1 Introduction and main result

This paper is concerned with the following biharmonic equations:

{ Δ 2 u Δ u + V ( x ) u = f ( x , u ) + λ ξ ( x ) | u | p 2 u , x R N , u H 2 ( R N ) ,
(1)

where Δ 2 :=Δ(Δ) is the biharmonic operator, VC( R N ), fC( R N ×R), ξ L 2 2 p ( R N ), λ>0, and 1p<2. There are many results for biharmonic equations, but most of them are on bounded domains; see [1]–[5]. In addition, biharmonic equations on unbounded domains also have captured a lot of interest; see [6]–[11] and the references therein. Many of these papers are devoted to the study of the existence and multiplicity of solutions for problem (1). In [6], [7], [9], [11], the authors considered the superlinear case; one considered the sublinear case in [8]–[10]. However, there are not many works focused on the asymptotically linear case. Motivated by the above facts, in the present paper, we shall study problem (1) with mixed nonlinearity, that is, a combination of superlinear and sublinear terms, or asymptotically linear and sublinear terms. So, the aim of the present paper is to unify and generalize the results of the above papers to a more general case. To the best of our knowledge, there have been no works concerning this case up to now, hence this is an interesting and new research problem. For related results, we refer the readers to [12]–[14] and the references therein.

More precisely, we make the following assumptions:

(V): VC( R N ,R) and inf R N V(x)>0, and there exists a constant l 0 >0 such that

lim | y | meas ( { x R N : | x y | l 0 , V ( x ) M } ) =0,M>0,

where meas() denotes the Lebesgue measure in R N ;

(F1): f(x,u)C( R N ×R,R), such that f(x,u)0 for all u<0 and x R N . Moreover, there exists b L ( R N , R + ) with | b | < 1 2 γ 2 2 γ 0 2 such that

lim | u | 0 + f ( x , u ) u =b(x)uniformly in x R N

and

f ( x , u ) u k b(x)for all u>0 and x R N ,

where γ 2 , γ 0 are defined in (3);

(F2): there exists q L ( R N , R + ) with | q | > 1 γ 2 2 γ 0 2 such that

lim | u | f ( x , u ) u k =q(x)uniformly in x R N ;

(F3): there exist two constants θ, d 0 satisfying θ>2 and 0 d 0 < θ 2 2 θ γ 2 2 γ 0 2 such that

F(x,u) 1 θ f(x,u)u d 0 u 2 for all u>0 and x R N ,

where F(x,u)= 0 u f(x,s)ds.

Before stating our result, we denote ξ ± =max{±ξ,0}. The main result of this paper is the following theorem.

Theorem 1.1

Suppose that (V), (F1)-(F3) are satisfied. ξ L 2 2 p ( R N ){0}with ξ + 0. In addition, for any real numberk1:

(I1): Ifk=1and μ <1with

μ =inf { R N ( | Δ u | 2 + | u | 2 + V ( x ) | u | 2 ) d x | u H 2 ( R N ) , R N q ( x ) u 2 d x = 1 } ,
(2)

then there exists Λ 0 >0such that, for every0<λ< Λ 0 , problem (1) has at least two solutions;

(I2): Ifk>1, then there exists Λ 0 >0such that, for every0<λ< Λ 0 , problem (1) has at least two solutions.

Remark 1.2

It is easy to check that f(x,u) is asymptotically linear at infinity in u when k=1 and f(x,u) is superlinear at infinity in u when k>1. Together with λ>0 and 1q<2, we see easily that our nonlinearity is a more general mixed nonlinearity, that is, a combination of sublinear, superlinear, and asymptotically linear terms. Therefore, our result unifies and sharply improves some recent results.

2 Variational setting and proof of the main result

Now we establish the variational setting for our problem (1). Let

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

equipped with the inner product

(u,v)= R N ( Δ u Δ v + u v + V ( x ) u v ) dx,u,vE,

and the norm

u= ( R N ( | Δ u | 2 + | u | 2 + V ( x ) | u | 2 ) d x ) 1 2 ,uE.

Lemma 2.1

([15])

Under assumptions (V), the embeddingE L s ( R N )is compact for anys[2, 2 ), where 2 = 2 N N 4 ifN5, 2 =ifN<5.

Clearly, E is continuously embedded into H 2 ( R N ) and from Lemma 2.1, there exist γ s >0 and γ 0 >0 such that

u s γ s u H 2 ( R N ) γ s γ 0 u,uE,2s< 2 .
(3)

Now, on E we define the following functional:

Φ(u)= 1 2 R N ( | Δ u | 2 + | u | 2 + V ( x ) u 2 ) dx R N F(x,u)dx λ p R N ξ(x) | u | p dx.
(4)

By a standard argument, it is easy to verify that Φ C 1 (E,R) and

Φ ( u ) , v = R N [ Δ u Δ v + u v + V ( x ) u v ] d x R N f ( x , u ) v d x λ R N ξ ( x ) | u | p 2 u v d x
(5)

for all u,vE.

Lemma 2.2

μ >0, and this is achieved by some ϕ 1 H 2 ( R N )with R N q ϕ 1 2 dx=1, where μ is given in (2).

Proof

By Lemma 2.1 and standard arguments, it is easy to prove this lemma, so we omit the proof here. □

Next, we give a useful theorem. It is the variant version of the mountain pass theorem, which allows us to find a ( C ) c sequence.

Theorem 2.3

([16])

Let E be a real Banach space, with dual space E , and suppose thatΦ C 1 (E,R)satisfies

max { Φ ( 0 ) , Φ ( e ) } μ<η inf u = ρ Φ(u),

for someμ<η, ρ>0andeEwithe>ρ. Let c ˆ ηbe characterized by

c ˆ = inf β Γ max 0 τ 1 Φ ( β ( τ ) ) ,

whereΓ={βC([0,1],E):β(0)=0,β(1)=e}is the set of continuous paths joining 0 and e, then there exists a sequence{ u n }Esuch that

Φ( u n ) c ˆ ηand ( 1 + u n ) Φ ( u n ) E 0as n.

Lemma 2.4

For any real numberk1, assume that (F1) and (F2) are satisfied, andξ L 2 2 p ( R N ){0}with ξ + 0. Then there exists Λ 0 >0such that, for everyλ(0, Λ 0 ), there exist two positive constants ρ, η such thatΦ(u) | u = ρ η>0.

Proof

For any ε>0, it follows from the conditions (F1) and (F2) that there exist C ε >0 and max{2,k}<r< 2 such that

F(x,u) | b | + ε 2 | u | 2 + C ε r | u | r ,for all uE.
(6)

Thus, from (3), (6), and the Sobolev inequality, we have, for all uE,

R N F ( x , u ) d x | b | + ε 2 R N u 2 d x + C ε r R N | u | r d x ( | b | + ε ) γ 2 2 γ 0 2 2 u 2 + C ε γ r r γ 0 r r u r ,

which implies that

Φ ( u ) = 1 2 u 2 R N F ( x , u ) d x λ p R N ξ ( x ) | u | p d x 1 2 u 2 ( | b | + ε ) γ 2 2 γ 0 2 2 u 2 C ε γ r r γ 0 r r u r λ γ 2 p γ 0 p p ξ 2 2 p u p = u p [ 1 2 ( 1 ( | b | + ε ) γ 2 2 γ 0 2 ) u 2 p C ε γ r r γ 0 r r u r p λ γ 2 p γ 0 p p ξ 2 2 p ] .
(7)

Take ε= 1 2 γ 2 2 γ 0 2 | b | and define

g(t)= 1 4 t 2 p C ε γ r r γ 0 r r t t p ,for t0.

It is easy to prove that there exists ρ>0 such that

max t 0 g(t)=g(ρ)= r 2 4 ( r p ) [ ( 2 p ) r 4 C ε γ r r γ 0 r ( r p ) ] 2 p r 2 .

Then it follows from (7) that there exists Λ 0 >0 such that, for every λ(0, Λ 0 ), there exists η>0 such that Φ(u) | u = ρ η. □

Lemma 2.5

For any real numberk1, assume that (F1), (F2) are satisfied, andξ L 2 2 p ( R N ){0}with ξ + 0. Letρ, Λ 0 >0be as in Lemma  2.4. Then we have the following results:

  1. (i)

    If k=1 and μ <1, then there exists eE with e>ρ such that Φ(e)<0 for all λ(0, Λ 0 );

  2. (ii)

    if k>1, then there exists eE with e>ρ such that Φ(e)<0 for all λ(0, Λ 0 ).

Proof

  1. (i)

    In the case k=1, since μ <1, we can choose a nonnegative function φE with

    R N q(x) φ 2 dx=1such that R N ( | Δ φ | 2 + | φ | 2 + V ( x ) | φ | 2 ) dx<1.

Therefore, from (F2) and Fatou’s lemma, we have

lim t + Φ ( t φ ) t 2 = 1 2 φ 2 lim t + R N F ( x , t φ ) t 2 φ 2 φ 2 d x lim t + λ p t 2 p R N ξ ( x ) | φ | p d x 1 2 φ 2 1 2 R N q ( x ) φ 2 d x = 1 2 ( φ 2 1 ) < 0 .

So, if Φ(tφ) as t+, then there exists eE with e>ρ such that Φ(e)<0.

  1. (ii)

    In the case k>1, since q L ( R N , R + ) with q + 0, we can choose a nonnegative function ωE such that R N q(x) ω k + 1 dx>0. Thus, from (F2) and Fatou’s lemma, we have

    lim t + Φ ( t ω ) t k + 1 = lim t + ω 2 2 t k 1 lim t + R N F ( x , t ω ) t k + 1 ω k + 1 ω k + 1 d x lim t + λ p t k + 1 p R N ξ ( x ) | ω | p d x 1 k + 1 R N q ( x ) ω k + 1 d x < 0 .

So, if Φ(tω) as t+, then there exists eE with e>ρ such that Φ(e)<0. This completes the proof. □

Next, we define

α= inf β Γ max 0 t 1 Φ ( β ( t ) ) ,

where Γ={βC([0,1],E):β(0)=0,β(1)=e}. Then by Theorem 2.3 and Lemmas 2.4 and 2.5, there exists a sequence { u n }E such that

Φ( u n )α>0and ( 1 + u n ) Φ ( u n ) E 0as n.
(8)

Lemma 2.6

For any real numberk1, assume that (V) and (F1)-(F3) are satisfied, andξ L 2 2 p ( R N ){0}with ξ + 0. Let Λ 0 >0be as in Lemma  2.4. Then{ u n }defined by (8) is bounded in E for allλ(0, Λ 0 ).

Proof

For n large enough, from (F2), (3), the Hölder inequality, and Lemma 2.4, we have

α + 1 Φ ( u n ) 1 θ Φ ( u n ) , u n = ( 1 2 1 θ ) u n 2 R N [ F ( x , u n ) 1 θ f ( x , u n ) u n ] d x λ ( 1 p 1 θ ) R N ξ ( x ) | u n | p d x θ 2 2 θ u n 2 d 0 R N u n 2 d x λ ( θ p ) p θ ξ 2 2 p u n 2 p θ 2 2 θ u n 2 d 0 γ 2 2 γ 0 2 u n 2 λ ( θ p ) γ 2 p γ 0 p p θ ξ 2 2 p u n p > ( θ 2 2 θ d 0 γ 2 2 γ 0 2 ) u n 2 λ ( θ p ) γ 2 p γ 0 p p θ ξ 2 2 p u n p ,

which implies that { u n } is bounded in E since 1p<2. □

Lemma 2.7

For any real numberk1, assume that (V) and (F1)-(F2) are satisfied, andξ L 2 2 p ( R N ){0}with ξ + 0. Let Λ 0 >0be as in Lemma  2.4. Then for everyλ(0, Λ 0 ), there exists u 0 Esuch that

Φ( u 0 )=inf { Φ ( u ) : u B ¯ ρ } <0

and u 0 is a nontrivial solution of problem (1).

Proof

Since ξ L 2 2 p ( R N ){0} with ξ + 0, we can choose a function ϕE such that

R N ξ(x) | ϕ | p dx>0.
(9)

By (9), for t>0, we have

Φ ( t ϕ ) = t 2 2 ϕ 2 R N F ( x , t ϕ ) d x λ t p p R N ξ ( x ) | ϕ | p d x t 2 2 ϕ 2 λ t p p R N ξ ( x ) | ϕ | p d x , for  t > 0  small enough .

Hence, θ 0 :=inf{Φ(u):u B ¯ ρ }<0. By Ekeland’s variational principle, there exists a minimizing sequence { u n } B ¯ ρ such that Φ( u n ) θ 0 and Φ ( u n )0 as n. Hence, Lemma 2.1 implies that there exists u 0 E such that Φ ( u 0 )=0 and Φ( u 0 )= θ 0 <0. □

Proof of Theorem 1.1

From Lemmas 2.1 and 2.6, there exists a constant u ˜ E such that, up to a subsequence,

u n u ˜ in E, u n u ˜ in  L s ( R N )  for s[2, 2 ).

By using a standard procedure, we can prove that u n u ˜ in E. Moreover, Φ( u ˜ )=α>0 and u ˜ is another nontrivial solution of problem (1). Therefore, combining with Lemma 2.7, we can prove that problem (1) has at least two nontrivial solutions u 0 , u ˜ E satisfying Φ( u 0 )<0 and Φ( u ˜ )>0. □