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Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems
J. Math. Anal. Appl. 383 (2011) 239–243
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems ✩ Anmin Mao ∗ , Shixia Luan School of Mathematical Sciences, Qufu Normal University, Shandong 273165, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 12 November 2010 Available online 14 May 2011 Submitted by J. Mawhin
This paper is concerned with nonlocal Kirchhoff’s equation
⎧ ⎪ ⎪ ⎨ − a + b |∇ u |2 u = f (x, u ) in Ω, ⎪ ⎪ ⎩
Keywords: Nonlocal problems Kirchhoff’s equation Variational method Invariant sets of descent flow
Ω
u=0
on ∂Ω,
via variational methods and invariant sets of descent flow. We obtain existence of signed and sign-changing solutions with asymptotically 3-linear bounded nonlinearity. © 2011 Elsevier Inc. All rights reserved.
1. Introduction and main result This paper is concerned with the existence of signed and sign-changing solutions for the following nonlocal Kirchhoff type problems
⎧ ⎪ ⎨ − a + b |∇ u |2 u = f (x, u ) in Ω, ⎪ ⎩
Ω
u=0
(1.1)
on ∂Ω,
where Ω is a smooth bounded domain in R N (N = 1, 2, or 3), a, b > 0. Below by | · |q we denote the usual L q -norm, C or C i stand for different positive constants. For the nonlinearity, we assume:
( f 1 ) f (x, t ) is locally Lipschitz continuous in t ∈ R, uniformly in x ∈ Ω ; ( f 2 ) | f (x, t )| C (|t | p −1 + 1) for some ∗
2< p<2 =
✩
*
2n , n −2
n 3,
+∞, n = 1, 2,
Supported by NSFC 10801088, Q2007A02. Corresponding author. E-mail address: [email protected] (A. Mao).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.05.021
©
2011 Elsevier Inc. All rights reserved.
240
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
and t f (x, t ) 0, which implies that
t F (x, t ) =
f (x, s) ds 0. 0
( f 3 ) f (x, u ) = g ∞ (x)u 3 + r (x, u ), where (i) r (x, u )/u 3 is bounded on Ω × R \ {0}, r (x, u ) = o(u 3 ) as |u | → ∞ uniformly in x, (ii) g ∞ ∈ C (Ω, R) is bounded, m({x ∈ Ω : g ∞ (x) = 0}) > 0 where m(·) is Lebesgue measure, and satisfies − v = g ∞ (x) v 3 , x ∈ Ω , has no nontrivial solution. f (x, u ) is called asymptotically 3-linear bounded provided that ( f 3 ) holds. Nonlocal problems have received growing attention in recent years (see [1–13]), and model several biological systems where u describes a process which depends on the average of itself, for example the population density, see [1]. Eq. (1.1) is related to the stationary analogue of the equation
2
utt − a + b
|∇ u |
u = g (x, u )
(1.2)
Ω
proposed by Kirchhoff [2] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early classical studies of Kirchhoff equations were Bernstein [3] and Pohožaev [4]. However, Eq. (1.2) received much attention only after Lions [5] proposed an abstract framework to the problem. Some interesting results can be found in [6,8,9] and the references therein. We list some recent results in this direction as follows. K. Perera and Z. Zhang [10] obtained the signed and sign-changing solutions of Eq. (1.1) by using the theory of Yang index and critical group if λ, μ is not an eigenvalue of (1.3) and (1.4) below, respectively, i.e.,
−u = λu in Ω, u=0
and
(1.3)
on ∂Ω,
−u 2 u = μu 3 in Ω, u=0
(1.4)
on ∂Ω,
where
lim
t →0
f (x, t ) at
= λ,
lim
|t |→+∞
f (x, t ) bt 3
= μ,
uniformly in x.
(1.5)
In [11], via invariant sets of descent flow, Z. Zhang and K. Perera revisited (1.1) in case of asymptotically 4-linear and obtained the following result. Theorem A (Asymptotically 4-linear). Problem (1.1) has a negative solution, a positive solution and a sign changing solution in the following cases: (i) ( f 1 ) and ( f 2 ) hold, and 1) λ2 is the second eigenvalue of (1.3),
∃λ > λ2 , 2)
F (x, u )
aλ 2
u2 ,
|u | small,
(1.6)
μ1 is the first eigenvalue of (1.4), μ < μ1 , where μ s.t. lim
f (x, t ) bt 3
|t |→+∞
= μ uniformly in x,
(1.7)
(ii) ( f 1 ), ( f 2 ) and (1.7) hold, μ > μ2 is not eigenvalue of (1.4), and
F (x, u )
a λ1 2
u2 ,
|u | small,
where λ1 is the first eigenvalue of (1.3), μ2 is the second eigenvalue of (1.4).
(1.8)
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
241
In [12], via variational methods and invariant sets of descent flow, A. Mao and Z. Zhang obtained sign-changing solution and multiple solutions of Eq. (1.1) in case of 4-superlinear. X. He and W. Zou [13] considered
⎧ ⎪ ⎨ − a + b |∇ u |2 u = λ f (x, u ) in Ω, ⎪ ⎩
Ω
u=0
on ∂Ω,
with f having an oscillation behavior, and obtained a sequence of positive solutions as the local minima of the energy functional. Motivated by [10,11], we intend to search for signed and sign-changing solutions of (1.1) with asymptotically 3-linear bounded nonlinearity. Case of asymptotically linear leads to the nonlinear eigenvalue problem (1.4) (see [11]), whose eigenvalues are the critical values of the functional
4
I (u ) = u ,
u ∈ S := u ∈ H =
H 01 (Ω):
4
|u | = 1 . Ω
As shown in [11], I satisfies the Palais–Smale condition (PS) and that the first eigenvalue μ1 > 0 obtained by minimizing I has an eigenfunction ψ > 0. Furthermore a second eigenvalue μ2 μ1 is defined in [11], so σdisc ( I ) = ∅. Note that Eq. (1.1) has a variational structure. More precisely, solutions of (1.1) are critical points of a C 1 functional Φ . Recall that a sequence (un ) ⊂ H , is said to be a Palais–Smale sequence of Φ provided that Φ(un ) is bounded and Φ (un ) → 0 in H ∗ . The compactness of such sequences strongly depends on the behavior of the nonlinear f . In order to control this asymptotic behaviors of f near zero and infinity, motivated by [14], we introduce the following quantities:
f (x, t )
f ∗ = lim inf inf
|t |→0 x∈Ω
at
f ∗∗ = lim inf inf
|t |→∞ x∈Ω
f ∗ = lim sup sup
,
f (x, t ) bt 3
f (x, t )
|t |→0 x∈Ω
f ∗∗ = lim sup sup
,
|t |→∞ x∈Ω
at
,
f (x, t ) bt 3
.
These quantities make sense when ( f 3 ) is satisfied. It is clear that
0 f ∗ f ∗,
0 f ∗∗ f ∗∗ .
We then assume:
( f 4 ) [ f ∗ , f ∗ ] ∩ [λ1 , +∞) = ∅; ( f 5 ) [ f ∗∗ , f ∗∗ ] ∩ [0, μ2 ] = ∅. We are ready to state our main result. Theorem 1.1. Assume ( f 1 )–( f 5 ) hold. Then problem (1.1) has a positive solutions, a negative solutions, and a sign-changing solution. Remark 1.1. Compared with (1.7) of Theorem A, the nonlinearity f (x, u ) here are more different from that in [10,11]. 2. Variational setting and proof
1
H = H 01 (Ω) is the usual Sobolev space endowed with norm u = ( Ω |∇ u |2 ) 2 . Since Ω is a bounded domain, it is known that H → L s (Ω) continuously for s ∈ [1, 2∗ ], compactly for s ∈ [1, 2∗ ), and there exists γs such that
|u |s γs u ,
∀u ∈ H .
Recall that a function u ∈ H is called a weak solution of (1.1) if
a + bu 2
∇u · ∇ v = Ω
f (x, u ) v ,
∀v ∈ H .
Ω
Weak solutions are the critical points of the C 1 functional
a
b
2
4
Φ(u ) := u 2 + u 4 − Ψ (u ), where
Ψ (u ) :=
F (x, u ), Ω
∀u ∈ H .
∀u ∈ H ,
242
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
Then
Φ (u ) v = a + bu 2
∇u · ∇ v − Ω
f (x, u ) v ,
∀u , v ∈ H .
Ω
They are also classical solutions if f (x, u ) is locally Lipschitz on Ω × R1 . Properties (especially, boundedness and compactness) of Palais–Smale sequences of the associated energy functional plays an essential role. As usual, main difficulty lies in the proof is to prove the compactness of Palais–Smale sequences. Lemma 2.1. Suppose ( f 1 )–( f 3 ) are satisfied. Then Φ satisfies (PS). Proof. Let (un )n be a sequence with (Φ(un ))n is bounded and Φ (un ) → 0 in H ∗ as n → ∞. We first show that (un )n is bounded in H . For this we suppose by contradiction that, passing to subsequence,
un → ∞ as n → ∞. We put v n :=
un u n .
(2.1)
Passing to a subsequence, we may assume that v n v ∈ H and
v n (x) → v (x)
a.e. on Ω.
(2.2)
We define gn : Ω → R by
gn (x) =
f (x,un (x)) , bun3 (x)
un (x) = 0,
(2.3)
un (x) = 0.
0,
Then gn ∈ L ∞ (Ω) and | gn | | g ∞ |∞ + | bu 3 |∞ . Hence, passing to a subsequence, we assume that gn converge in the weak∗ topology to some function g˜ ∈ L ∞ (Ω). Here the weak∗ topology refers to the identification of L ∞ (Ω) with the topological dual of L 1 (Ω). We claim that v n → v in H and v is a weak solution of the equation r (x,u )
− v = g (x) v 3 ,
x ∈ Ω.
Indeed, let w ∈ H , passing to the limit in
(Φ (un ), w ) = (a + bun 2 )un
∇ vn · ∇ w − Ω
gn (x) Ω
v n3 1 + bua 2 n
w
(2.4)
which gives
∇ v n · ∇ w − g˜ v 3 w = 0
(2.5)
Ω
and passing to the limit in (2.4) with w = v n − v shows that v = 1, so v n → v and v satisfies
− v = g˜ v 3 ,
x ∈ Ω.
(2.6)
By the uniqueness of limit, g ∞ = g˜ , this contradicts ( f 3 ). Thus we conclude that (un )n is bounded in H . Passing again to a subsequence, we may assume that un u in H ∗ . Set w n := un − u, then w n 0 and w n → 0 in L s (Ω) for s ∈ [2, 2∗ ). In order to establish strong convergence it suffice to show that w n → 0.
Φ (un ) − Φ (u ), w n +
(x, un ) − f (x, u ) w n = a + bun 2 w n 2 +
Ω
∇ u · w n bun 2 − u 2
Ω
implies that w n → 0. We conclude that un → u in H , and this finishes the proof.
2
Let X = C 01 (Ω) with the norm
u X = max sup D α u (x), 0|α |1 x∈Ω
which is densely embedded in H . By the elliptic regularity theory, the critical point set K := {u ∈ H: Φ (u ) = 0} ⊂ X under the assumption that (x, u ) → f (x, u ) is continuous in Ω × R1 , and locally Lipschitz continuous for u uniformly in x ∈ Ω . Let Φ˜ = Φ| X , we know Φ˜ is a C 2−0 functional defined on X .
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
243
Definition 2.1. (See [15].) We call M is an invariant set of descent flow of Φ˜ if M ⊂ X and {u (t , u 0 ): u 0 ∈ M , t ∈ [0, T (u 0 ))} ⊂ M. Note that for any pseudo gradient vector, X is an invariant set of descent flow of Φ˜ . Let P H := {u ∈ H: u 0 almost everywhere}, then P := P H ∩ X = {u ∈ C 01 (Ω): u 0} is a positive cone in X , which has nonempty interior points P˚ . As shown in [11], P , P˚ are invariant sets of Φ˜ , the negative cone − P and its interior − P˚ are also invariant sets of Φ˜ . Definition 2.2. (See [15].) Assume M ⊂ D are invariant sets of descent flow of Φ˜ . We call
C D ( M ) := u 0 ∈ D: u (t , u 0 ) ∈ M for some t ∈ 0, T (u 0 ) is the invariant set of descent flow expanded by D. Setting
C ( M ) := C X ( M ) = u 0 ∈ X: u (t , u 0 ) ∈ M for some t ∈ [0, T (u 0 )) ⊃ M . We call M is complete if C ( M ) = M (which implies that {u (t , u 0 ): u 0 ∈ X \ M , t ∈ [0, T (u 0 ))} ∩ M = ∅).
˜ M ) = c > −∞ and M is closed in X , then Φ˜ achieves the infimum at a Lemma 2.2. (See [15].) Let M ⊂ X be an invariant set. If inf Φ( critical point in M. Now we give the proof of main result. Proof of Theorem 1.1. It follows from ( f 2 )( f 4 ) that
F (x, u ) For
a λ1 2
u 2 + C |u | p .
(2.7)
μ > μ2 and ε ∈ (0, μ − μ2 ), ( f 2 )–( f 5 ) imply that F (x, u )
b(μ − ε /2) 4
u4 − C .
(2.8)
Hence the (3.6)–(3.9) of [11] are satisfied. Using Lemma 2.2 and proceeding similarly as in the asymptotically case (iii) of Theorem 1.1 of [11], we can obtain a positive solutions, a negative solutions, and a sign-changing solution of Eq. (1.1). 2 References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (7) (1997) 4619–4627. G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. S. Bernstein, Sur une classe d’équations fonctinnelles aux dérivées partielles, Bull. Acad. Sci. URSS Sér. [Izv. Akad. Nauk SSSR] 4 (1940) 17–26. S.I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (138) (1975) 152–166, 168 (in Russian). J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978, pp. 284–346. A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1) (1996) 305–330. C.O. Alves, F.J.S.A. Corra, T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (1) (2005) 85–93. M.M. Cavalcanti, V.N.Domingos Cavalcanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (6) (2001) 701–730. P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (2) (1992) 247–262. K. Perera, Z.T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (1) (2006) 246–255. Z.T. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2) (2006) 456–463. A.M. Mao, Z.T. Zhang, Sign-changing and multiple solutions for a class of Kirchhoff type problems without P.S. condition, Nonlinear Anal. 70 (2009) 1275–1287. X.M. He, W.M. Zou, Imfinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009) 1407–1414. Z.L. Liu, J.B. Su, T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations 231 (2006) 501–512. J.X. Sun, On some problems about nonlinear operators, PhD thesis, Shandong University, Jinan, 1984.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems ✩ Anmin Mao ∗ , Shixia Luan School of Mathematical Sciences, Qufu Normal University, Shandong 273165, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 12 November 2010 Available online 14 May 2011 Submitted by J. Mawhin
This paper is concerned with nonlocal Kirchhoff’s equation
⎧ ⎪ ⎪ ⎨ − a + b |∇ u |2 u = f (x, u ) in Ω, ⎪ ⎪ ⎩
Keywords: Nonlocal problems Kirchhoff’s equation Variational method Invariant sets of descent flow
Ω
u=0
on ∂Ω,
via variational methods and invariant sets of descent flow. We obtain existence of signed and sign-changing solutions with asymptotically 3-linear bounded nonlinearity. © 2011 Elsevier Inc. All rights reserved.
1. Introduction and main result This paper is concerned with the existence of signed and sign-changing solutions for the following nonlocal Kirchhoff type problems
⎧ ⎪ ⎨ − a + b |∇ u |2 u = f (x, u ) in Ω, ⎪ ⎩
Ω
u=0
(1.1)
on ∂Ω,
where Ω is a smooth bounded domain in R N (N = 1, 2, or 3), a, b > 0. Below by | · |q we denote the usual L q -norm, C or C i stand for different positive constants. For the nonlinearity, we assume:
( f 1 ) f (x, t ) is locally Lipschitz continuous in t ∈ R, uniformly in x ∈ Ω ; ( f 2 ) | f (x, t )| C (|t | p −1 + 1) for some ∗
2< p<2 =
✩
*
2n , n −2
n 3,
+∞, n = 1, 2,
Supported by NSFC 10801088, Q2007A02. Corresponding author. E-mail address: [email protected] (A. Mao).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.05.021
©
2011 Elsevier Inc. All rights reserved.
240
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
and t f (x, t ) 0, which implies that
t F (x, t ) =
f (x, s) ds 0. 0
( f 3 ) f (x, u ) = g ∞ (x)u 3 + r (x, u ), where (i) r (x, u )/u 3 is bounded on Ω × R \ {0}, r (x, u ) = o(u 3 ) as |u | → ∞ uniformly in x, (ii) g ∞ ∈ C (Ω, R) is bounded, m({x ∈ Ω : g ∞ (x) = 0}) > 0 where m(·) is Lebesgue measure, and satisfies − v = g ∞ (x) v 3 , x ∈ Ω , has no nontrivial solution. f (x, u ) is called asymptotically 3-linear bounded provided that ( f 3 ) holds. Nonlocal problems have received growing attention in recent years (see [1–13]), and model several biological systems where u describes a process which depends on the average of itself, for example the population density, see [1]. Eq. (1.1) is related to the stationary analogue of the equation
2
utt − a + b
|∇ u |
u = g (x, u )
(1.2)
Ω
proposed by Kirchhoff [2] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early classical studies of Kirchhoff equations were Bernstein [3] and Pohožaev [4]. However, Eq. (1.2) received much attention only after Lions [5] proposed an abstract framework to the problem. Some interesting results can be found in [6,8,9] and the references therein. We list some recent results in this direction as follows. K. Perera and Z. Zhang [10] obtained the signed and sign-changing solutions of Eq. (1.1) by using the theory of Yang index and critical group if λ, μ is not an eigenvalue of (1.3) and (1.4) below, respectively, i.e.,
−u = λu in Ω, u=0
and
(1.3)
on ∂Ω,
−u 2 u = μu 3 in Ω, u=0
(1.4)
on ∂Ω,
where
lim
t →0
f (x, t ) at
= λ,
lim
|t |→+∞
f (x, t ) bt 3
= μ,
uniformly in x.
(1.5)
In [11], via invariant sets of descent flow, Z. Zhang and K. Perera revisited (1.1) in case of asymptotically 4-linear and obtained the following result. Theorem A (Asymptotically 4-linear). Problem (1.1) has a negative solution, a positive solution and a sign changing solution in the following cases: (i) ( f 1 ) and ( f 2 ) hold, and 1) λ2 is the second eigenvalue of (1.3),
∃λ > λ2 , 2)
F (x, u )
aλ 2
u2 ,
|u | small,
(1.6)
μ1 is the first eigenvalue of (1.4), μ < μ1 , where μ s.t. lim
f (x, t ) bt 3
|t |→+∞
= μ uniformly in x,
(1.7)
(ii) ( f 1 ), ( f 2 ) and (1.7) hold, μ > μ2 is not eigenvalue of (1.4), and
F (x, u )
a λ1 2
u2 ,
|u | small,
where λ1 is the first eigenvalue of (1.3), μ2 is the second eigenvalue of (1.4).
(1.8)
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
241
In [12], via variational methods and invariant sets of descent flow, A. Mao and Z. Zhang obtained sign-changing solution and multiple solutions of Eq. (1.1) in case of 4-superlinear. X. He and W. Zou [13] considered
⎧ ⎪ ⎨ − a + b |∇ u |2 u = λ f (x, u ) in Ω, ⎪ ⎩
Ω
u=0
on ∂Ω,
with f having an oscillation behavior, and obtained a sequence of positive solutions as the local minima of the energy functional. Motivated by [10,11], we intend to search for signed and sign-changing solutions of (1.1) with asymptotically 3-linear bounded nonlinearity. Case of asymptotically linear leads to the nonlinear eigenvalue problem (1.4) (see [11]), whose eigenvalues are the critical values of the functional
4
I (u ) = u ,
u ∈ S := u ∈ H =
H 01 (Ω):
4
|u | = 1 . Ω
As shown in [11], I satisfies the Palais–Smale condition (PS) and that the first eigenvalue μ1 > 0 obtained by minimizing I has an eigenfunction ψ > 0. Furthermore a second eigenvalue μ2 μ1 is defined in [11], so σdisc ( I ) = ∅. Note that Eq. (1.1) has a variational structure. More precisely, solutions of (1.1) are critical points of a C 1 functional Φ . Recall that a sequence (un ) ⊂ H , is said to be a Palais–Smale sequence of Φ provided that Φ(un ) is bounded and Φ (un ) → 0 in H ∗ . The compactness of such sequences strongly depends on the behavior of the nonlinear f . In order to control this asymptotic behaviors of f near zero and infinity, motivated by [14], we introduce the following quantities:
f (x, t )
f ∗ = lim inf inf
|t |→0 x∈Ω
at
f ∗∗ = lim inf inf
|t |→∞ x∈Ω
f ∗ = lim sup sup
,
f (x, t ) bt 3
f (x, t )
|t |→0 x∈Ω
f ∗∗ = lim sup sup
,
|t |→∞ x∈Ω
at
,
f (x, t ) bt 3
.
These quantities make sense when ( f 3 ) is satisfied. It is clear that
0 f ∗ f ∗,
0 f ∗∗ f ∗∗ .
We then assume:
( f 4 ) [ f ∗ , f ∗ ] ∩ [λ1 , +∞) = ∅; ( f 5 ) [ f ∗∗ , f ∗∗ ] ∩ [0, μ2 ] = ∅. We are ready to state our main result. Theorem 1.1. Assume ( f 1 )–( f 5 ) hold. Then problem (1.1) has a positive solutions, a negative solutions, and a sign-changing solution. Remark 1.1. Compared with (1.7) of Theorem A, the nonlinearity f (x, u ) here are more different from that in [10,11]. 2. Variational setting and proof
1
H = H 01 (Ω) is the usual Sobolev space endowed with norm u = ( Ω |∇ u |2 ) 2 . Since Ω is a bounded domain, it is known that H → L s (Ω) continuously for s ∈ [1, 2∗ ], compactly for s ∈ [1, 2∗ ), and there exists γs such that
|u |s γs u ,
∀u ∈ H .
Recall that a function u ∈ H is called a weak solution of (1.1) if
a + bu 2
∇u · ∇ v = Ω
f (x, u ) v ,
∀v ∈ H .
Ω
Weak solutions are the critical points of the C 1 functional
a
b
2
4
Φ(u ) := u 2 + u 4 − Ψ (u ), where
Ψ (u ) :=
F (x, u ), Ω
∀u ∈ H .
∀u ∈ H ,
242
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
Then
Φ (u ) v = a + bu 2
∇u · ∇ v − Ω
f (x, u ) v ,
∀u , v ∈ H .
Ω
They are also classical solutions if f (x, u ) is locally Lipschitz on Ω × R1 . Properties (especially, boundedness and compactness) of Palais–Smale sequences of the associated energy functional plays an essential role. As usual, main difficulty lies in the proof is to prove the compactness of Palais–Smale sequences. Lemma 2.1. Suppose ( f 1 )–( f 3 ) are satisfied. Then Φ satisfies (PS). Proof. Let (un )n be a sequence with (Φ(un ))n is bounded and Φ (un ) → 0 in H ∗ as n → ∞. We first show that (un )n is bounded in H . For this we suppose by contradiction that, passing to subsequence,
un → ∞ as n → ∞. We put v n :=
un u n .
(2.1)
Passing to a subsequence, we may assume that v n v ∈ H and
v n (x) → v (x)
a.e. on Ω.
(2.2)
We define gn : Ω → R by
gn (x) =
f (x,un (x)) , bun3 (x)
un (x) = 0,
(2.3)
un (x) = 0.
0,
Then gn ∈ L ∞ (Ω) and | gn | | g ∞ |∞ + | bu 3 |∞ . Hence, passing to a subsequence, we assume that gn converge in the weak∗ topology to some function g˜ ∈ L ∞ (Ω). Here the weak∗ topology refers to the identification of L ∞ (Ω) with the topological dual of L 1 (Ω). We claim that v n → v in H and v is a weak solution of the equation r (x,u )
− v = g (x) v 3 ,
x ∈ Ω.
Indeed, let w ∈ H , passing to the limit in
(Φ (un ), w ) = (a + bun 2 )un
∇ vn · ∇ w − Ω
gn (x) Ω
v n3 1 + bua 2 n
w
(2.4)
which gives
∇ v n · ∇ w − g˜ v 3 w = 0
(2.5)
Ω
and passing to the limit in (2.4) with w = v n − v shows that v = 1, so v n → v and v satisfies
− v = g˜ v 3 ,
x ∈ Ω.
(2.6)
By the uniqueness of limit, g ∞ = g˜ , this contradicts ( f 3 ). Thus we conclude that (un )n is bounded in H . Passing again to a subsequence, we may assume that un u in H ∗ . Set w n := un − u, then w n 0 and w n → 0 in L s (Ω) for s ∈ [2, 2∗ ). In order to establish strong convergence it suffice to show that w n → 0.
Φ (un ) − Φ (u ), w n +
(x, un ) − f (x, u ) w n = a + bun 2 w n 2 +
Ω
∇ u · w n bun 2 − u 2
Ω
implies that w n → 0. We conclude that un → u in H , and this finishes the proof.
2
Let X = C 01 (Ω) with the norm
u X = max sup D α u (x), 0|α |1 x∈Ω
which is densely embedded in H . By the elliptic regularity theory, the critical point set K := {u ∈ H: Φ (u ) = 0} ⊂ X under the assumption that (x, u ) → f (x, u ) is continuous in Ω × R1 , and locally Lipschitz continuous for u uniformly in x ∈ Ω . Let Φ˜ = Φ| X , we know Φ˜ is a C 2−0 functional defined on X .
A. Mao, S. Luan / J. Math. Anal. Appl. 383 (2011) 239–243
243
Definition 2.1. (See [15].) We call M is an invariant set of descent flow of Φ˜ if M ⊂ X and {u (t , u 0 ): u 0 ∈ M , t ∈ [0, T (u 0 ))} ⊂ M. Note that for any pseudo gradient vector, X is an invariant set of descent flow of Φ˜ . Let P H := {u ∈ H: u 0 almost everywhere}, then P := P H ∩ X = {u ∈ C 01 (Ω): u 0} is a positive cone in X , which has nonempty interior points P˚ . As shown in [11], P , P˚ are invariant sets of Φ˜ , the negative cone − P and its interior − P˚ are also invariant sets of Φ˜ . Definition 2.2. (See [15].) Assume M ⊂ D are invariant sets of descent flow of Φ˜ . We call
C D ( M ) := u 0 ∈ D: u (t , u 0 ) ∈ M for some t ∈ 0, T (u 0 ) is the invariant set of descent flow expanded by D. Setting
C ( M ) := C X ( M ) = u 0 ∈ X: u (t , u 0 ) ∈ M for some t ∈ [0, T (u 0 )) ⊃ M . We call M is complete if C ( M ) = M (which implies that {u (t , u 0 ): u 0 ∈ X \ M , t ∈ [0, T (u 0 ))} ∩ M = ∅).
˜ M ) = c > −∞ and M is closed in X , then Φ˜ achieves the infimum at a Lemma 2.2. (See [15].) Let M ⊂ X be an invariant set. If inf Φ( critical point in M. Now we give the proof of main result. Proof of Theorem 1.1. It follows from ( f 2 )( f 4 ) that
F (x, u ) For
a λ1 2
u 2 + C |u | p .
(2.7)
μ > μ2 and ε ∈ (0, μ − μ2 ), ( f 2 )–( f 5 ) imply that F (x, u )
b(μ − ε /2) 4
u4 − C .
(2.8)
Hence the (3.6)–(3.9) of [11] are satisfied. Using Lemma 2.2 and proceeding similarly as in the asymptotically case (iii) of Theorem 1.1 of [11], we can obtain a positive solutions, a negative solutions, and a sign-changing solution of Eq. (1.1). 2 References [1] [2] [3] [4] [5]
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