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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

MULTIPLE SOLUTIONS FOR KIRCHHOFF TYPE PROBLEM NEAR RESONANCE

SHU-ZHI SONG, CHUN-LEI TANG, SHANG-JIE CHEN

Abstract. Based on Ekeland’s variational principle and the mountain pass theorem, we show the existence of three solutions to the Kirchhoff type problem

a + b

Z

|∇u|2dx

∆u = bµu3+ f (x, u) + h(x), in Ω, u = 0, on ∂Ω.

Where the parameter µ is sufficiently close, from the left, to the first nonlinear eigenvalue.

1. Introduction and statement of main result

The articles shows the existence of multiple solutions for the Kirchhoff type problem with Dirichlet boundary condition,

− a + b

Z

|∇u|2dx

∆u = bµu3+ f (x, u) + h(x), in Ω, u = 0, on ∂Ω,

(1.1)

where Ω is a bounded domain in RN (N = 1, 2, 3) with a smooth boundary ∂Ω, a ≥ 0, b > 0 are real constants and µ is a nonnegative parameter. Assume that f ∈ C( ¯Ω × R, R) satisfies the sublinear growth condition:

(F1) lim|t|→∞f (x,t)bt3 = 0, uniformly for x ∈ Ω.

Problem (1.1) can be looked on as a perturbed problem which was first studied by Mawhin and Schmit[6], related to the two-point boundary value equation

− u00− λu = f (x, u) + h, u(0) = u(π) = 0. (1.2) Specifically, on the assumption: λ < λ1 is sufficiently near to λ11 is the first eigenvalue of the corresponding linear problem) and f is bounded and satisfies a sign condition, the existence of three solutions to equation (1.2) was proved in [6].

Later, various papers related to the result appeared. We mention for example, [1, 3, 4, 5, 8]. Ma, Ramos and Sanchez [3] considered the boundary-value problem for

∆u + λu + f (x, u) = h(x)

2010 Mathematics Subject Classification. 35J61, 35C06, 35J20.

Key words and phrases. Near resonance; mountain pass theorem; Kirchhoff type;

Ekeland’s variational principle.

c

2015 Texas State University - San Marcos.

Submitted March 23, 2015. Published August 17, 2015.

1

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defined on a bounded open set Ω ⊂ RN. As the parameter λ is sufficiently close to λ1

from the left, there exist three solutions on both Dirchlet boundary conditions and Neumann boundary conditions. In addition, similar to the results in the linear case, the existence of three solutions was proved to the perturbed p-Laplacian equation in a bounded domain. Further consideration to the perturbed p-Laplacian equation in a bounded domain can be found in [4]. As for extension to the the perturbed p-Laplacian equation in the whole space RN, we refer to [5]. These results were also extended to some elliptic systems with the Dirichlet boundary conditions, refer to [8]. More recently, the authors in [1] extended these conclusions to some degenerate quasilinear elliptic systems with the Dirichlet boundary conditions. By analogy to the results mentioned above, we expect that problem (1.1) has at least three solutions as the parameter µ < µ1 is sufficiently close to µ1. Here µ1 is the first eigenvalue of the eigenvalue problem

−kuk2∆u = µu3, in Ω, u = 0, on ∂Ω.

Let H = H01(Ω) be the Hilbert space equipped with the norm kuk = (R

|∇u|2dx)1/2 and kukLs= (R

|u|sdx)1s denote the norm of Ls(Ω). As shown in [9] and [13], the first nonlinear eigenvalue µ1 > 0 is simple and has a eigenfunction ψ1 > 0 with kψ1kL4= 1. Specifically, µ1 can be characterized by

µ1= infkuk4: u ∈ H, Z

|u|4dx = 1 . (1.3)

Now we are in a position to state our result.

Theorem 1.1. Suppose f satisfies (F1) and the following conditions:

(F2)

lim

|t|→∞

Z

F (x, tψ1) −a 2

√µ1t2= +∞, uniformly for x ∈ Ω,

where F (x, t) =Rt

0f (x, s)ds.

(H1) h ∈ L2(Ω) and R

h(x)ψ1(x)dx = 0.

Then (1.1) has at least three solutions if µ < µ1 is sufficiently close to µ1.

Many authors have studied the Kirchhoff type equation in a bounded domain by applying variational methods. For example, they consider Kirchhoff type problem

− a + b

Z

|∇u|2dx

∆u = g(x, u), in Ω, u = 0, on ∂Ω,

(1.4)

assuming that

lim

|t|→∞

4G(x, t)

bt4 = µ, uniformly in x ∈ Ω (1.5) where G(x, t) = Rt

0g(x, s)ds. For the case µ < µ1 in (1.5), the Euler func- tional corresponding to (1.4) is coercive. For the case µ = µ1 in (1.5), that is, problem (1.4) is resonance at the first nonlinear eigenvalue µ1, the Euler func- tional corresponding to (1.4) is still coercive, together with the assumption that lim|t|→∞[g(x, t)t − 4G(x, t)] = +∞. So, the existence of weak solution for equa- tion (1.4) is obtained based on the Least Action Principle (refer to [11, 12, 13]).

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Furthermore, provided g with some conditions at zero, positive solution was ob- tained based on the topological degree argument (refer to [2]), multiple solutions are found by means of invariant sets of descent flow method (refer to [11, 13]), or the Local Linking Theorem (refer to [12]). Our result is different from the results in [2, 11, 12, 13] since we deal with the perturbation problem near to µ1 and all hypotheses on f are just at infinity.

2. Proof of main result

We begin with some standard facts upon the variational formulation of problem (1.1). Let Iµ: H 7→ R be the functional defined by

Iµ(u) = a

2kuk2+b

4kuk4−bµ 4

Z

|u|4dx − Z

F (x, u)dx − Z

hudx.

Since f satisfies the sublinear growth condition (F1), it is not difficult to verify that Iµ ∈ C1(H, R). Furthermore, finding weak solutions of (1.1) is equivalent to finding critical points of functional Iµ in H.

Since Ω is a bounded domain in RN (N = 1, 2, 3), the embedding H ,→ Ls(Ω) is continuous for s ∈ [1, 2], compact for s ∈ [1, 2),

2 = ( 2N

N −2, N = 3, +∞, N = 1, 2.

Hence, for s ∈ [1, 2], there exists τs> 0 such that

kukLs≤ τskuk, ∀u ∈ H. (2.1)

We will prove the result by using Ekeland’s variational principle [7, Theorem 4.1] and a mountain pass theorem [10]. For the convenience of readers, we state the mountain pass theorem as follows.

Theorem 2.1 ([10, Corollary 1]). Consider a real Banach space X and a function I ∈ C1(X, R). If the (P S) condition holds and if I has two different local mininum points, then I possesses a third critical point.

To prove our theorem, using critical point theory, we need the Palais-Smale compactness.

Lemma 2.2. Assume that (F1) holds. Then any bounded (P S) sequence of Iµ has a convergent subsequence in H.

Proof. Let {un} ⊂ H be a bounded (P S) sequence of Iµ; that is,

kunk ≤ c, |Iµ(un)| ≤ c, kIµ0(un)k → 0, (2.2) where c denotes positive constant. By the reflexivity of H, we can assume that there exists u ∈ H such that

un* u weakly in H, (2.3)

un→ u strongly in Lp(Ω) (1 ≤ p < 2). (2.4) It follows from (F1) that for any ε > 0, there exits Mε> 0 such that

|f (x, t)| ≤ bε|t|3+ Mε, ∀(x, t) ∈ Ω × R. (2.5)

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We can now put together the results in (2.1), (2.2), (2.4) and (2.5) to conclude that

Z

f (x, un)(u − un)dx ≤

Z

|f (x, un)||u − un|dx

≤ Z

(bε|un|3+ Mε)|u − un|dx

≤ bεkunk3L4ku − unkL4+ Mε|Ω|1/2ku − unkL2

≤ bτ43εkunk3ku − unkL4+ Mε|Ω|1/2ku − unkL2

≤ c(ku − unkL4+ ku − unkL2) → 0, as n → ∞, (2.6)

where c = max{bτ43ε, Mε|Ω|1/2}, and |Ω| is the measure of Ω. Similarly, we may deduce that

Z

(|un|2un(u − un) − |u|2u(u − un))dx → 0, as n → ∞. (2.7) From (2.2) and (2.4), we have

hIµ0(un) − Iµ0(u), u − uni → 0, as n → ∞,

which combining with (2.6), (2.7), implies kunk → kuk as n → ∞. It follows from

(2.3) that un→ u in H. 

Set

V =v ∈ H : Z

ψ31vdx = 0 .

From the simplicity of µ1 we have H = span{ψ1} ⊕ V . We introduce the quantity µV = infkuk4: u ∈ V, kuk4L4= 1 .

Then

kuk4≥ µVkuk4L4, ∀u ∈ V, (2.8) and we have the following result.

Lemma 2.3. µ1< µV.

Proof. It is evident from (1.3) that µ1 ≤ µV. Assume, by contradiction, that µ1 = µV. Then there exists a sequence {un} ⊆ V such that kunkL4 = 1 for all n ≥ 1, and kunk4→ µV = µ1. Since the sequence {un} is bounded in H, we may assume that

un* u weakly in H, un→ u strongly in L4(Ω). (2.9) Thus, one has

kukL4 = lim

n→+∞kunkL4 = 1, µ1≤ kuk4≤ lim inf

n→∞ kunk4= lim

n→∞kunk4= µ1. So, kukL4= 1 and kuk4= µ1. This implies u = ±ψ1.

On the other hand, from {un} ⊆ V it follows that R

ψ13un = 0 for all n ≥ 1.

Combining this with (2.9) and H¨older’s inequality, we have

Z

ψ13udx =

Z

ψ31udx − Z

ψ31undx =

Z

ψ13(u − un)dx

≤ Z

ψ13(u − un) dx

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≤ kψ1k3L4kun− ukL4 → 0, as n → ∞.

This is in direct contradiction to the fact u = ±ψ1. Hence µ1< µV.  Proof of Theorem 1.1. We shall divide the proof into four steps.

Step 1. The functional Iµ is bounded below in H and V and even coercive in H and V . More specifically, there is a constant α, independent of µ, such that infV Iµ≥ α. From (2.5), we obtain

|F (x, t)| ≤ bε

4|t|4+ Mε|t|, ∀(x, t) ∈ Ω × R. (2.10) It follows from (2.1), (2.10) and H¨older inequality that

Iµ(u) ≥ a

2kuk2+b

4kuk4−b(µ + ε) 4

Z

u4dx − (Mε|Ω|1/2+ khkL2)kukL2

≥ b

4(1 −µ + ε µ1

)kuk4− τ2(Mε|Ω|1/2+ khkL2)kuk, ∀u ∈ H .

Note that µ < µ1. Then, for 0 < ε < µ1− µ, Iµis bounded below and even coercive in H. Similarly, for 0 < ε < µV − µ1, (2.1), (2.8), (2.10) and H¨older inequality lead to

Iµ1(v) ≥ a

2kvk2+b

4kvk4−b(µ1+ ε) 4

Z

v4dx − (Mε|Ω|1/2+ khkL2)kvkL2

≥ b

4(1 −µ1+ ε µV

)kvk4− τ2(Mε|Ω|1/2+ khkL2)kvk, ∀v ∈ V,

which implies that Iµ1 is bounded below and coercive in V . Noting that Iµ≥ Iµ1

for all µ < µ1, we deduce Iµ is coercive in V and inf

V Iµ≥ α := inf

V Iµ1.

Step 2. If µ < µ1 is sufficiently close to µ1, there exist two constants t, t+ with t < 0 < t+ such that Iµ(t±ψ1) < α. Noting kψ1kL4 = 1, kψ1k4 = µ1 and then combining this with (H1), for t ∈ R, we have

Iµ(tψ1) = at2

2 kψ1k2+bt4

4 kψ1k4−bµt4 4

Z

ψ41dx − Z

F (x, tψ1)dx

= b(µ1− µ)

4 t4−Z

F (x, tψ1)dx −a 2

õ1t2 . From (F2), taking a constant t+ with t+> 0 large enough, we obtain

Z

F (x, t+ψ1)dx −a 2

√µ1(t+)2> −α + 1.

The above inequality reduces to

Iµ(t+ψ1) ≤ b(µ1− µ)

4 (t+)4+ α − 1.

Consequently, for −b(t+1)4 < µ < µ1, we obtain Iµ(t+ψ1) < α. The same conclusion holds for a constant t with t< 0.

Step 3. Two solutions are obtained based on the coerciveness of Iµ and Ekeland’s variational principle. Set

Θ±= {u ∈ H : u = ±tψ1+ v with t > 0, v ∈ V }.

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When µ < µ1is sufficiently close to µ1, from step 1 and step 2, Iµis bounded below in Θ+ with

−∞ < c+:= inf

Θ+

Iµ< α.

In Θ+, if we apply Ekeland’s variational principle to Iµ, there exists a sequence {un} ⊂ Θ+ such that Iµ(un) → c+and Iµ0(un) → 0 as n → ∞. By the coerciveness of Iµin H, we deduce that {un} is bounded. So, {un} is a sequence satisfying (2.2) so that Lemma 2.2 implies {un} has a convergent subsequence, say {un} itself.

Noting that V = ∂Θ+ and infVIµ ≥ α (step 1), we conclude that {un} converges to an interior point u+∈ Θ+, that is, the infimum is attained in Θ+. Therefore, Iµ

has a critical point u+ as a local minimum in Θ+. Similarly, we obtain a critical point u of Iµ as a local minimum in Θ. Note that Θ+∩ Θ = ∅ which implies u+6= u, that is, Iµ has two different local minimum points.

Step 4. It follows from Theorem 2.1 that Iµ has a third solution. By Lemma 2.2, we see Iµ satisfies (P S) condition. It follows from step 3 that u+, u are two different local minimum points. Consequently, Theorem 2.1 shows that Iµ has a

third critical point. 

Acknowledgments. This work is supported by National Natural Science Founda- tion of China (No. 11471267), and by Science and Technology Researching Program of Chongqing Educational Committee of China (Grant No.KJ130703).

References

[1] Y. C. An, X. Lu, H. M. Suo; Existence and multiplicity results for a degenerate quasilinear elliptic system near resonance. Bound. Value Probl. 2014, 2014:184.

[2] Z .P. Liang, F. Y. Li, J. P. Shi; Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 31 (2014), no. 1, 155–167.

[3] T. F. Ma, M. Ramos, L. Sanchez; Multiple solutions for a class of nonlinear boundary value problems near resonance: a variational approach. Proceedings of the Second World Congress of Nonlinear Analysts, Part 6 (Athens, 1996). Nonlinear Anal. 30 (1997), no. 6, 3301–3311.

[4] T. F. Ma, M. Ramos; Three solutions of a quasilinear elliptic problem near resonance. Math.

Slovaca 47(1997), 451-457.

[5] T. F. Ma, M. Pelicer; Perturbations near resonance for the p-Laplacian in RN. Abstr. Appl.

Anal. 7 (2002), no. 6, 323–334.

[6] J. Mawhin, K. Schmitt; Nonlinear eigenvalue problems with the parameter near resonance.

Ann. Polon. Math. 51 (1990), 241–248.

[7] J. Mawhin, M. Willem; Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989.

[8] Z. Q. Ou, C. L. Tang; Existence and multiplicity results for some elliptic systems at reso- nance. Nonlinear Anal. 71 (2009), no. 7-8, 2660–2666.

[9] K. Perera, Z. T. Zhang; Nontrivial solutions of Kirchhoff-type problems via the Yang index.

J. Differential Equations 221 (2006), no. 1, 246–255.

[10] P. Pucci, J. Serrin; A mountain pass theorem. J. Differential Equations 60 (1985), no. 1, 142–149.

[11] Y. Yang, J. H. Zhang; Positive and negative solutions of a class of nonlocal problems. Non- linear Anal. 73 (2010), no. 1, 25–30.

[12] Y. Yang, J. H. Zhang; Nontrivial solutions of a class of nonlocal problems via local linking theory. Appl. Math. Lett. 23 (2010), no. 4, 377–380.

[13] Z. T. Zhang, K. Perera; Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317 (2006), no. 2, 456–463.

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Shu-Zhi Song

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China E-mail address: sjrdj@163.com

Chun-Lei Tang (corresponding author)

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China E-mail address: tangcl@swu.edu.cn, Tel +8613883159865

Shang-Jie Chen

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

E-mail address: chensj@ctbu.edu.cn, 11183356@qq.com

References

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