas(W ) > 0. Define the function h(s) = min(s+ , ak ), where s+ = max(s, 0), and set wk = h(uGk ). Obviously, h is Lipschitz continuous and h(0) = 0. Then, due to Marcus and Mizel [13],
w ek ∈ W 1,p (RN )∩ W 1,p (RN ) ⊆ E. Moreover, wk is G-invariant (radially symmetric) because of uGk ; thus wk ∈ X . In addition, wk ∈ Sk . We introduce the following two sets: +
−
W1 = {x ∈ W : uGk (x) < 0} and W2 = {x ∈ W : uGk (x) > ak }.
(3.4)
Then, W = W1 ∪ W2 , and we have that
wk (x) =
uGk (x)
if x ∈ RN \ W if x ∈ W1 if x ∈ W2 .
0 a k
(3.5)
Moreover,
Z [|wk |p(x) − |uGk |p(x) ]dx − [F (x, wk ) − F (x, uGk )]dx p(x) p(x) W ZW ZW Z 1 1 1 p(x) G p(x) G p(x) =− |∇ uk | dx − |uk | dx + [ak − (uGk )p(x) ]dx W p(x) W1 p(x) W2 p(x) Z Z G − [F (x, 0) − F (x, uk )]dx − [F (x, ak ) − F (x, uGk )]dx.
Φ (wk ) − Φ (uGk ) = −
Z
1
|∇ uGk |p(x) dx +
W1
1
W2
)]dx = 0. By (S0 ), one has that F (x, ak ) ≥ F (x, s) for every s ∈ [ak , bk ] and almost all x ∈ RN . In conclusion, every term of the above expression is nonpositive. On the other hand, because IG (wk ) ≥ IG (uGk ) = infSk IG , Note that
R
W1
[F (x, 0) − F (x,
Z
uGk
then every term should be zero. In particular,
Z
1 W1
p(x)
uGk p(x) dx
| |
Z
1
= W2
p(x)
[apk(x) − (uGk )p(x) ]dx = 0.
These equalities imply that meas(W1 ) = meas(W2 ) = 0, so meas(W ) = 0.
(3.6)
Proposition 3.4. uGk is a local minimum point of IG in X for every k ∈ N. Proof. Indeed, otherwise there would be a sequence {un } ⊂ X which converges to uGk and I (un ) < I (uGk ) for all n ∈ N. From this inequality it follows that un ∈Sk for any n ∈ N. Since un → uGk in X , then due to Theorem 2.2, un → uGk in e X as well. min{−r ,b −a }
k k In particular, for every 0 < δ < , there exists nδ ∈ N such that |un − uGk |∞ < δ for every n ≥ nδ . By using 2 Proposition 3.3 and taking into account the choice of the number δ , we conclude that
r < un (x) < bk
for all x ∈ RN , n ≥ nδ ,
which clearly contradicts the fact that un ∈Sk .
(3.7)
Proposition 3.5. mk < 0 for all k ∈ N and limk→+∞ mk = 0.
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G. Dai / Nonlinear Analysis 71 (2009) 1133–1139 p−
Proof. Taking into account condition (F0 ), we can find a sequence {ηk }k∈N ⊂ [0, bk ] such that F (x, ηk ) ≥ h0 ηk for almost all x ∈ RN . Now, we consider a function ωk ∈ X defined by
0 ηk ωk (x) = 2ηk (γ − |x|) γ
if x ∈ RN \ B(0, γ ) γ if x ∈ B 0, 2
if x ∈ B(0, γ ) \ B 0,
γ 2
(3.8)
.
Without loss of generality, we may assume that ηk ≤ min{γ , 1}. Obviously, ωk (x) ∈ Sk . We have IG (ωk ) ≤ −
γ B(0, 2 ) +
≤
=
+
Z
F (x, ηk )dx +
p−
2p ηk ωγ n
− p− γ p − n p k − p− p 2n
ωγ η γ
p− γ p
−
+
2n
+
1−
−
1
1−
p−
2p ηk ωγ n
ηkp ωγ n p−
1
+
2n
Z −
−
ηkp ωγ n
p− p−
γ B(0, 2 )
−
h0 ηk dx
−
[2p (2n − 1) + γ p 2n − h0 p− γ p ]
−
ωγ n bp + − − ≤ − p−k n [2p (2n − 1) + γ p 2n − h0 p− γ p ]. p γ 2 Hence, since h0 >
+ − 2p (2n −1)+γ p 2n , − γ p p−
this forces
−
ωγ n bpk + − − [2p (2n − 1) + γ p 2n − h0 p− γ p ] < 0. IG (ωk ) ≤ − p− γ p 2n
(3.9)
Now we will prove that limk→+∞ mk = 0. Due to Proposition 3.3, for every x ∈ RN one has 0 > IG (uGk ) ≥ −Φ (uGk ) ≥ −ak
Z
sup |f (x, t )|dx.
(3.10)
RN |t |≤a1
Since the sequence {ak } tends to zero, then mk → 0 as k → +∞.
Proof of Theorem 1.1 concluded. Since uGk are local minima of IG , they are critical points of IG , and thus radially symmetric weak solutions of (1.1). Due to Proposition 3.5, there are infinitely many distinct uGk . Moreover, we have +
kuGk kp p+
Z ≤ mk + RN
F (x,
uGk
(x))dx ≤ ak
which proves that limk→+∞ kuGk kE = 0.
Z
sup f (x, t )dx,
(3.11)
RN |t |≤a1
4. Proof of Theorem 1.2 The proof of Theorem 1.2 is similar to that of Theorem 1.1; consequently, we only outline the proof. We assume that the hypotheses of Theorem 1.2 are fulfilled. Let us fix a number r < 0 arbitrarily, and for every k ∈ N, consider the set Tk = {u ∈ X : r ≤ u(x) ≤ bk a.e.x ∈ RN },
(4.1)
The first part of the proof is similar to that of Theorem 1.1. Indeed, we can prove that the functional IG is bounded from below on Tk and its infimum on Tk is attained (see Proposition 3.2). Moreover, if uGk (x) ∈ Tk is chosen such that IG (uGk ) = infTk IG , then 0 ≤ uGk ≤ ak for all x ∈ RN (see Proposition 3.3), and uGk is a local minimum point of IG in X (see Proposition 3.4). Instead of Proposition 3.5, we prove:
ek = IG (uGk ) = infTk IG . Then limk→+∞ m ek = −∞. Proposition 4.1. Let m p+
Proof. Taking into account condition (F∞ ), we can find a sequence {ηk }k∈N ⊂ [0, bk ] such that F (x, ηk ) ≥ h0 ηk for almost all x ∈ RN . Now, if we consider a function ωk ∈ X defined in (3.8), without loss of generality, we may assume
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1139
that ηk ≥ max{γ , 1}. Obviously, ωk (x) ∈ Sk . We have IG (ωk ) ≤ −
+
Z γ B(0, 2 )
F (x, ηk )dx +
p+
2p ηk ωγ n
1−
p− γ p
+
+
2n
≤
=
+ n p k + p− p 2 n
≤
ωγ n bpk + + + [2p (2n − 1) + γ p 2n − h0 p− γ p ]. + p− γ p 2 n
η ωγ n γ
ωγ η γ
1−
1 2n
+
η ωγ n p−
Z −
ηkp ωγ n p− p+
γ
B(0, 2 )
+
+
+
+ p+ 2p k + p− p
p+ k
1
h0 ηk dx
+
[2p (2n − 1) + γ p 2n − h0 p− γ p ]
+
Hence, since h0 > p+
+ + 2p (2n −1)+γ p 2n , + − p γ p +
this forces +
+
lim bk [2p (2n − 1) + γ p 2n − h0 p− γ p ] = −∞.
(4.2)
k→+∞
Proof of Theorem 1.2 concluded. It is clear that we have infinitely many pairwise distinct local minimum points uGk of IG with uGk ∈ Tk . Now we will prove that kuGk kE → +∞ as k → +∞. Arguing by contradiction, assume that there exists a subsequence {uGnk } of {uGk } which is bounded in E. Thus, it is also bounded in e X due to Theorem 2.1. In particular we can find m0 ∈ N such that uGnk ∈ Tm0 for all k ∈ N. For every nk ≥ m0 one has
em0 ≥ m enk = inf IG = IG (uGnk ) ≥ inf IG = m em0 , m Tnk
(4.3)
Tm0
enk = m em0 for all nk ≥ m0 , contradicting Proposition 4.1. This concludes our proof. which proves that m
Acknowledgments Thanks are offered to Professor X.L. Fan for his valuable guidance. The author also wishes to express his gratitude to the referee for reading the paper carefully and making several corrections and remarks. References [1] G. Dai, Infinitely many solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian, Nonlinear Anal. (2008). doi:10.1016/j.na.2008.03.009. [2] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drábek, J. Rákosník, FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp.38–58. [3] L.C. Evans, Partial Differential Equations, AMS, Providence, RI, 1998. [4] X.L. Fan, On the sub–supersolution methods for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007) 665–682. [5] X.L. Fan, X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , Nonlinear Anal. 59 (2004) 173–188. [6] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W k,p(x) (Ω ), J. Math. Anal. Appl. 262 (2001) 749–760. [7] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003) 1843–1852. [8] X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005) 306–317. [9] X.L. Fan, D. Zhao, On the Spaces Lp(x) and W m,p(x) , J. Math. Anal. Appl. 263 (2001) 424–446. [10] P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, in: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Work- shop, Jyväskylä, 2003, pp. 85–93. [11] A. Kristály, G. Moroşanu, S. Tersian, Quasilinear elliptic problems in RN involving oscillatory nonlinearities, J. Differential Equations 235 (2007) 366–375. [12] A. Kristály, Infinitely many solutions for a differential inclusion problem in RN , J. Differential Equations 220 (2006) 511–530. [13] M. Marcus, V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979) 217–229. [14] M. R u˘zi˘cka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [15] 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. [16] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [17] V.V. Zhikov, V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, (Translated from the Russian by G.A. Yosifian). [18] V.V. Zhikov, On some variational problems, Russian J. Math. Phys. 5 (1997) 105–116. [19] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9 (1987) 33–66.
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Infinitely many solutions for a p(x)-Laplacian equation in RN I Guowei Dai ∗ Department of Mathematics, Lanzhou University, Lanzhou, 730000, PR China
article
info
a b s t r a c t This paper deals with a p(x)-Laplacian equation in RN . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose W 1,p(x) (RN )-norms tend to zero (to infinity, respectively) under weaker hypotheses about nonlinearity at zero (at infinity, respectively). © 2008 Elsevier Ltd. All rights reserved.
Article history: Received 22 May 2008 Accepted 5 November 2008 MSC: 35D05 35J20 35J70 Keywords: p(x)-Laplacian Variational method Infinitely many solutions
1. Introduction In this paper we study the following problem:
−div(|∇ u|p(x)−2 ∇ u) + |u|p(x)−2 u = f (x, u) in RN , u ∈ W 1,p(x) (RN ),
(1.1)
where p(x) = p(|x|) ∈ C (RN ) with 2 ≤ N < p− := infRN p(x) ≤ p+ := supRN p(x) < +∞, and f : RN × R → R satisfies the Carathéodory condition and is radial respect to x, with f (x, 0) = 0 and for each r > 0 sup|t |≤r |f (x, t )| ∈ L1 (RN ).
The operator −div(|∇ u|p(x)−2 ∇ u) is said to be p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been receiving considerable attention in recent years. These problems are interesting in applications (see e.g. [14,19]) and raise many difficult mathematical problems. We refer the reader to [2,10,15,17,18] for an overview of and references on this subject, and to [1,4–9] for the study of the p(x)-Laplacian equations and the corresponding variational problems. In order to describe our results precisely, we recall some basic concepts. For simplicity we write E = W 1,p(x) (Ω ), and employ B(y, ρ) to denote the open N-dimensional balls with center y ∈ RN and radius ρ > 0 in RN and the letter C to denote any constant (the exact value of C may change from line to line). The energy functional I : E → R associated with problem (1.1) I (u) =
Z
1 RN
p(x)
(|∇ u|p(x) + |u|p(x) )dx −
Z RN
F (x, u)dx
I Research supported by the National Natural Science Foundation of China (10671084).
∗
Tel.: +86 931 8794115. E-mail address: [email protected].
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.11.037
(1.2)
1134
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
Ru
is well defined, where F (x, u) = 0 f (x, t )dt. Since f (x, 0) = 0 we may assume that f is extended to the whole real line with zero on (−∞, 0] for almost all x ∈ RN . Now, we are in a position to state our main results. First, we make the following assumptions. (S0 ) There are two sequences {ak }k∈N , {bk }k∈N such that 0 < bk+1 < ak < bk , limk→+∞ bk = 0, and f (x, s) ≤ 0 for almost all x ∈ RN and s ∈ [ak , bk ], k ∈ N.
(F0 ) There exist a sequence {ηk }k∈N ⊂ [0, bk ] and a constant h0 > γ
+ − 2p (2n −1)+γ p 2n − − p γ p
p−
such that F (x, ηk ) ≥ h0 ηk for almost all
x ∈ RN , and F (x, u(x)) ≥ 0 for a.e. x ∈ B(0, γ ) \ B(0, 2 ), where γ is a positive constant. (S∞ ) There are two sequences {ak }k∈N , {bk }k∈N such that 0 < ak < bk < ak+1 , limk→+∞ bk = +∞, and f (x, s) ≤ 0 for almost all x ∈ RN and s ∈ [ak , bk ], k ∈ N.
(F∞ ) There exist a sequence {ηk }k∈N ⊂ [0, bk ] and a constant h0 > γ
+ + 2p (2n −1)+γ p 2n + − p γ p
p+
such that F (x, ηk ) ≥ h0 ηk for almost
all x ∈ RN , and F (x, u(x)) ≥ 0 for a.e. x ∈ B(0, γ ) \ B(0, 2 ), where γ is a positive constant. We are interested in the existence of infinitely many solutions for problem (1.1). Our main results are the following two theorems, which are generalizations of Theorems (1.1) and (1.2) of [11]. Theorem 1.1. Assume f satisfies (S0 ) and (F0 ). Then, there exists a radially symmetric sequence {uGk } ⊂ X of nonnegative, homoclinic weak solutions of (1.1) such that lim I (uGk ) = 0 and
k→+∞
lim kuGk kE = 0.
k→+∞
Theorem 1.2. Assume f satisfies (S∞ ) and (F∞ )). Then, there exists a radially symmetric sequence {uGk } ⊂ X of nonnegative, homoclinic weak solutions of (1.1) such that lim I (uGk ) = −∞ and
lim kuGk kE = +∞.
k→+∞
k→+∞
Example 1.1. A simple function which satisfies (F0 ) and (S0 ) is F (x, s) = α(x)sq(x) sin 1 < q(x) ≤ p− , α ∈ L∞ (RN ) and essinfRN α ≥ h0 .
1 s
Example 1.2. A simple function which satisfies (F∞ ) and (S∞ ) is F (x, s) = α(x)sq(x) sin p+ ≤ q(x), α ∈ L∞ (RN ) and essinfRN α ≥ h0 .
for s > 0, and f (x, 0) = 0, where
1 s
for s > 0, and f (x, 0) = 0, where
Remark 1.1. It is possible to handle the case where the nonlinear term f locally essentially bounded (not necessarily continuous) with respect to the second argument u in Theorems 1.1 and 1.2; in such a case, a differential inclusion problem is formulated instead of (1.1). Remark 1.2. Note that in this paper we take G = O(N ) which is a particular form of [11], where the authors work within a very general framework, considering instead of O(N ) the form GN = {G ⊆ O(N ) : G = O(N1 ) × · · · × O(Nk ), k ≥ 1, N1 + · · · + Nk = N , Nj ≥ 2, j = 1, . . . , k}. Remark 1.3. In [11], the nonlinearity oscillates at zero (at infinity, respectively), while, our condition (F0 ) ((F∞ ), respectively) is weaker than it, even in the case of constant exponent. The aim of the present paper is to improve and generalize the main results of [11]. f is independent of x in [11], while in the variable exponent case we need to consider f depending on x in our problem (1.1), which raises some essential difficulties. On the other hand, the p(x)-Laplacian possesses more complex nonlinearities, which is the second difficulty. This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and give two key embedding results. In Sections 3 and 4, we prove Theorems 1.1 and 1.2, respectively. 2. Preliminaries 2.1. Variable exponent Sobolev space 1,p(x)
In order to discuss problem (1.1), we need some theories on W0
(Ω ) which we call the variable exponent Sobolev 1,p(x)
space, where Ω is a open subset of R . Firstly we state some basic properties of spaces W0 (Ω ) which will be used later (for details, see [9]). Denote by S(Ω ) the set of all measurable real functions defined on Ω . Two functions in S(Ω ) are considered as the same element of S(Ω ) when they are equal almost everywhere. Write N
Lp(x) (Ω ) =
u ∈ S(Ω ) :
Z Ω
|u(x)|p(x) dx < +∞
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1135
with the norm
) Z u(x) p(x) = inf λ > 0 : λ dx ≤ 1 , Ω (
|u|Lp(x) (Ω ) = |u|p(x) and
W 1,p(x) (Ω ) = u ∈ Lp(x) (Ω ) : |∇ u| ∈ Lp(x) (Ω )
with the norm
kukW 1,p(x) (Ω ) = |u|Lp(x) (Ω ) + |∇ u|Lp(x) (Ω ) . 1,p(x)
Denote by W0
(Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ) . 1,p(x)
Proposition 2.1 (See [9]). The spaces Lp(x) (Ω ) , W 1,p(x) (Ω ) and W0 Proposition 2.2 (See [9]). Set ρ(u) =
R
Ω
(Ω ) are separable and reflexive Banach spaces.
|u(x)|p(x) dx. For any u ∈ Lp(x) (Ω ), then:
(1) For u 6= 0, |u|p(x) = λ ⇔ ρ( λu ) = 1 (2) |u|p(x) < 1(= 1; > 1) ⇔ ρ(u) < 1(= 1; > 1). p− p+ (3) If |u|p(x) > 1, then |u|p(x) ≤ ρ (u) ≤ |u|p(x) . (4) (5) (6)
p+
p−
If |u|p(x) < 1, then |u|p(x) ≤ ρ (u) ≤ |u|p(x) . limk→+∞ |uk |p(x) = 0 ⇐⇒ limk→+∞ ρ(uk ) = 0 |uk |p(x) → +∞ ⇐⇒ ρ(uk ) → +∞. 1,p(x)
Proposition 2.3 (See [9]). In W0
(Ω ) the Poincaré inequality holds, that is, there exists a positive constant C such that
∀u ∈ W01,p(x) (Ω ) .
|u|Lp(x) (Ω ) ≤ C |∇ u|Lp(x) (Ω ) ,
Proposition 2.4 (See [7,9]). The conjugate space of Lp(x) (Ω ) is Lp (x) (Ω ), where p(1x) + R 0 v ∈ Lp (x) (Ω ) , Ω |uv| dx ≤ 2 |u|p(x) |v|p0 (x) . 0
1 p0 (x)
= 1. For any u ∈ Lp(x) (Ω ) and
2.2. A key embedding result The action of the orthogonal group G = O(N ) on W 1,p(x) RN can be defined by (gu)(x) = u(g −1 x) for every g ∈ O(N ), x ∈ RN . It is clear that this group acts linearly and isometrically; in particular kgukE = kukE for every g ∈ O(N ) and u ∈ E. Defining the subspace of radially symmetric functions of E by
Wr1,p(x) (RN ) = {u ∈ W 1,p(x) RN : gu = u
∀g ∈ O(N )},
we can state the following crucial result, which is a generalization of Theorem 3.1 in [12]. From now on, we write X = 1,p(x) Wr (RN ) and e X = L∞ (RN ) for simplicity. Theorem 2.1. The embedding E ,→ e X is continuous whenever 2 ≤ N < p− ≤ p+ < +∞. Proof. The proof is similar to that for the case of constant exponent (see Morrey’s inequality in [3]), so we only outline the proof. We only need to prove that there exists a constant C , depending only on p+ , p− and N, such that sup |u| ≤ C kukE . RN
Firstly choose any ball B(x, r ) ⊂ RN . Using the same method as in [3], we can claim there exists a constant C , depending only on N, for any y ∈ B(x, r ), such that
Z B(x,r )
|u(x) − u(y)|dy ≤ C
Z B(x,r )
|∇ u(y)| dy. |x − y|N −1
(2.5)
Now fix x ∈ RN . We apply inequality (2.5) and Proposition 2.4 as follows:
|u(x)| ≤
1
|B(x, 1)| Z
Z B(x,1)
|u(x) − u(y)|dy +
1
|B(x, 1)|
Z B(x,1)
|u(y)|dy
|∇ u(y)| dy + C |u|Lp(x) (B(x,1)) | x − y|N −1 B(x,1) 1 ≤ C |∇ u|Lp(x) (RN ) + C |u|Lp(x) (RN ) . |x − y|(N −1) p0 (x) ≤C
L
(B(x,1))
1136
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
p(x) Since p(x) ≥ p− > N implies (N − 1) p(x)−1 < N, we have
ρ
1
Z
1
=
|x − y|N −1
B(x,1)
Using Proposition 2.2, one has |
|x − y|(N −1)p0 (x) 1
|0 |x−y|(N −1) p (x)
dy < +∞.
< +∞. Therefore, we obtain |u(x)| ≤ C kukE . This ends the proof.
Theorem 2.2. The embedding X ,→ e X is compact whenever 2 ≤ N < p− ≤ p+ < +∞. Proof. Let un be a bounded sequence in X . Up to a subsequence, un * u in X for some u ∈ X . Let ρ > 0 be an arbitrarily fixed number. Due to the radially symmetric properties of u and un , we have
kun − ukW 1,p(x) (B(g1 y,ρ)) = kun − ukW 1,p(x) (B(g2 y,ρ))
(2.1)
for every g1 , g2 ∈ O(N ) and y ∈ RN . For a fixed y ∈ RN , we can define m(y, ρ) = sup{n ∈ N : ∃gi ∈ O(N ), i ∈ N such that B(gi y, ρ) ∩ B(gj y, ρ) = ∅, ∀i 6= j}. By virtue of (2.1), for every y ∈ RN and n ∈ N, we have
k un − uk E ≤ ≤ m(y, ρ)
kun − ukW 1,p(x) (B(y,ρ))
sup kun kE + kukE n∈N
m(y, ρ)
.
The right-hand side does not depend on n, and m(y, ρ) → +∞ whenever |y| → +∞. Thus, for every ε > 0 there exists Rε > 0 such that for every y ∈ RN with |y| ≥ Rε one has
kun − ukW 1,p(x) (B(y,ρ)) < ε(2Sρ )−1 ∀n ∈ N,
(2.2)
where Sρ > 0 is the embedding constant of W 1,p(x) (BN (y, ρ)) ,→ C 0 (B[y, ρ]), where B[y, ρ] denotes the closure of B(y, ρ). Thus, in view of (2.2), one has that sup kun − ukC 0 (B[y,ρ]) ≤
|y|≥Rε
ε 2
∀n ∈ N.
(2.3)
On the other hand, since un * u in X , then by the compact embedding of X ,→ C 0 (B[0, Rε ]) (because X is a continuous 1,p(x) embedding Wr (B[0, Rε ]) and Wr1,p(x) (B[0, Rε ]) ,→ C 0 (B[0, Rε ]) is compact). It follows that there exists nε ∈ N such that
kun − ukC 0 (BN [0,Rε ]) ≤
ε 2
∀ n ≥ nε .
(2.4)
Combining (2.3) with (2.4), one concludes that kun − uke X < ε for every n ≥ nε . This ends the proof.
3. Proof of Theorem 1.1 Due to the principle of symmetric criticality of Palais (see [16]), the critical points of IG are critical points of I as well, where IG stands for the restriction of I to X (note that the fixed point space of the action G on the space E is exactly X ). Proposition 3.1. Functional IG is sequentially weakly lower semicontinuous on X . Proof. The function
1 (|∇ u|p(x) RRN p(x)
R
+ |u|p(x) )dx is clearly sequentially weakly lower semicontinuous on X . Let us prove
that the function Φ = RN F (x, u)dx is sequentially weakly continuous. Suppose the contrary, i.e. let {un } ⊂ X be a sequence which converges weakly to u ∈ X but Φ (un ) 9 Φ (u) as n → +∞. Therefore, up to a subsequence, one can find a number ε0 > 0 such that
|Φ (un ) − Φ (u)| ≥ ε0 > 0 ∀n ∈ N.
(3.1)
By Theorem 2.2, for every n ∈ N one has 0 < θn < 1 such that
|Φ (un ) − Φ (u)| ≤
Z RN
|f (x, u + θn (un − u))| · |un − u|dx
≤ | sup |f (x, t )||1 · |un − u|∞ → 0 as n → +∞, |t |≤Mn
where Mn = |u|∞ + |un |∞ < +∞, a contradiction to (3.1).
Let us fix a number r < 0 arbitrarily, and for every k ∈ N, consider the set Sk = {u ∈ X : r ≤ u(x) ≤ bk
a.e. x ∈ RN },
where bk is from (S0 ). Proposition 3.2. The functional IG is bounded from below on Sk and its infimum mk on Sk is attained at uGk ∈ Sk .
(3.2)
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1137
Proof. Firstly, for every u ∈ Sk , we have
Z
1
(|∇ u|p(x) + |u|p(x) )dx − Φ (u) Z ≥ −Φ (u) ≥ − max{bk , −r } sup
IG (u) =
RN
p(x)
RN |t |≤max{bk ,−r }
|f (x, t )|dx > −∞.
Thus, IG is bounded from below on Sk . It is clear that Sk is convex. Moreover, we have closed X due to Theorem 2.1. Thus Sk is weakly closed in X . Let mk = infSk IG , and {un } be a sequence in Sk such that mk ≤ IG (un ) ≤ mk + 1n for all n ∈ N. Then, if kun kSk ≤ 1, we are done; otherwise, we have −
kun kpSk
≤ mk + 1 + max{bk , −r }
p+
Z sup
RN |t |≤max{bk ,−r }
f (x, t )dx
(3.3)
for all n ∈ N, and thus {un } is bounded in Sk . So, up to a subsequence, {un } weakly converges in Sk to some uGk ∈ Sk . By Proposition 3.1, IG is sequentially weakly lower semicontinuous in Sk , which implies that I (uGk ) = mk = infSk IG . Proposition 3.3. 0 ≤ uGk (x) ≤ ak a.e. x ∈ RN . Proof. Let W = {x ∈ RN : uGk (x)∈[0, ak ]} and suppose that meas(W ) > 0. Define the function h(s) = min(s+ , ak ), where s+ = max(s, 0), and set wk = h(uGk ). Obviously, h is Lipschitz continuous and h(0) = 0. Then, due to Marcus and Mizel [13],
w ek ∈ W 1,p (RN )∩ W 1,p (RN ) ⊆ E. Moreover, wk is G-invariant (radially symmetric) because of uGk ; thus wk ∈ X . In addition, wk ∈ Sk . We introduce the following two sets: +
−
W1 = {x ∈ W : uGk (x) < 0} and W2 = {x ∈ W : uGk (x) > ak }.
(3.4)
Then, W = W1 ∪ W2 , and we have that
wk (x) =
uGk (x)
if x ∈ RN \ W if x ∈ W1 if x ∈ W2 .
0 a k
(3.5)
Moreover,
Z [|wk |p(x) − |uGk |p(x) ]dx − [F (x, wk ) − F (x, uGk )]dx p(x) p(x) W ZW ZW Z 1 1 1 p(x) G p(x) G p(x) =− |∇ uk | dx − |uk | dx + [ak − (uGk )p(x) ]dx W p(x) W1 p(x) W2 p(x) Z Z G − [F (x, 0) − F (x, uk )]dx − [F (x, ak ) − F (x, uGk )]dx.
Φ (wk ) − Φ (uGk ) = −
Z
1
|∇ uGk |p(x) dx +
W1
1
W2
)]dx = 0. By (S0 ), one has that F (x, ak ) ≥ F (x, s) for every s ∈ [ak , bk ] and almost all x ∈ RN . In conclusion, every term of the above expression is nonpositive. On the other hand, because IG (wk ) ≥ IG (uGk ) = infSk IG , Note that
R
W1
[F (x, 0) − F (x,
Z
uGk
then every term should be zero. In particular,
Z
1 W1
p(x)
uGk p(x) dx
| |
Z
1
= W2
p(x)
[apk(x) − (uGk )p(x) ]dx = 0.
These equalities imply that meas(W1 ) = meas(W2 ) = 0, so meas(W ) = 0.
(3.6)
Proposition 3.4. uGk is a local minimum point of IG in X for every k ∈ N. Proof. Indeed, otherwise there would be a sequence {un } ⊂ X which converges to uGk and I (un ) < I (uGk ) for all n ∈ N. From this inequality it follows that un ∈Sk for any n ∈ N. Since un → uGk in X , then due to Theorem 2.2, un → uGk in e X as well. min{−r ,b −a }
k k In particular, for every 0 < δ < , there exists nδ ∈ N such that |un − uGk |∞ < δ for every n ≥ nδ . By using 2 Proposition 3.3 and taking into account the choice of the number δ , we conclude that
r < un (x) < bk
for all x ∈ RN , n ≥ nδ ,
which clearly contradicts the fact that un ∈Sk .
(3.7)
Proposition 3.5. mk < 0 for all k ∈ N and limk→+∞ mk = 0.
1138
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139 p−
Proof. Taking into account condition (F0 ), we can find a sequence {ηk }k∈N ⊂ [0, bk ] such that F (x, ηk ) ≥ h0 ηk for almost all x ∈ RN . Now, we consider a function ωk ∈ X defined by
0 ηk ωk (x) = 2ηk (γ − |x|) γ
if x ∈ RN \ B(0, γ ) γ if x ∈ B 0, 2
if x ∈ B(0, γ ) \ B 0,
γ 2
(3.8)
.
Without loss of generality, we may assume that ηk ≤ min{γ , 1}. Obviously, ωk (x) ∈ Sk . We have IG (ωk ) ≤ −
γ B(0, 2 ) +
≤
=
+
Z
F (x, ηk )dx +
p−
2p ηk ωγ n
− p− γ p − n p k − p− p 2n
ωγ η γ
p− γ p
−
+
2n
+
1−
−
1
1−
p−
2p ηk ωγ n
ηkp ωγ n p−
1
+
2n
Z −
−
ηkp ωγ n
p− p−
γ B(0, 2 )
−
h0 ηk dx
−
[2p (2n − 1) + γ p 2n − h0 p− γ p ]
−
ωγ n bp + − − ≤ − p−k n [2p (2n − 1) + γ p 2n − h0 p− γ p ]. p γ 2 Hence, since h0 >
+ − 2p (2n −1)+γ p 2n , − γ p p−
this forces
−
ωγ n bpk + − − [2p (2n − 1) + γ p 2n − h0 p− γ p ] < 0. IG (ωk ) ≤ − p− γ p 2n
(3.9)
Now we will prove that limk→+∞ mk = 0. Due to Proposition 3.3, for every x ∈ RN one has 0 > IG (uGk ) ≥ −Φ (uGk ) ≥ −ak
Z
sup |f (x, t )|dx.
(3.10)
RN |t |≤a1
Since the sequence {ak } tends to zero, then mk → 0 as k → +∞.
Proof of Theorem 1.1 concluded. Since uGk are local minima of IG , they are critical points of IG , and thus radially symmetric weak solutions of (1.1). Due to Proposition 3.5, there are infinitely many distinct uGk . Moreover, we have +
kuGk kp p+
Z ≤ mk + RN
F (x,
uGk
(x))dx ≤ ak
which proves that limk→+∞ kuGk kE = 0.
Z
sup f (x, t )dx,
(3.11)
RN |t |≤a1
4. Proof of Theorem 1.2 The proof of Theorem 1.2 is similar to that of Theorem 1.1; consequently, we only outline the proof. We assume that the hypotheses of Theorem 1.2 are fulfilled. Let us fix a number r < 0 arbitrarily, and for every k ∈ N, consider the set Tk = {u ∈ X : r ≤ u(x) ≤ bk a.e.x ∈ RN },
(4.1)
The first part of the proof is similar to that of Theorem 1.1. Indeed, we can prove that the functional IG is bounded from below on Tk and its infimum on Tk is attained (see Proposition 3.2). Moreover, if uGk (x) ∈ Tk is chosen such that IG (uGk ) = infTk IG , then 0 ≤ uGk ≤ ak for all x ∈ RN (see Proposition 3.3), and uGk is a local minimum point of IG in X (see Proposition 3.4). Instead of Proposition 3.5, we prove:
ek = IG (uGk ) = infTk IG . Then limk→+∞ m ek = −∞. Proposition 4.1. Let m p+
Proof. Taking into account condition (F∞ ), we can find a sequence {ηk }k∈N ⊂ [0, bk ] such that F (x, ηk ) ≥ h0 ηk for almost all x ∈ RN . Now, if we consider a function ωk ∈ X defined in (3.8), without loss of generality, we may assume
G. Dai / Nonlinear Analysis 71 (2009) 1133–1139
1139
that ηk ≥ max{γ , 1}. Obviously, ωk (x) ∈ Sk . We have IG (ωk ) ≤ −
+
Z γ B(0, 2 )
F (x, ηk )dx +
p+
2p ηk ωγ n
1−
p− γ p
+
+
2n
≤
=
+ n p k + p− p 2 n
≤
ωγ n bpk + + + [2p (2n − 1) + γ p 2n − h0 p− γ p ]. + p− γ p 2 n
η ωγ n γ
ωγ η γ
1−
1 2n
+
η ωγ n p−
Z −
ηkp ωγ n p− p+
γ
B(0, 2 )
+
+
+
+ p+ 2p k + p− p
p+ k
1
h0 ηk dx
+
[2p (2n − 1) + γ p 2n − h0 p− γ p ]
+
Hence, since h0 > p+
+ + 2p (2n −1)+γ p 2n , + − p γ p +
this forces +
+
lim bk [2p (2n − 1) + γ p 2n − h0 p− γ p ] = −∞.
(4.2)
k→+∞
Proof of Theorem 1.2 concluded. It is clear that we have infinitely many pairwise distinct local minimum points uGk of IG with uGk ∈ Tk . Now we will prove that kuGk kE → +∞ as k → +∞. Arguing by contradiction, assume that there exists a subsequence {uGnk } of {uGk } which is bounded in E. Thus, it is also bounded in e X due to Theorem 2.1. In particular we can find m0 ∈ N such that uGnk ∈ Tm0 for all k ∈ N. For every nk ≥ m0 one has
em0 ≥ m enk = inf IG = IG (uGnk ) ≥ inf IG = m em0 , m Tnk
(4.3)
Tm0
enk = m em0 for all nk ≥ m0 , contradicting Proposition 4.1. This concludes our proof. which proves that m
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