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High energy solutions for the fourth-order elliptic equations in R N

Abstract

In the present paper, we study the following fourth-order elliptic equations: Δ 2 uΔu+V(x)u=f(x,u), for x R N , u(x) H 2 ( R N ), where VC( R N ,R), fC( R N ×R,R). Under more relaxed assumptions on f(x,u), by using some special techniques, a new existence result of high energy solutions is obtained via the symmetric mountain pass theorem.

1 Introduction and main results

In this paper, we consider the following fourth-order elliptic equations:

{ Δ 2 u Δ u + V ( x ) u = f ( x , u ) , for  x R N , u ( x ) H 2 ( R N ) ,
(1.1)

where Δ 2 :=Δ(Δ) is the biharmonic operator, VC( R N ,R), fC( R N ×R,R) satisfy some further conditions.

When Ω is a smooth bounded domain in R N , the problem

{ Δ 2 u + c Δ u = f ( x , u ) , in  Ω , u = Δ u = 0 , on  Ω ,
(1.2)

arises in many applications from mathematical physics, which is usually used to describe some phenomena appearing in various physical, engineering, and other sciences. In [1], [2], Lazer and Mckenna firstly proposed and studied the problem (1.2) of periodic oscillations and traveling waves in a suspension bridge. It was pointed out in [3]–[6] that the problem (1.2) furnishes a good model to the study of the static deflection of an elastic plate in a fluid. Moreover, Ahmed and Harbi presented that the problem (1.2) can also be applied to engineering, such as communication satellites, space shuttles and space stations equipped with large antennas mounted on long flexible beams in [7]. Further, there is a large quantity of studies on the existence and multiplicity of solutions in the literature, we refer the readers to [8]–[21] and the references therein.

Recently, Lü [22] considered a class of biharmonic elliptic systems with Sobolev critical exponent on a bounded domain

{ Δ 2 u = F u ( u , v ) + λ | u | q 2 u , x  in  Ω ; Δ 2 v = F v ( u , v ) + δ | v | q 2 v , x  in  Ω ; u = u n , v = v n , x  on  Ω ;
(1.3)

where Ω R N (N5) is a bounded domain with smooth boundary Ω, 0Ω, F C 1 ( R 2 , R + ) is homogeneous of degree 2 ( 2 = 2 N N 4 denotes the Sobolev critical exponent). The existence and multiplicity results of nontrivial solutions for the system (1.3) were obtained by using the variational methods and the Nehari manifold.

The above mentioned interesting studies for the fourth-order elliptic equations are based on boundness of Ω R N . Some authors had studied the fourth-order elliptic equations in the whole space R N . In [23], Chabrowski and Marcos do Ó considered the following fourth-order elliptic problems in R N :

{ Δ 2 u λ g ( x ) u = f ( x ) | u | p 2 u , in  R N ; u D 2 , 2 ( R N ) { 0 } ,
(1.4)

where λ>0 and p= 2 N N 4 the Sobolev critical exponent. By employing the mountain pass theorem and the constrained minimization together with the concentration-compactness principle, the existence of two solutions were obtained. In [24], Pimenta and Soares considered the following superlinear fourth-order elliptic equations:

{ ϵ 4 Δ 2 u + V ( x ) u = f ( u ) , in  R N ; u H 2 ( R N ) ,
(1.5)

where ϵ>0, N5, and f, and V satisfy some certain assumptions. Under a weak version of the Ambrosetti-Rabinowitz condition, by means of variational methods, a family of solutions was proved to exist and to concentrate at a point in the limit. Very recently, some authors studied the following fourth-order elliptic equations in R N :

{ Δ 2 u Δ u + λ V ( x ) u = f ( x , u ) , in  R N ; u H 2 ( R N ) ,
(1.6)

where λ1 is a parameter and fC( R N ,R). For the potential V(x), they usually assumed one of the following assumptions.

(V1):VC( R N ,R) satisfies infV(x) V 0 >0 and for each M>0, meas{x R N :V(x)M}<+, where V 0 is a constant and meas denote the Lebesgue measure in R N .

( V 1 ):VC( R N ,R) satisfies infV(x) V 0 >0 and there exists a constant b>0 such that the set {x R N :V(x)b} is nonempty and meas{x R N :V(x)b}<+, where V 0 is a constant and meas denote the Lebesgue measure in R N .

Yin and Wu [25] obtained three new results of the existence of infinitely many high energy solutions for the equation (1.6) with λ=1 and V satisfying (V1). Soon after, Ye and Tang [26] unified and improved their results by means of fountain theorem. To be precise, they assumed the following assumptions.

(f1): There exist C 1 >0 and p(2, 2 ) such that

| f ( x , t ) | C 1 ( | t | + | t | p 1 ) ,(x,t) R N ×R,

where 2 = 2 N N 4 , if N>4; 2 =+, if N4.

( f 2 ): There exists L 0 >0 such that inf x R N , | t | = L 0 F(x,t)>0, where F(x,t)= 0 t f(x,s)ds.

( f 3 ): There exists μ 0 >2 such that

μ 0 F(x,t)f(x,t)t,for a.e. x R N  and |t| L 0 .

(f4):f(x,t)=f(x,t) for all (x,t) R N ×R.

We restate the corresponding result in [26] as follows.

Theorem A

(see [26], Theorem 1.1])

Assume that (V1), (f1), ( f 2 )-( f 3 ), and (f4) hold. Then the problem (1.1) possesses infinitely many solutions{ u k }such that

1 2 R N ( | Δ u k | 2 + | u k | 2 + V ( x ) u k 2 ) dx R N F(x, u k )dx+,as k.

More recently, Liu et al. in [27] studied the existence and multiplicity of nontrivial solutions of (1.6) for large λ and V satisfying ( V 1 ). These results had been subsequently unified and improved by Ye and Tang in [28]. For the sublinear case of the fourth-order elliptic equation (1.1), by using the genus properties in critical point theory, infinitely many small negative energy solutions were also established; we refer the readers to [26], [28], [29].

Remark 1

The condition (V1), which contains the coercivity condition: V(x) as |x|, was first introduced by Bartsch and Wang in [30] to ensure the compactness of embedding of the working space. Furthermore, when replacing (V1) by the weaker condition ( V 1 ), this situation becomes more delicate due to the lack of compactness. Although the compactness of embedding of the working space fails, thanks to the fact that the parameter λ can be taken sufficiently large, two main theorems as regards the existence and multiplicity of nontrivial solutions of (1.6) were obtained via variational methods successfully.

Motivated by the spirit of [25]–[28], we shall consider the fourth-order elliptic equations with Bartsch type potential (that is, V satisfies the condition (V1)) and superlinear nonlinearity case in the whole space R N , and we are interested in high energy solutions of the problem (1.1). Under more relaxed assumptions on the nonlinear term f, we will present a new proof technique to construct high energy solutions for the problem (1.1).

Next, we give some notations. Define the function space

H 2 ( R N ) := { u L 2 ( R N ) : | u | , Δ u L 2 ( R N ) }

with the standard inner product and the norm

u , v H 2 = R N (ΔuΔv+uv+uv)dx, u H 2 = u , u H 2 1 2

whenever u,v H 2 ( R N ). Let

E:= { u L 2 ( R N ) : R N ( ( Δ u ) 2 + | u | 2 + V ( x ) u 2 ) d x < + } .

Then E is a Hilbert space with the inner product and the norm

u , v E = R N ( Δ u Δ v + u v + V ( x ) u ) dx, u E = u , u E 1 2 .

Obviously, the embedding

E L s ( R N ) ,2s 2

is continuous. Hence, for any s[2, 2 ], there is a constant a s >0 such that

u L s a s u E .
(1.7)

It is well known that a weak solution for the problem (1.1) is a critical point of the following functional I defined on E:

I(u)= 1 2 R N ( | Δ u | 2 + | u | 2 + V ( x ) u 2 ) dx R N F(x,u)dx,uE.
(1.8)

We say that a weak solutions sequence { u n }E for the problem (1.1) is a sequence of high energy solutions if the energy I( u n )+ as n.

In order to reduce the statements of our result, we make the following assumptions.

(f2): There exists 2<θ< 2 such that lim inf | t | + F ( x , t ) | t | θ >0 uniformly in x R N .

(f3): There exist μ>2, L>0, and a[0, ( μ 2 ) V 0 4 ) such that

μF(x,t)f(x,t)ta | t | 2 ,for a.e. x R N  and |t|L.

Now, we can state our result about the existence of a sequence of high energy solutions for the fourth-order elliptic equations (1.1) in R N .

Theorem 1.1

Assume that (V1), (f1)-(f4) hold. Then the problem (1.1) possesses a sequence of high energy solutions in E.

Remark 2

From Remark 1.1 in [26], we know that the condition (f1) is much weaker than the combination of the usual subcritical condition and the asymptotically linear condition near zero. Furthermore, conditions (f2)-(f3) are much weaker than ( f 2 )-( f 3 ). Indeed, for any x R N , zR, define

h(t):=F ( x , t 1 z ) t μ 0 ,t[1,+).

Then, for |z| L 0 and t[1, | z | L 0 ], ( f 3 ) implies that

h (t)= t μ 0 1 [ μ 0 F ( x , t 1 z ) t 1 z f ( x , t 1 z ) ] 0.

Hence, h(1)h( | z | L 0 ). Therefore, it follows from ( f 2 ) that

F(x,z)F ( x , L 0 z | z | ) ( | z | L 0 ) μ 0 c | z | μ 0 ,x R N  and |z| L 0 ,
(1.9)

where c= ( 1 L 0 ) μ 0 inf x R N , | t | = L 0 F(x,t)>0. Hence, (1.9) implies that (f2) holds by taking θ= μ 0 . Further, it follows from ( f 3 ) that (f3) holds by taking L= L 0 , μ= μ 0 . Consequently, (f2)-(f3) are much weaker than ( f 2 )-( f 3 ). Thus, Theorem 1.1 sharply improves Theorem A and, of course, unites and improves the results of [25].

2 Some lemmas

In order to apply variational techniques, we first state the key compactness result.

Lemma 2.1

(Lemma 3.4 in [31])

Under the assumption (V1), the embedding

E L s ( R N ) ,2s< 2

is compact.

The following lemma was proved by [26], Lemma 2.1].

Lemma 2.2

Assume that (V1) and (f1) hold. Then I is well defined on E, I C 1 (E,R), and for anyu,vE,

I ( u ) , v = R N ( Δ u Δ v + u v + V ( x ) u v ) dx R N f(x,u)vdx.
(2.1)

Moreover, Ψ :E E is compact, whereΨ(u)= R N F(x,u)dx.

Recall that we say I satisfies the (PS) condition at the level cR ((PS) c condition for short) if any sequence { u n }E along with I( u n )c and I ( u n )0 as n possesses a convergent subsequence. If I satisfies the (PS) c condition for each cR, then we say that I satisfies the (PS) condition.

Lemma 2.3

Let assumption (V1) and (f1) hold. Then any bounded Palais-Smale sequence of I has a strongly convergent subsequence in E.

Proof

Let { u n }E be any bounded Palais-Smale sequence of I. Then, up to a subsequence, there exists cR such that

I( u n )c, I ( u n )0and sup n u n E <+.
(2.2)

Since the embedding

E L s ( R N ) ,2s< 2

is compact, going if necessary to a subsequence, we can assume that there is a uE such that

{ u n u , weakly in  E ; u n u , strongly in  L s ( R N ) ; u n ( x ) u ( x ) , a.e. in  R N .
(2.3)

In view of (2.1), one has

I ( u n ) I ( u ) , u n u = R N ( Δ u n Δ ( u n u ) + u n ( u n u ) + V ( x ) u n ( u n u ) ) d x R N f ( x , u n ) ( u n u ) d x R N ( Δ u Δ ( u n u ) u ( u n u ) V ( x ) u ( u n u ) ) d x + R N f ( x , u ) ( u n u ) d x = u n u E 2 R N [ f ( x , u n ) f ( x , u ) ] ( u n u ) d x .
(2.4)

By (f1), using the Hölder inequality, we can conclude

| R N [ f ( x , u n ) f ( x , u ) ] ( u n u ) d x | C 1 R N [ | u n | + | u | + | u n | p 1 + | u | p 1 ] | u n u | d x C 1 ( u n L 2 + u L 2 ) u n u L 2 + C 1 ( u n L p p 1 + u L p p 1 ) u n u L p .

Therefore, it follows from (2.3) that

R N [ f ( x , u n ) f ( x , u ) ] ( u n u)dx0,as n.
(2.5)

Moreover, combining (2.2) with (2.3), then

I ( u n ) I ( u ) , u n u 0,as n.
(2.6)

Consequently, (2.4), (2.5) together with (2.6) imply that

u n u, strongly in E as n.

This completes the proof. □

Lemma 2.4

Let assumptions (V1), (f1), and (f3) hold. Then any Palais-Smale sequence of I is bounded.

Proof

Let { u n }E be any Palais-Smale sequence of I, then, up to a subsequence, there exists c 1 R such that

I( u n ) c 1 ,and I ( u n )0.
(2.7)

The combination of (1.7), (1.8), (2.1), (2.7), (V1) with (f3) implies

c 1 + 1 + u n E I ( u n ) 1 μ I ( u n ) , u n = μ 2 2 μ R N ( | Δ u n | 2 + | u n | 2 d x + R N V ( x ) u n 2 ) d x + R N F ˜ ( x , u n ) d x μ 2 2 μ R N ( | Δ u n | 2 + | u n | 2 d x + R N V ( x ) u n 2 ) d x a μ R N u n 2 d x + A n F ˜ ( x , u n ) d x μ 2 2 μ R N ( | Δ u n | 2 + | u n | 2 d x + R N V ( x ) u n 2 ) d x a μ V 0 R N V ( x ) u n 2 d x + A n F ˜ ( x , u n ) d x μ 2 4 μ u n E 2 + ( μ 2 4 μ a μ V 0 ) R N V ( x ) u n 2 d x + A n F ˜ ( x , u n ) d x = μ 2 4 μ u n E 2 + ( μ 2 ) V 0 4 a 4 μ V 0 R N V ( x ) u n 2 d x + A n F ˜ ( x , u n ) d x ,
(2.8)

where F ˜ (x, u n )= 1 μ f(x, u n ) u n F(x, u n ) and A n ={x R N :| u n |L}. For x R N and | u n |L, by (f1), one has

| F ˜ ( x , u n ) | 1 μ | f ( x , u n ) | | u n | + | F ( x , u n ) | 1 μ C 1 ( | u n | 2 + | u n | p ) + C 1 ( 1 2 | u n | 2 + 1 p | u n | p ) C 1 ( 1 + | u n | p 2 ) | u n | 2 C 1 ( 1 + L p 2 ) | u n | 2 .

Take M>max{ 4 C 1 ( 1 + L p 2 ) μ V 0 ( μ 2 ) V 0 4 a , V 0 }, then

F ˜ (x, u n ) ( μ 2 ) V 0 4 a 4 μ V 0 M | u n | 2 ,x R N ,| u n |L.
(2.9)

Let A ˜ ={x R N :V(x)M}, R(n)= ( μ 2 ) V 0 4 a 4 μ V 0 R N V(x) u n 2 dx+ A n F ˜ (x, u n )dx. By (V1) and (2.9), we can conclude

R ( n ) ( μ 2 ) V 0 4 a 4 μ V 0 | u n | L ( V ( x ) M ) | u n | 2 d x ( μ 2 ) V 0 4 a 4 μ V 0 A ˜ A n ( V ( x ) M ) L 2 d x ( μ 2 ) V 0 4 a 4 μ V 0 ( V 0 M ) L 2 meas ( A ˜ A n ) ( μ 2 ) V 0 4 a 4 μ V 0 ( V 0 M ) L 2 meas ( A ˜ ) .
(2.10)

Note that meas( A ˜ )<+ due to (V1), and it follows from (2.8) and (2.10) that

c 1 + 1 + u n E μ 2 4 μ u n E 2 + R ( n ) μ 2 4 μ u n E 2 + ( μ 2 ) V 0 4 a 4 μ V 0 ( V 0 M ) L 2 meas ( A ˜ ) ,

which implies { u n }E is bounded in E. Hence the proof is completed. □

Remark 3

We present a new proof technique to verify the boundedness of Palais-Smale sequences, which is much clearer and simpler than the previous literature.

3 Proof of Theorem 1.1

In this section we will use the classical symmetric mountain pass theorem of Rabinowitz instead of the fountain theorem in [26] to obtain high energy solutions for the problem (1.1) and prove Theorem 1.1. First of all, we present some notations.

In view of E L 2 ( R N ) and L 2 ( R N ) being a separable Hilbert space, E has a countable orthogonal basis { e j } j = 1 . Let

E k = j = 1 k X j , Z k = E k ,

where X j =span{ e j }. Thus, E= E k Z k and E k is finite dimensional.

Lemma 3.1

Let the assumption (V1) hold. Define

η(k):= sup u Z k , u E = 1 u L 2 ,kN,

then there exists k 0 Nsuch that0<η( k 0 ) ( 1 2 C 1 ) 1 2 .

Proof

Firstly, η(k) is convergent since η(k)0 and η(k) is decreasing in k. Furthermore, for any kN, by the definition of η(k), there exists u k Z k such that

u k E =1and u k L 2 > η ( k ) 2 .
(3.1)

For any vE, v= n = 1 a n e n , it has

| u k , v E | = | u k , n = 1 a n e n E | u k E n = k + 1 a n e n E n = k + 1 a n e n E 0

as k, which implies that u k 0 weakly in E. By virtue of Lemma 2.1, we can conclude

u k 0strongly in  L 2 ( R N ) .
(3.2)

The combination (3.1) with (3.2) implies that η(k)0, as k. Then there exists k 0 N such that 0<η( k 0 ) ( 1 2 C 1 ) 1 2 . Hence the proof is completed. □

Lemma 3.2

Let assumptions (V1) and (f1) hold, then there exist some constants ρ, α such thatI(u)αwheneveru Z k 0 with u E =ρ.

Proof

For any u Z k 0 , by Lemma 3.1, we have

u L 2 η( k 0 ) u E and0<η( k 0 ) ( 1 2 C 1 ) 1 2 .
(3.3)

By (1.7), (1.8), (f1), (3.3), and the Hölder inequality, one has

I ( u ) = 1 2 u E 2 R N F ( x , u ) d x 1 2 u E 2 C 1 2 ( u L 2 2 + u L p p ) 1 2 u E 2 C 1 2 η 2 ( k 0 ) u E 2 C 1 2 a p p u E p 1 2 u E [ 1 2 u E C 1 a p p u E p 1 ] .

Set

l(t)= 1 2 t C 1 a p p t p 1 ,t0.

Note that p>2, we can conclude that there exists a constant ρ>0 such that

l(ρ)= max t 0 l(t)>0.

Therefore,

I(u) 1 2 ρl(ρ)=:α>0,

whenever u Z k 0 with u E =ρ. This completes the proof. □

Lemma 3.3

Let assumptions (f1)-(f2) hold, then for each finite dimensional subspace E ˜ E, there is anr=r( E ˜ )>0such thatI | E ˜ B r <0.

Proof

By the assumptions (f1)-(f2), there exist two positive constants C 2 and C 3 such that

F(x,z) C 2 | z | θ C 3 | z | 2 ,(x,z) R N ×R.
(3.4)

For any finite dimensional subspace E ˜ E, by the equivalence of norms in the finite dimensional space, there exists a constant β s >0 such that

u L s β s u E ,u E ˜ ,
(3.5)

for 2s< 2 . Therefore, the combination of (1.7)-(1.8) with (3.4)-(3.5) implies

I ( u ) = 1 2 u E 2 R N F ( x , u ) d x 1 2 u E 2 C 2 u L θ θ + C 3 u L 2 2 1 2 u E 2 C 2 β θ θ u E θ + C 3 a 2 2 u E 2

for all u E ˜ . Note that θ>2, hence there is an r=r( E ˜ )>0 such that I | E ˜ B r <0. This completes the proof. □

Next, we shall prove our Theorem 1.1. To begin with, for convenience of notation, we state the classical symmetric mountain pass theorem as follows.

Theorem 3.4

([32], Theorem 9.12])

Let E be an infinite dimensional Banach space, and letI C 1 (E,R)be even and satisfy the (PS) condition andI(0)=0. IfE=YZ, Y is finite dimensional, and I satisfies

(I1):there exist constantsρ,α>0such thatI | B ρ Z α, and

(I2):for each finite dimensional subspace E ˜ E, there isr=r( E ˜ )such thatI0on E ˜ B r ,

then I possesses an unbounded sequence of critical values.

Proof of Theorem 1.1

The proof is to verify I satisfies all the conditions of Theorem 3.4. Set Y= E k 0 , Z= Z k 0 , then E=YZ and Y is finite dimensional. First, I satisfies (I1) and (I2) in Theorem 3.4 by Lemmas 3.2 and 3.3, respectively. Second, I satisfies the (PS) condition by virtue of Lemmas 2.3 and 2.4. Finally, I(0)=0, I is even on E due to (f4) and I C 1 (E,R) by Lemma 2.2. Hence, the conclusion directly follows from Theorem 3.4. The proof is completed. □

Remark 4

Compared with Theorem A (see [26], Theorem 1.1]), on one hand, the assumptions imposed on f are much weaker. On the other hand, we present a new proof technique to verify the boundedness of Palais-Smale sequences, and we apply the classical symmetric mountain pass theorem of Rabinowitz instead of the fountain theorem in [26] to obtain high energy solutions for the problem (1.1). Hence, it is very different.

References

  1. Lazer AC, Mckenna PJ: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 1990, 32(4):537-578. 10.1137/1032120

    Article  MathSciNet  Google Scholar 

  2. Lazer AC, Mckenna PJ: Global bifurcation and a theorem of Tarantello. J. Math. Anal. Appl. 1994, 181(3):648-655. 10.1006/jmaa.1994.1049

    Article  MathSciNet  Google Scholar 

  3. Tarantella G: A note on a semilinear elliptic problem. Differ. Integral Equ. 1992, 5(3):561-565.

    Google Scholar 

  4. Mckenna PJ, Walter W: Traveling waves in a suspension bridge. SIAM J. Appl. Math. 1990, 50: 703-715. 10.1137/0150041

    Article  MathSciNet  Google Scholar 

  5. Chen Y, Mckenna PJ: Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations. J. Differ. Equ. 1997, 135: 325-355. 10.1006/jdeq.1996.3155

    Article  MathSciNet  Google Scholar 

  6. Pao CV: On fourth-order elliptic boundary value problems. Proc. Am. Math. Soc. 2000, 128: 1023-1030. 10.1090/S0002-9939-99-05430-1

    Article  MathSciNet  Google Scholar 

  7. Ahmed NU, Harbi H: Mathematical analysis of dynamic models of suspension bridges. SIAM J. Appl. Math. 1998, 58: 853-874. 10.1137/S0036139996308698

    Article  MathSciNet  Google Scholar 

  8. Micheletti AM, Pistoia A: Multiplicity results for a fourth-order semilinear elliptic problem. Nonlinear Anal. 1998, 31(7):895-908. 10.1016/S0362-546X(97)00446-X

    Article  MathSciNet  Google Scholar 

  9. Micheletti AM, Pistoia A: Nontrivial solutions for some fourth order semilinear elliptic problems. Nonlinear Anal. 1998, 34(4):509-523. 10.1016/S0362-546X(97)00596-8

    Article  MathSciNet  Google Scholar 

  10. Zhang J: Existence results for some fourth-order nonlinear elliptic problems. Nonlinear Anal. 2001, 45(1):29-36. 10.1016/S0362-546X(99)00328-4

    Article  MathSciNet  Google Scholar 

  11. Xu G, Zhang J: Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity. J. Math. Anal. Appl. 2003, 281(2):633-640. 10.1016/S0022-247X(03)00170-7

    Article  MathSciNet  Google Scholar 

  12. Zhang J, Li S: Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. 2005, 60(2):221-230. 10.1016/j.na.2004.07.047

    Article  MathSciNet  Google Scholar 

  13. Wang W, Zang A, Zhao P: Multiplicity of solutions for a class of fourth-order elliptic equations. Nonlinear Anal. 2009, 70(12):4377-4385. 10.1016/j.na.2008.10.020

    Article  MathSciNet  Google Scholar 

  14. Zhou J, Wu X: Sign-changing solutions for some fourth-order nonlinear elliptic problems. J. Math. Anal. Appl. 2008, 342(1):542-558. 10.1016/j.jmaa.2007.12.020

    Article  MathSciNet  Google Scholar 

  15. Wang Y, Shen Y: Infinitely many sign-changing solutions for a class of biharmonica equation without symmetry. Nonlinear Anal. 2009, 71: 967-977. 10.1016/j.na.2008.11.052

    Article  Google Scholar 

  16. Yang Y, Zhang J: Existence of solutions for some fourth-order nonlinear elliptic problems. J. Math. Anal. Appl. 2009, 351(1):128-137. 10.1016/j.jmaa.2008.08.023

    Article  MathSciNet  Google Scholar 

  17. Liu X, Huang Y: On sign-changing solutions for a fourth-order asymptotically linear elliptic problems. Nonlinear Anal. 2010, 72(5):2271-2276. 10.1016/j.na.2009.11.001

    Article  MathSciNet  Google Scholar 

  18. Qian AX, Li SJ: Multiple solutions for a fourth-order asymptotically linear elliptic problems. Acta Math. Sin. Engl. Ser. 2006, 22(4):1121-1126. 10.1007/s10114-005-0665-7

    Article  MathSciNet  Google Scholar 

  19. An Y, Liu R: Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equations. Nonlinear Anal. 2008, 68(11):3325-3331. 10.1016/j.na.2007.03.028

    Article  MathSciNet  Google Scholar 

  20. Zhang J, Wei Z: Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems. Nonlinear Anal. 2011, 74: 7474-7485. 10.1016/j.na.2011.07.067

    Article  MathSciNet  Google Scholar 

  21. Zhang J, Wei Z: Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin. J. Math. Anal. Appl. 2011, 383: 291-306. 10.1016/j.jmaa.2011.05.030

    Article  MathSciNet  Google Scholar 

  22. Lü D: Multiple solutions for a class of biharmonic elliptic system with Sobolev critical exponent. Nonlinear Anal. 2011, 74(17):6371-6382. 10.1016/j.na.2011.06.018

    Article  MathSciNet  Google Scholar 

  23. Chabrowski J, Marcos do Ó J:On some fourth-order semilinear elliptic problems in R N . Nonlinear Anal. 2002, 49: 861-884. 10.1016/S0362-546X(01)00144-4

    Article  MathSciNet  Google Scholar 

  24. Pimenta MTO, Soares SHM: Existence and concentration of solutions for a class biharmonic equation. J. Math. Anal. Appl. 2012, 390: 274-289. 10.1016/j.jmaa.2012.01.039

    Article  MathSciNet  Google Scholar 

  25. Yin Y, Wu X: High energy solutions and nontrivial solutions for fourth-order elliptic equations. J. Math. Anal. Appl. 2011, 375: 699-705. 10.1016/j.jmaa.2010.10.019

    Article  MathSciNet  Google Scholar 

  26. Ye Y, Tang C: Infinitely many solutions for fourth-order elliptic equations. J. Math. Anal. Appl. 2012, 394: 841-854. 10.1016/j.jmaa.2012.04.041

    Article  MathSciNet  Google Scholar 

  27. Lin J, Chen S, Wu X:Existence and multiplicity of solutions for a class of fourth-order elliptic equations in R N . J. Math. Anal. Appl. 2012, 395: 608-615. 10.1016/j.jmaa.2012.05.063

    Article  MathSciNet  Google Scholar 

  28. Ye Y, Tang C:Existence and multiplicity of solutions for fourth-order elliptic equations in R N . J. Math. Anal. Appl. 2013, 406: 335-351. 10.1016/j.jmaa.2013.04.079

    Article  MathSciNet  Google Scholar 

  29. Zhang W, Tang X, Zhang J: Infinitely many solutions for fourth-order elliptic equations with general potentials. J. Math. Anal. Appl. 2013, 407: 359-368. 10.1016/j.jmaa.2013.05.044

    Article  MathSciNet  Google Scholar 

  30. Bartsch T, Wang ZQ:Existence and multiplicity results for some superlinear elliptic problems on R N . Commun. Partial Differ. Equ. 1995, 20: 1725-1741. 10.1080/03605309508821149

    Article  MathSciNet  Google Scholar 

  31. Zou WM, Schechter M: Critical Point Theory and Its Applications. Springer, New York; 2006.

    Google Scholar 

  32. Rabinowitz PH: Minimax Methods in Critical Point Theory with Application to Differential Equations. Am. Math. Soc., Providence; 1986.

    Google Scholar 

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Acknowledgements

This work is partly supported the Foundation of Education of Commission of Yunnan Province (2014Z153, 2012Z123C), and the Youth Program of Yunnan Provincial Science and Technology Department (2013FD046).

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Correspondence to Bitao Cheng.

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Cheng, B. High energy solutions for the fourth-order elliptic equations in R N . Bound Value Probl 2014, 199 (2014). https://doi.org/10.1186/s13661-014-0199-y

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