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Infinitely many positive solutions for Kirchhoff-type problems
Nonlinear Analysis 70 (2009) 1407–1414 www.elsevier.com/locate/na
Infinitely many positive solutions for Kirchhoff-type problemsI Xiaoming He a,b,∗ , Wenming Zou b a College of Science, Central University for Nationalities, Beijing 100081, PR China b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China
Received 31 December 2006; accepted 14 February 2008
Abstract This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem R −(a + b Ω |∇u|2 dx)∆u = λ f (x, u) in Ω and u = 0 on ∂Ω , where Ω is a smooth bounded domain of R N , a, b > 0, λ > 0 and f : Ω × R → R is a Carath´eodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in L ∞ (Ω ) with f being more general than that of [K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255; Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463]. c 2008 Elsevier Ltd. All rights reserved.
MSC: 35J60; 35J25 Keywords: Nonlocal problems; Kirchhoff’s equation; Positive solutions; Local minimum; Variational methods
1. Introduction In this paper we consider the problem Z −(a + b |∇u|2 dx)∆u = λ f (x, u) in Ω , Ω u = 0 on ∂Ω ,
(Pλ )
where Ω is a smooth bounded domain in R N , a, b > 0 and λ > 0, f (x, u) : Ω × R → R is a Carath´eodory function and there exists t ∗ > 0 such that sup f (·, t) ∈ L ∞ (Ω ).
(1.1)
t∈[0,t ∗ ]
Recently, Perera and Zhang [1] obtained a nontrivial solution of (P1 ) by using the theory of Yang index and critical group if λ, µ is not an eigenvalue of (1.2) and (1.3) below, respectively, i.e., −∆u = λu in Ω , (1.2) u=0 on ∂Ω , I Supported by NSFC 10571096, 10001019, SRF for ROCS, SEM and China Postdoctoral Science Foundation. ∗ Corresponding author at: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China.
E-mail addresses: [email protected] (X. He), [email protected] (W. Zou). c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.02.021
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
and
−kuk2 ∆u = µu 3 u=0
in Ω , on ∂Ω ,
(1.3)
where f (x, t) satisfies lim
t→0
f (x, t) = λ, at
lim
|t|→+∞
f (x, t) =µ bt 3
uniformly in x.
(1.4)
Almost at the same time, Zhang and Perera [2] revisited (P1 ) via invariant sets of descent flow and obtained the following existence result (see Theorem 1.1 [2]). Theorem A. Problem (P1 ) has a positive solution, a non-negative solution and a sign changing solution in the following cases: (H1) f (x, t) is locally Lipschitz continuous in t, uniformly in Ω , and subcritical: | f (x, t)| ≤ C(|t| p−1 + 1) for some 2N , N ≥ 3, p < 4 and 2 < p < 2∗ = N − 2 ∞, N = 1, 2, aλ 2 t , |t| small. 2 Here C is a positive constant and λ2 is the second eigenvalue of (1.2). (H2) f (x, t) is asymptotically 4-linear: ∃λ > λ2 ;
lim
|t|→+∞
F(x, t) ≥
f (x, t) =µ bt 3
uniformly in x,
(1.5)
(1.6)
µ < µ1 , and (1.5) holds, where µ1 is the first eigenvalue of (1.3). (H3) (1.6) holds, µ > µ2 is not an eigenvalue of (1.3), and aλ1 2 t , |t| small, 2 Rt where F(x, t) = 0 f (x, s)ds. (H4) f (x, t) is 4-superlinear:
(1.7)
F(x, t) ≤
∃ν > 4 : ν F(x, t) ≤ t f (x, t),
for |t| large,
(1.8)
and (1.7) holds. Problem (Pλ ) is related to the stationary analogue of the equation Z 2 u tt − a + b |∇u| dx ∆u = g(x, u)
(1.9)
Ω
proposed by Kirchhoff [11] as an existence 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, we refer to [7,12] for early classical studies. Eq. (1.9) received much attention only after Lions [10] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [6,8,9]. More recently, Alves et al. [4] and Ma and Rivera [3] obtained positive solutions of such problems by variational methods. Similar nonlocal problems also model several physical and biological systems where u describes a process which depends on the average of itself, for example the population density, see [5,13,14]. The existence of infinitely many solutions for problem (Pλ ) with a = 1 and b = 0 has been widely investigated. The most classical results on this topic are essentially based on Ljusternik–Schnirelman theory. In them, the crucial role is played by the oddness of the nonlinearity. Moreover, in order to check the Palais–Smale condition or some of its variant, one has to impose certain conditions that do not satisfy an oscillating behavior of the nonlinearity, we
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
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refer to [2,20–24] as typical papers of this kind. In [15,16], Omari and Zanolin proved the existence of an unbounded sequence of weak solutions for the p-Laplacian problem −∆ p u = λ f (u) in Ω , (1.10) u=0 on ∂Ω under the following conditions F(t) F(t) = 0 and lim sup p = +∞, (1.11) tp t t→0+ Rt where F(u) = 0 f (s)ds. They showed problem (1.10) has a sequence {u n } of non-zero nonnegative weak solutions, satisfying maxΩ u n → 0 as n → +∞. Annello and Cordaro [19] recently weakened the condition (1.11) and obtained infinitely many positive solutions of (1.10). The main ideal of [19] is based on the general approach proposed in [17], which yields weak solutions by searching for local minima of the underlying energy functional. This technique was suggested earlier in the papers of Raymond [18] and Ricceri [17]. Motivated by the papers mentioned above, in this paper we intend to search for positive solutions for problem (Pλ ) as the local minima of the energy functional corresponding to (Pλ ). We consider a class of nonlinearity f (x, ·) different from [1,2], and prove that there exists a sequence of a.e. positive weak solutions of (Pλ ) converging to zero in L ∞ (Ω ). If u is a weak solution of (Pλ ), we say that u is a.e. positive if m({x ∈ Ω : u(x) ≤ 0}) = 0, where m(·) is the Lebesgue measure. To the best of our knowledge, little is known about the multiplicity results for problem (Pλ ) with f (x, ·) having an oscillating behavior. Our result improves the main results of [1,2]. The paper is divided into three sections, including this introduction. In Section 2, we state and prove our main result. In Section 3, we give an example to illustrate our result. lim inf
t→0+
2. Main result R 1 Let H = H01 (Ω ) be the usual Sobolev space endowed with norm kuk = ( Ω |∇u|2 ) 2 . We recall that a function u ∈ H is called a weak solution of (Pλ ) if Z Z 2 f (x, u)vdx, ∀v ∈ H. (2.1) ∇u∇vdx = λ (a + bkuk ) Ω
Ω
Weak solutions are the critical points of the C 1 functional Z b a F(x, u)dx, u ∈ H. I (u) = kuk2 + kuk4 − λ 2 4 Ω
(2.2)
They are also classic solutions if f is locally Lipschitz on Ω × R. Theorem 2.1. Suppose that the function f satisfies the following conditions. (i) For every n ∈ N, there exist ξn , ξn0 ∈ R with 0 ≤ ξn < ξn0 and limn→+∞ ξn0 = 0 such that for a.e. x ∈ Ω , Z ξn Z t f (x, s)ds = sup f (x, s)ds. t∈[ξn ,ξn0 ] 0
0
(ii) There exists a non-empty open set S ⊆ Ω , a constant M ≥ 0 and a sequence {tn }n∈N ⊂ R+ \ {0} with limn→+∞ tn = 0, such that Rt ess inf 0n f (x, s)ds x∈S lim = +∞ n→+∞ tn2 and Z tn Z t ess inf inf f (x, s)ds ≥ −Mess inf f (x, s)ds . x∈S
t∈[0,tn ] 0
x∈S
0
Then, for every λ > 0, problem (Pλ ) has a sequence {u n } of a.e. positive weak solutions strongly convergent to zero and such that limn→+∞ maxΩ¯ u n = 0.
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Proof. By (1.1), there exist a > 0 and t ∗ > 0 such that | f (x, t)| ≤ a, for every 0 ≤ t ≤ t ∗ and a.e. x ∈ Ω . Moreover, conditions (i) and (ii) imply that, f (x, 0) = 0 for a.e. x ∈ Ω . Without loss of generality, we suppose that, for every n ∈ N, max{ξn0 , tn } ≤ t ∗ . Let λ > 0, and define f (x, t ∗ ), if t > t ∗ ; g(x, t) = f (x, t), if 0 ≤ t ≤ t ∗ ; 0, if t < 0. Hence, we have (2.3)
|g(x, t)| ≤ d for a.e. x ∈ Ω and t ∈ R. Now we consider the following problem Z |∇u|2 dx)∆u = λg(x, u) in Ω , −(a + b Ω u = 0 on ∂Ω .
(2.4)
The weak solutions of (2.4) are the critical points of the functional Z a b 2 4 Iµ (u) = µ kuk + kuk − G(x, u)dx, u ∈ H, (2.5) 2 4 Ω Ru where G(x, u) = 0 g(x, t)dt, µ = 1/λ. By (2.3), it is easy to see that Iµ is well defined, weakly sequentially lower semicontinuous and Gˆateaux differentiable in H . For a fixed n ∈ N, the subspace E n = {u ∈ H : 0 ≤ u n (x) ≤ ξn0 , a.e. in Ω }. is a closed and convex, and it is also weakly closed. Moreover, for each u ∈ E n , one has Iµ (u) ≥ −dm(Ω )ξn0 . Hence Iµ is lower bounded in E n , so we can set αn = inf E n Iµ . For ∀k ∈ N, there exist vk ∈ E n such that αn ≤ Iµ (vk ) ≤ αn + k1 , then it follows that Z a b 1 2 4 G(x, vk )dx + Iµ (vk ) kvk k + kvk k = 2 4 µ Ω Z 1 1 ≤ dvk dx + αn + µ Ω k 1 ≤ (dm(Ω )ξn0 + αn + 1), µ which implies that {vk } is norm bounded in H . Therefore, there exists a subsequence {vkm }, weakly convergent to u n ∈ E n , being E n weakly closed. At this point, we obtain that Iµ (u n ) = αn in view of the weakly sequentially lower semi-continuity of Iµ . Now we claim that u n (x) ∈ (0, ξn ] a.e. in Ω . In fact, set ξn , if t > ξn ; if 0 < t ≤ ξn ; τ (t) = t, 0, if t ≤ 0. Define the map T : H → H as follows T u := τ (u)
for ∀u ∈ H.
The operator T is continuous in H . Furthermore, for ∀u ∈ H, T u ∈ E n . Set v ∗ = T u n and X = {x ∈ Ω : u n 6∈ (0, ξn ]}. Then for a.e., x ∈ X , we have ξn < u n (x) ≤ ξn0 ,
or
u n (x) = 0.
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
But by (i) we have Z u n (x) Z g(x, t)dt ≤ and
0 |∇v ∗ |
1411
v ∗ (x)
g(x, t)dt
0
= 0. Therefore
! Z Z v∗ a b Iµ (v ∗ ) − Iµ (u n ) = µ kv ∗ k2 + kv ∗ k4 − g(x, t)dt dx 2 4 Ω 0 Z Z u n a b −µ ku n k2 + ku n k4 + g(x, t)dt dx 2 4 Ω 0 ! 2 ! Z Z Z v ∗ (x) Z a b 2 2 |∇u n | dx − g(x, t)dt dx = −µ |∇u n | dx + 2 X 4 X X u n (x) 2 ! Z Z b a 2 2 . |∇u n | dx + |∇u n | dx ≤ −µ 2 X 4 X R R Since v ∗ ∈ E n , it follows that Iµ (v ∗ ) − Iµ (u n ) ≥ 0. Then X |∇u n |2 dx = 0. Hence kv ∗ − u n k2 = Ω |∇v ∗ − ∇u n |2 = R 2 ∗ X |∇u n | dx = 0 which means that u n (x) = v (x) ∈ (0, ξn ] a.e. in Ω . Let u ∈ H, and T be the operator defined above. Set
X 0 = {x ∈ Ω : u(x) 6∈ (0, ξn ]}. We have that if x ∈ Ω \ X 0 , then Z u(x) g(x, t)dt = 0. T u(x)
Furthermore, if x ∈ X 0 , then one has the following three cases. (C1 ) If u(x) ≤ 0, then Z u(x) Z g(x, t)dt =
u(x)
g(x, t)dt = 0.
0
T u(x)
(C2 ) If ξn < u(x) ≤ ξn0 , then by (i), Z u(x) g(x, t)dt ≤ 0. T u(x)
(C3 ) If u(x) > ξn0 , then Z u(x) Z g(x, t)dt =
u(x)
ξn
T u(x)
Z g(x, t)dt ≤
u(x) ξn
ddt = d(u(x) − ξn ).
Let q > 1 and fix it, then the constant A = sup
ξ ≥ξn0
d(ξ − ξn ) (ξ − ξn )q+1
is finite, we have, for a.e. x ∈ Ω , Z u(x) g(x, t)dt ≤ A|u(x) − T u(x)|q+1 . T u(x)
Then Z Ω
Z
u(x) T u(x)
! g(x, t)dt dx ≤ AL q+1 ku − T ukq+1 ,
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
where R sup
L=
u∈H \{0}
Ω
|u|q+1 dx kuk
1 q+1
.
Therefore, ! Z Z u(x) a b a b 2 4 2 4 Iµ (u) − Iµ (T u) = µ kuk + kuk − µ kT uk + kT uk − g(x, t)dt dx 2 4 2 4 Ω T u(x) ! Z 2 Z Z Z u(x) aµ bµ = |∇u|2 dx − |∇u|2 dx + g(x, t)dt dx 2 X 4 X Ω T u(x) ! Z 2 Z Z Z u(x) aµ bµ 2 2 = |∇u − ∇(T u)| dx − |∇u − ∇(T u)| dx + g(x, t)dt dx 2 Ω 4 Ω Ω T u(x)
bµ aµ ku − T uk2 + ku − T uk4 − AL q+1 ku − T ukq+1 . 2 4 Since T u ∈ E n , we have Iµ (T u) ≥ I (u n ). Then a bµ Iµ (u) ≥ Iµ (u n ) + ku − T uk2 µ+ ku − T uk2 − AL q+1 ku − T ukq−1 2 4 aµ − AL q+1 ku − T ukq−1 . ≥ Iµ (u n ) + ku − T uk2 2 Since T is continuous and q > 1, there exists β > 0 such that, for every u ∈ H with aµ ku − u n k < β, ku − T ukq−1 ≤ . 4AL q+1 Then if ku − u n k < β, we have aµ Iµ (u) ≥ Iµ (u n ) + ku − T uk2 ≥ Iµ (u n ), 4 this implies that u n is a local minimum of Iµ . For every n ∈ N and u ∈ E n , we have that ≥
Iµ (u) ≥ −dm(Ω )ξn0 . Then, since −dm(Ω )ξn0 ≤ αn ≤ 0, it is easy to see that lim αn = 0.
n→+∞
From u n ∈ E n with αn = Iµ (u n ), it follows that ! ! Z Z u n (x) a b 1 2 4 ku n k + ku n k = g(x, t)dt dx + Iµ (u n ) 2 4 µ Ω 0 ! ! Z Z u n (x) 1 = g(x, t)dt dx + αn µ Ω 0 ≤
1 (dm(Ω )ξn0 + αn ). µ
Therefore, limn→+∞ ku n k2 = 0. Now we prove that αn < 0, for every n ∈ N. For this, fix n ∈ N, and choose a compact set K ⊂ S with m(K ) = (M + 1)m(S \ K ) and a function v ∈ H such that if x ∈ K ; 1, v(x) = 0 ≤ v(x) ≤ 1, if x ∈ S \ K ; 0, if x ∈ Ω \ S.
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
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By the former condition of (ii), there exist k˜ ∈ N such that tk ≤ ξn0 and some positive constant C such that ! Z tk (M + 1) atk2 b 4 4 2 2 ess inf g(x, t)dt > Ctk ≥ µ kvk + tk kvk x∈S 0 m(K ) 2 4 ˜ Next, taking into account the latter condition of (ii), for every k ≥ k, ˜ we have for every k ∈ N with k ≥ k. R R t v(x) R Rt R R t v(x) − Ω 0k g(x, t)dt dx − K ( 0k g(x, t)dt)dx − S\K ( 0k g(x, t)dt)dx = 4 2 4 2 2 btk R btk atk R atk 2 2 2 4 Ω |∇v| dx 2 Ω |∇v| dx + 4 2 kvk + 4 kvk Rt R R R tk inf inf 0 g(x, t)dt)dx − K (ess inf 0 g(x, t)dt)dx S\K (ess x∈S t∈[0,tk ] x∈S − ≤ btk4 btk4 atk2 atk2 2 4 2 4 2 kvk + 4 kvk 2 kvk + 4 kvk R R tk R R tk − K (ess inf 0 g(x, t)dt)dx M S\K (ess inf 0 g(x, t)dt)dx x∈S x∈S ≤ + 4 2 atk2 bt at btk4 k k 2 4 2 4 2 kvk + 4 kvk 2 kvk + 4 kvk R tk −(1/(M + 1))m(K )ess inf 0 g(x, t)dt x∈S = 4 atk2 bt k 2 4 2 kvk + 4 kvk < −µ. Whence tk v ∈ E n and Iµ (tk v) < 0, which means αn < 0. Hence, there exists a subsequence of u n of pairwise distinct elements. Such a subsequence is a sequence of weak solutions of (Pλ ). Moreover, we have 0 = ess inf u n (x) < ess sup u n (x) ≤ t ∗ x∈Ω
x∈Ω
for every n ∈ N. Then it is a sequence of weak solutions for (Pλ ).
3. An example In this section we present an example of an application of Theorem 2.1, when neither of conditions (H1)–(H4) is satisfied. Example 3.1. Let us consider problem (Pλ ) with a, b > 0 constants and λ > 0 any parameter. The nonlinearity f : Ω × R → R is defined by ( 2 1 1 1 if t > 0, (1 + |x|2 ) 15t 3 sin(1/t 3 ) − 3t 3 cos(1/t 3 ) f (x, t) = 0 if t ≤ 0, 1
where x = (x1 , x2 , . . . , xn ) ∈ Ω , |x| = (x12 + x22 + · · · + xn2 ) 2 . Thus we have, for every x ∈ Ω , ( 5 1 2 3 3 F(x, t) = (1 + |x| )9t sin(1/t ) if t > 0, 0 if t ≤ 0. Taking ξn = 0, ξn0 = (π/2 + 2nπ )−3 , it is easy to see that condition (i) is satisfied. To check that condition (ii) holds, let 8 tn = 3 π (1 + 4n)3 for each n ∈ N. Then F(x, tn ) 9 = (1 + |x|2 ) lim √ = +∞ uniformly in x. lim 2 n→+∞ n→+∞ tn tn
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
Moreover, we have inf F(x, t) ≥ −F(x, tn )
t∈[0,tn ]
for every x ∈ Ω .
Then the latter condition of (ii) is also satisfied for M = 1. Hence Theorem 2.1 ensures that there exists a sequence of a.e. positive and pairwise distinct solutions to problem (Pλ ) (with a, b > 0, arbitrarily constants and λ > 0) that tend to zero. Remark 3.1. We note that in Example 3.1, the nonlinearity f (x, t) does not satisfy (1.4), and lim inf
t→0+
F(x, t) = −∞ uniformly in x. t2
Also, by a simple computation, it is easy to see that neither of conditions (H1)–(H4) in Theorem A is satisfied in Example 3.1. So the results of [1,2] are invalid for Example 3.1. Acknowledgement The authors thank the referee for his/her carefully reading the manuscript and valuable comments and suggestions. References [1] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255. [2] Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463. [3] T.F. Ma, J.E. Mu˜noz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003) 243–248. [4] C.O. Alves, F.J.S.A. Correa, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. [5] D. Andrade, T.F. Ma, An operator equation suggested by a class of nonlinear stationary problems, Commun. Appl. Nonl. Anal. 4 (1997) 65–71. [6] A. Arosio, S. Pannizi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305–330. [7] S. Bernstein, Sur une classe d’´equations fonctionnelles aux d´eriv´ees partielles, Bull. Acad. Sci. URRS. S´er. Math. [Izv. Akad. Nauk SSSR] 4 (1940) 17–26. [8] M.M. Cavalcanti, V.N. Cavacanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001) 701–730. [9] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992) 247–262. [10] J.-L. Lions, On some quations 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: NorthHolland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346. [11] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [12] S.I. Pohoˇzaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (1975) 152–168. [13] M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997) 4619–4627. [14] M. Chipot, J.-F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Mod´el. Math. Anal. Num´er. 26 (1992) 447–467. [15] P. Omari, F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations 21 (1996) 721–733. [16] P. Omari, F. Zanolin, An elliptic problem with arbitrarily small positive solution, Electron, J. Diff. Conf. 5 (2000) 301–308. [17] B. Ricceri, A genaral variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [18] J. Saint Raymond, On the multiplicity of the solutions of the equations −∆u = λ f (u), J. Differential Equations 180 (2002) 65–88. [19] G. Anello, G. Cordaro, Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian, Proc. Roy. Soc. Edinburgh 132 (A) (2002) 511–519. [20] T. Bartsch, Z. Liu, T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc. (3) 91 (2005) 129–152. [21] T. Bartsch, Z. Liu, On superlinear elliptic p-Laplacian eqaution, J. Differential Equations 198 (2004) 149–175. [22] M. Schechter, W. Zou, Superlinear problem, Pacific J. Math. 214 (2004) 145–160. [23] M. Schechter, W. Zou, Infinitely many sign-changing solutions for perturbed elliptic equations with Hardy potentials, J. Funct. Anal. 228 (2005) 1–38. [24] M. Schechter, Z.-Q. Wang, W. Zou, New liking theorem and sign-changing solutions, Comm. Partial Differential Equations 29 (2004) 471–488.
Infinitely many positive solutions for Kirchhoff-type problemsI Xiaoming He a,b,∗ , Wenming Zou b a College of Science, Central University for Nationalities, Beijing 100081, PR China b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China
Received 31 December 2006; accepted 14 February 2008
Abstract This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem R −(a + b Ω |∇u|2 dx)∆u = λ f (x, u) in Ω and u = 0 on ∂Ω , where Ω is a smooth bounded domain of R N , a, b > 0, λ > 0 and f : Ω × R → R is a Carath´eodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in L ∞ (Ω ) with f being more general than that of [K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255; Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463]. c 2008 Elsevier Ltd. All rights reserved.
MSC: 35J60; 35J25 Keywords: Nonlocal problems; Kirchhoff’s equation; Positive solutions; Local minimum; Variational methods
1. Introduction In this paper we consider the problem Z −(a + b |∇u|2 dx)∆u = λ f (x, u) in Ω , Ω u = 0 on ∂Ω ,
(Pλ )
where Ω is a smooth bounded domain in R N , a, b > 0 and λ > 0, f (x, u) : Ω × R → R is a Carath´eodory function and there exists t ∗ > 0 such that sup f (·, t) ∈ L ∞ (Ω ).
(1.1)
t∈[0,t ∗ ]
Recently, Perera and Zhang [1] obtained a nontrivial solution of (P1 ) by using the theory of Yang index and critical group if λ, µ is not an eigenvalue of (1.2) and (1.3) below, respectively, i.e., −∆u = λu in Ω , (1.2) u=0 on ∂Ω , I Supported by NSFC 10571096, 10001019, SRF for ROCS, SEM and China Postdoctoral Science Foundation. ∗ Corresponding author at: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China.
E-mail addresses: [email protected] (X. He), [email protected] (W. Zou). c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.02.021
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
and
−kuk2 ∆u = µu 3 u=0
in Ω , on ∂Ω ,
(1.3)
where f (x, t) satisfies lim
t→0
f (x, t) = λ, at
lim
|t|→+∞
f (x, t) =µ bt 3
uniformly in x.
(1.4)
Almost at the same time, Zhang and Perera [2] revisited (P1 ) via invariant sets of descent flow and obtained the following existence result (see Theorem 1.1 [2]). Theorem A. Problem (P1 ) has a positive solution, a non-negative solution and a sign changing solution in the following cases: (H1) f (x, t) is locally Lipschitz continuous in t, uniformly in Ω , and subcritical: | f (x, t)| ≤ C(|t| p−1 + 1) for some 2N , N ≥ 3, p < 4 and 2 < p < 2∗ = N − 2 ∞, N = 1, 2, aλ 2 t , |t| small. 2 Here C is a positive constant and λ2 is the second eigenvalue of (1.2). (H2) f (x, t) is asymptotically 4-linear: ∃λ > λ2 ;
lim
|t|→+∞
F(x, t) ≥
f (x, t) =µ bt 3
uniformly in x,
(1.5)
(1.6)
µ < µ1 , and (1.5) holds, where µ1 is the first eigenvalue of (1.3). (H3) (1.6) holds, µ > µ2 is not an eigenvalue of (1.3), and aλ1 2 t , |t| small, 2 Rt where F(x, t) = 0 f (x, s)ds. (H4) f (x, t) is 4-superlinear:
(1.7)
F(x, t) ≤
∃ν > 4 : ν F(x, t) ≤ t f (x, t),
for |t| large,
(1.8)
and (1.7) holds. Problem (Pλ ) is related to the stationary analogue of the equation Z 2 u tt − a + b |∇u| dx ∆u = g(x, u)
(1.9)
Ω
proposed by Kirchhoff [11] as an existence 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, we refer to [7,12] for early classical studies. Eq. (1.9) received much attention only after Lions [10] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [6,8,9]. More recently, Alves et al. [4] and Ma and Rivera [3] obtained positive solutions of such problems by variational methods. Similar nonlocal problems also model several physical and biological systems where u describes a process which depends on the average of itself, for example the population density, see [5,13,14]. The existence of infinitely many solutions for problem (Pλ ) with a = 1 and b = 0 has been widely investigated. The most classical results on this topic are essentially based on Ljusternik–Schnirelman theory. In them, the crucial role is played by the oddness of the nonlinearity. Moreover, in order to check the Palais–Smale condition or some of its variant, one has to impose certain conditions that do not satisfy an oscillating behavior of the nonlinearity, we
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
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refer to [2,20–24] as typical papers of this kind. In [15,16], Omari and Zanolin proved the existence of an unbounded sequence of weak solutions for the p-Laplacian problem −∆ p u = λ f (u) in Ω , (1.10) u=0 on ∂Ω under the following conditions F(t) F(t) = 0 and lim sup p = +∞, (1.11) tp t t→0+ Rt where F(u) = 0 f (s)ds. They showed problem (1.10) has a sequence {u n } of non-zero nonnegative weak solutions, satisfying maxΩ u n → 0 as n → +∞. Annello and Cordaro [19] recently weakened the condition (1.11) and obtained infinitely many positive solutions of (1.10). The main ideal of [19] is based on the general approach proposed in [17], which yields weak solutions by searching for local minima of the underlying energy functional. This technique was suggested earlier in the papers of Raymond [18] and Ricceri [17]. Motivated by the papers mentioned above, in this paper we intend to search for positive solutions for problem (Pλ ) as the local minima of the energy functional corresponding to (Pλ ). We consider a class of nonlinearity f (x, ·) different from [1,2], and prove that there exists a sequence of a.e. positive weak solutions of (Pλ ) converging to zero in L ∞ (Ω ). If u is a weak solution of (Pλ ), we say that u is a.e. positive if m({x ∈ Ω : u(x) ≤ 0}) = 0, where m(·) is the Lebesgue measure. To the best of our knowledge, little is known about the multiplicity results for problem (Pλ ) with f (x, ·) having an oscillating behavior. Our result improves the main results of [1,2]. The paper is divided into three sections, including this introduction. In Section 2, we state and prove our main result. In Section 3, we give an example to illustrate our result. lim inf
t→0+
2. Main result R 1 Let H = H01 (Ω ) be the usual Sobolev space endowed with norm kuk = ( Ω |∇u|2 ) 2 . We recall that a function u ∈ H is called a weak solution of (Pλ ) if Z Z 2 f (x, u)vdx, ∀v ∈ H. (2.1) ∇u∇vdx = λ (a + bkuk ) Ω
Ω
Weak solutions are the critical points of the C 1 functional Z b a F(x, u)dx, u ∈ H. I (u) = kuk2 + kuk4 − λ 2 4 Ω
(2.2)
They are also classic solutions if f is locally Lipschitz on Ω × R. Theorem 2.1. Suppose that the function f satisfies the following conditions. (i) For every n ∈ N, there exist ξn , ξn0 ∈ R with 0 ≤ ξn < ξn0 and limn→+∞ ξn0 = 0 such that for a.e. x ∈ Ω , Z ξn Z t f (x, s)ds = sup f (x, s)ds. t∈[ξn ,ξn0 ] 0
0
(ii) There exists a non-empty open set S ⊆ Ω , a constant M ≥ 0 and a sequence {tn }n∈N ⊂ R+ \ {0} with limn→+∞ tn = 0, such that Rt ess inf 0n f (x, s)ds x∈S lim = +∞ n→+∞ tn2 and Z tn Z t ess inf inf f (x, s)ds ≥ −Mess inf f (x, s)ds . x∈S
t∈[0,tn ] 0
x∈S
0
Then, for every λ > 0, problem (Pλ ) has a sequence {u n } of a.e. positive weak solutions strongly convergent to zero and such that limn→+∞ maxΩ¯ u n = 0.
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
Proof. By (1.1), there exist a > 0 and t ∗ > 0 such that | f (x, t)| ≤ a, for every 0 ≤ t ≤ t ∗ and a.e. x ∈ Ω . Moreover, conditions (i) and (ii) imply that, f (x, 0) = 0 for a.e. x ∈ Ω . Without loss of generality, we suppose that, for every n ∈ N, max{ξn0 , tn } ≤ t ∗ . Let λ > 0, and define f (x, t ∗ ), if t > t ∗ ; g(x, t) = f (x, t), if 0 ≤ t ≤ t ∗ ; 0, if t < 0. Hence, we have (2.3)
|g(x, t)| ≤ d for a.e. x ∈ Ω and t ∈ R. Now we consider the following problem Z |∇u|2 dx)∆u = λg(x, u) in Ω , −(a + b Ω u = 0 on ∂Ω .
(2.4)
The weak solutions of (2.4) are the critical points of the functional Z a b 2 4 Iµ (u) = µ kuk + kuk − G(x, u)dx, u ∈ H, (2.5) 2 4 Ω Ru where G(x, u) = 0 g(x, t)dt, µ = 1/λ. By (2.3), it is easy to see that Iµ is well defined, weakly sequentially lower semicontinuous and Gˆateaux differentiable in H . For a fixed n ∈ N, the subspace E n = {u ∈ H : 0 ≤ u n (x) ≤ ξn0 , a.e. in Ω }. is a closed and convex, and it is also weakly closed. Moreover, for each u ∈ E n , one has Iµ (u) ≥ −dm(Ω )ξn0 . Hence Iµ is lower bounded in E n , so we can set αn = inf E n Iµ . For ∀k ∈ N, there exist vk ∈ E n such that αn ≤ Iµ (vk ) ≤ αn + k1 , then it follows that Z a b 1 2 4 G(x, vk )dx + Iµ (vk ) kvk k + kvk k = 2 4 µ Ω Z 1 1 ≤ dvk dx + αn + µ Ω k 1 ≤ (dm(Ω )ξn0 + αn + 1), µ which implies that {vk } is norm bounded in H . Therefore, there exists a subsequence {vkm }, weakly convergent to u n ∈ E n , being E n weakly closed. At this point, we obtain that Iµ (u n ) = αn in view of the weakly sequentially lower semi-continuity of Iµ . Now we claim that u n (x) ∈ (0, ξn ] a.e. in Ω . In fact, set ξn , if t > ξn ; if 0 < t ≤ ξn ; τ (t) = t, 0, if t ≤ 0. Define the map T : H → H as follows T u := τ (u)
for ∀u ∈ H.
The operator T is continuous in H . Furthermore, for ∀u ∈ H, T u ∈ E n . Set v ∗ = T u n and X = {x ∈ Ω : u n 6∈ (0, ξn ]}. Then for a.e., x ∈ X , we have ξn < u n (x) ≤ ξn0 ,
or
u n (x) = 0.
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
But by (i) we have Z u n (x) Z g(x, t)dt ≤ and
0 |∇v ∗ |
1411
v ∗ (x)
g(x, t)dt
0
= 0. Therefore
! Z Z v∗ a b Iµ (v ∗ ) − Iµ (u n ) = µ kv ∗ k2 + kv ∗ k4 − g(x, t)dt dx 2 4 Ω 0 Z Z u n a b −µ ku n k2 + ku n k4 + g(x, t)dt dx 2 4 Ω 0 ! 2 ! Z Z Z v ∗ (x) Z a b 2 2 |∇u n | dx − g(x, t)dt dx = −µ |∇u n | dx + 2 X 4 X X u n (x) 2 ! Z Z b a 2 2 . |∇u n | dx + |∇u n | dx ≤ −µ 2 X 4 X R R Since v ∗ ∈ E n , it follows that Iµ (v ∗ ) − Iµ (u n ) ≥ 0. Then X |∇u n |2 dx = 0. Hence kv ∗ − u n k2 = Ω |∇v ∗ − ∇u n |2 = R 2 ∗ X |∇u n | dx = 0 which means that u n (x) = v (x) ∈ (0, ξn ] a.e. in Ω . Let u ∈ H, and T be the operator defined above. Set
X 0 = {x ∈ Ω : u(x) 6∈ (0, ξn ]}. We have that if x ∈ Ω \ X 0 , then Z u(x) g(x, t)dt = 0. T u(x)
Furthermore, if x ∈ X 0 , then one has the following three cases. (C1 ) If u(x) ≤ 0, then Z u(x) Z g(x, t)dt =
u(x)
g(x, t)dt = 0.
0
T u(x)
(C2 ) If ξn < u(x) ≤ ξn0 , then by (i), Z u(x) g(x, t)dt ≤ 0. T u(x)
(C3 ) If u(x) > ξn0 , then Z u(x) Z g(x, t)dt =
u(x)
ξn
T u(x)
Z g(x, t)dt ≤
u(x) ξn
ddt = d(u(x) − ξn ).
Let q > 1 and fix it, then the constant A = sup
ξ ≥ξn0
d(ξ − ξn ) (ξ − ξn )q+1
is finite, we have, for a.e. x ∈ Ω , Z u(x) g(x, t)dt ≤ A|u(x) − T u(x)|q+1 . T u(x)
Then Z Ω
Z
u(x) T u(x)
! g(x, t)dt dx ≤ AL q+1 ku − T ukq+1 ,
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
where R sup
L=
u∈H \{0}
Ω
|u|q+1 dx kuk
1 q+1
.
Therefore, ! Z Z u(x) a b a b 2 4 2 4 Iµ (u) − Iµ (T u) = µ kuk + kuk − µ kT uk + kT uk − g(x, t)dt dx 2 4 2 4 Ω T u(x) ! Z 2 Z Z Z u(x) aµ bµ = |∇u|2 dx − |∇u|2 dx + g(x, t)dt dx 2 X 4 X Ω T u(x) ! Z 2 Z Z Z u(x) aµ bµ 2 2 = |∇u − ∇(T u)| dx − |∇u − ∇(T u)| dx + g(x, t)dt dx 2 Ω 4 Ω Ω T u(x)
bµ aµ ku − T uk2 + ku − T uk4 − AL q+1 ku − T ukq+1 . 2 4 Since T u ∈ E n , we have Iµ (T u) ≥ I (u n ). Then a bµ Iµ (u) ≥ Iµ (u n ) + ku − T uk2 µ+ ku − T uk2 − AL q+1 ku − T ukq−1 2 4 aµ − AL q+1 ku − T ukq−1 . ≥ Iµ (u n ) + ku − T uk2 2 Since T is continuous and q > 1, there exists β > 0 such that, for every u ∈ H with aµ ku − u n k < β, ku − T ukq−1 ≤ . 4AL q+1 Then if ku − u n k < β, we have aµ Iµ (u) ≥ Iµ (u n ) + ku − T uk2 ≥ Iµ (u n ), 4 this implies that u n is a local minimum of Iµ . For every n ∈ N and u ∈ E n , we have that ≥
Iµ (u) ≥ −dm(Ω )ξn0 . Then, since −dm(Ω )ξn0 ≤ αn ≤ 0, it is easy to see that lim αn = 0.
n→+∞
From u n ∈ E n with αn = Iµ (u n ), it follows that ! ! Z Z u n (x) a b 1 2 4 ku n k + ku n k = g(x, t)dt dx + Iµ (u n ) 2 4 µ Ω 0 ! ! Z Z u n (x) 1 = g(x, t)dt dx + αn µ Ω 0 ≤
1 (dm(Ω )ξn0 + αn ). µ
Therefore, limn→+∞ ku n k2 = 0. Now we prove that αn < 0, for every n ∈ N. For this, fix n ∈ N, and choose a compact set K ⊂ S with m(K ) = (M + 1)m(S \ K ) and a function v ∈ H such that if x ∈ K ; 1, v(x) = 0 ≤ v(x) ≤ 1, if x ∈ S \ K ; 0, if x ∈ Ω \ S.
X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
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By the former condition of (ii), there exist k˜ ∈ N such that tk ≤ ξn0 and some positive constant C such that ! Z tk (M + 1) atk2 b 4 4 2 2 ess inf g(x, t)dt > Ctk ≥ µ kvk + tk kvk x∈S 0 m(K ) 2 4 ˜ Next, taking into account the latter condition of (ii), for every k ≥ k, ˜ we have for every k ∈ N with k ≥ k. R R t v(x) R Rt R R t v(x) − Ω 0k g(x, t)dt dx − K ( 0k g(x, t)dt)dx − S\K ( 0k g(x, t)dt)dx = 4 2 4 2 2 btk R btk atk R atk 2 2 2 4 Ω |∇v| dx 2 Ω |∇v| dx + 4 2 kvk + 4 kvk Rt R R R tk inf inf 0 g(x, t)dt)dx − K (ess inf 0 g(x, t)dt)dx S\K (ess x∈S t∈[0,tk ] x∈S − ≤ btk4 btk4 atk2 atk2 2 4 2 4 2 kvk + 4 kvk 2 kvk + 4 kvk R R tk R R tk − K (ess inf 0 g(x, t)dt)dx M S\K (ess inf 0 g(x, t)dt)dx x∈S x∈S ≤ + 4 2 atk2 bt at btk4 k k 2 4 2 4 2 kvk + 4 kvk 2 kvk + 4 kvk R tk −(1/(M + 1))m(K )ess inf 0 g(x, t)dt x∈S = 4 atk2 bt k 2 4 2 kvk + 4 kvk < −µ. Whence tk v ∈ E n and Iµ (tk v) < 0, which means αn < 0. Hence, there exists a subsequence of u n of pairwise distinct elements. Such a subsequence is a sequence of weak solutions of (Pλ ). Moreover, we have 0 = ess inf u n (x) < ess sup u n (x) ≤ t ∗ x∈Ω
x∈Ω
for every n ∈ N. Then it is a sequence of weak solutions for (Pλ ).
3. An example In this section we present an example of an application of Theorem 2.1, when neither of conditions (H1)–(H4) is satisfied. Example 3.1. Let us consider problem (Pλ ) with a, b > 0 constants and λ > 0 any parameter. The nonlinearity f : Ω × R → R is defined by ( 2 1 1 1 if t > 0, (1 + |x|2 ) 15t 3 sin(1/t 3 ) − 3t 3 cos(1/t 3 ) f (x, t) = 0 if t ≤ 0, 1
where x = (x1 , x2 , . . . , xn ) ∈ Ω , |x| = (x12 + x22 + · · · + xn2 ) 2 . Thus we have, for every x ∈ Ω , ( 5 1 2 3 3 F(x, t) = (1 + |x| )9t sin(1/t ) if t > 0, 0 if t ≤ 0. Taking ξn = 0, ξn0 = (π/2 + 2nπ )−3 , it is easy to see that condition (i) is satisfied. To check that condition (ii) holds, let 8 tn = 3 π (1 + 4n)3 for each n ∈ N. Then F(x, tn ) 9 = (1 + |x|2 ) lim √ = +∞ uniformly in x. lim 2 n→+∞ n→+∞ tn tn
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X. He, W. Zou / Nonlinear Analysis 70 (2009) 1407–1414
Moreover, we have inf F(x, t) ≥ −F(x, tn )
t∈[0,tn ]
for every x ∈ Ω .
Then the latter condition of (ii) is also satisfied for M = 1. Hence Theorem 2.1 ensures that there exists a sequence of a.e. positive and pairwise distinct solutions to problem (Pλ ) (with a, b > 0, arbitrarily constants and λ > 0) that tend to zero. Remark 3.1. We note that in Example 3.1, the nonlinearity f (x, t) does not satisfy (1.4), and lim inf
t→0+
F(x, t) = −∞ uniformly in x. t2
Also, by a simple computation, it is easy to see that neither of conditions (H1)–(H4) in Theorem A is satisfied in Example 3.1. So the results of [1,2] are invalid for Example 3.1. Acknowledgement The authors thank the referee for his/her carefully reading the manuscript and valuable comments and suggestions. References [1] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255. [2] Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463. [3] T.F. Ma, J.E. Mu˜noz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003) 243–248. [4] C.O. Alves, F.J.S.A. Correa, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. [5] D. Andrade, T.F. Ma, An operator equation suggested by a class of nonlinear stationary problems, Commun. Appl. Nonl. Anal. 4 (1997) 65–71. [6] A. Arosio, S. Pannizi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305–330. [7] S. Bernstein, Sur une classe d’´equations fonctionnelles aux d´eriv´ees partielles, Bull. Acad. Sci. URRS. S´er. Math. [Izv. Akad. Nauk SSSR] 4 (1940) 17–26. [8] M.M. Cavalcanti, V.N. Cavacanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001) 701–730. [9] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992) 247–262. [10] J.-L. Lions, On some quations 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: NorthHolland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346. [11] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [12] S.I. Pohoˇzaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (1975) 152–168. [13] M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997) 4619–4627. [14] M. Chipot, J.-F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Mod´el. Math. Anal. Num´er. 26 (1992) 447–467. [15] P. Omari, F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations 21 (1996) 721–733. [16] P. Omari, F. Zanolin, An elliptic problem with arbitrarily small positive solution, Electron, J. Diff. Conf. 5 (2000) 301–308. [17] B. Ricceri, A genaral variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [18] J. Saint Raymond, On the multiplicity of the solutions of the equations −∆u = λ f (u), J. Differential Equations 180 (2002) 65–88. [19] G. Anello, G. Cordaro, Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian, Proc. Roy. Soc. Edinburgh 132 (A) (2002) 511–519. [20] T. Bartsch, Z. Liu, T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc. (3) 91 (2005) 129–152. [21] T. Bartsch, Z. Liu, On superlinear elliptic p-Laplacian eqaution, J. Differential Equations 198 (2004) 149–175. [22] M. Schechter, W. Zou, Superlinear problem, Pacific J. Math. 214 (2004) 145–160. [23] M. Schechter, W. Zou, Infinitely many sign-changing solutions for perturbed elliptic equations with Hardy potentials, J. Funct. Anal. 228 (2005) 1–38. [24] M. Schechter, Z.-Q. Wang, W. Zou, New liking theorem and sign-changing solutions, Comm. Partial Differential Equations 29 (2004) 471–488.