In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:

{ − div ( a ( | ∇ u | ) ∇ u ) = f ( x , u ) , in  Ω ; u = 0 , on  ∂ Ω ,

where Ω⊂RN is a bounded domain with a smooth boundary ∂Ω. The function a is such that p : R → R defined by

p ( t ) = { a ( | t | ) t , t ≠ 0 ; 0 , t = 0 ,

is an increasing homeomorphism from R onto itself and the continuous function f ( x , t ) ∈ C ( Ω ¯ × R , R ) satisfies f ( x , 0 ) = 0 , x ∈ Ω ¯ . Especially, when a ( t ) = | t | p − 2 , the problem (1.1) is the well-known p-Laplacian equation. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem (1.1) possesses more complicated nonlinearities. For example, it is inhomogeneous and has an important physical background, e.g.,

(a) nonlinear elasticity: P ( t ) = ( 1 + t 2 ) γ − 1 , γ > 1 2 ;

(b) plasticity: P ( t ) = t α ( log ( 1 + t ) ) β , α ≥ 1 , β > 0 ;

(c) generalized Newtonian fluids: P ( t ) = ∫ 0 t s 1 − α ( sinh − 1 s ) β d s , 0 ≤ α ≤ 1 , β>0.

So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers 123456789. In paper 9, Fang and Tan discussed the problem (1.1) under the conditions that f ( x , t ) was odd in t. They got the first result that when h + < p − , and f ( x , t ) ≥ C t q − 1 for 0 < t < δ , q < p − , the problem (1.1) had a sequence of solutions by genus theory. The second result is that when f(x,t) satisfies 0 < α F ( x , t ) ≤ t f ( x , t ) , ∀ x ∈ Ω ¯ , t ≠ 0 , α > p + and f ( x , t ) = o ( p ( | t | ) ) as | t | → 0 , the problem (1.1) has infinitely many pairs of solutions which correspond to the positive critical values by the symmetric mountain pass theorem.

Motivated by their results, in this note, we discuss the problem (1.1) when f(x,t) is still odd in t but it satisfies weaker conditions than 9; and furthermore, we need not know the behaviors of f(x,t) near the zero. If h + > p − , we can get multiplicity of solutions by a version of the symmetric mountain pass theorem.

The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.

Obviously, the problem (1.1) allows a nonhomogeneous function p in the differential operator defining the problem (1.1). To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem (1.1) as follows.

Let

P ( t ) = ∫ 0 t p ( s ) d s , P ˜ ( t ) = ∫ 0 t p − 1 ( s ) d s , t ∈ R ,

then P and P ˜ are complementary N-functions (see 10), which define the Orlicz spaces L P : = L P ( Ω ) and L P ˜ : = L P ˜ ( Ω ) respectively.

Throughout this paper, we assume the following condition on P:

( p ) 1 < p − : = inf t > 0 t p ( t ) P ( t ) ≤ p + : = sup t > 0 t p ( t ) P ( t ) < + ∞ .

Under the condition (p), the Orlicz space L P coincides with the set (equivalence classes) of measurable functions u : Ω → R such that

∫ Ω P ( | u | ) d x < + ∞ ,

and is equipped with the (Luxemburg) norm, i.e.,

| u | P : = inf { k > 0 : ∫ Ω P ( | u | k ) d x < 1 } .

We will denote by W 1 , P ( Ω ) the corresponding Orlicz-Sobolev space with the norm

∥ u ∥ W 1 , P ( Ω ) : = | u | P + ∥ ∇ u ∥ P

and define W 0 1 , P ( Ω ) as the closure of C 0 ∞ in W1,P(Ω). In this note, we will use the following equivalent norm on W01,P(Ω):

∥ u ∥ : = inf { k > 0 : ∫ Ω P ( | ∇ u | k ) d x < 1 } .

Now, we introduce the Orlicz-Sobolev conjugate P ∗ of P, which is given by

P ∗ − 1 ( t ) : = ∫ 0 t p − 1 ( τ ) τ N + 1 N d τ ,

where we suppose that

lim t → 0 ∫ t 1 p − 1 ( τ ) τ N + 1 N d τ < + ∞ , lim t → ∞ ∫ 1 t p − 1 ( τ ) τ N + 1 N d τ = + ∞ .

Let p ∗ − : = inf t > 0 t P ∗ ′ ( t ) P ∗ ( t ) , p ∗ + : = sup t > 0 t P ∗ ′ ( t ) P ∗ ( t ) . Throughout this paper, we assume that p + < p ∗ − . Now, we will make the following assumptions on f(x,t).

( f ∗ ) There exists an odd increasing homeomorphism h from R to R, and nonnegative constants c 1 , c 2 such that

| f ( x , t ) | ≤ c 1 + c 2 h ( | t | ) , ∀ t ∈ R , ∀ x ∈ Ω ¯ ,

and lim t → + ∞ H ( t ) P ∗ ( k t ) = 0 , ∀ k > 0 , where

H ( t ) : = ∫ 0 t h ( s ) d s .

Let

H ˜ ( t ) : = ∫ 0 t h − 1 ( s ) d s ,

then we can obtain complementary N-functions which define corresponding Orlicz spaces L H and L H ∗ .

Similar to the condition (p), we also assume the following condition on H:

( h ) 1 < h − : = inf t > 0 t h ( t ) H ( t ) ≤ h + : = sup t > 0 t h ( t ) H ( t ) < + ∞ .

In order to prove our results, we now state some useful lemmas.

Lemma 2.1 10

Under the condition (p), the spaces L P ( Ω ) , W01,P(Ω) and W1,P(Ω) are separable and reflexive Banach spaces.

Lemma 2.2 10

Under the condition (f∗), the embedding W 0 1 , P ( Ω ) ↪ L H ( Ω ) is compact.

Lemma 2.3 2

Let ρ ( u ) = ∫ Ω P ( u ) d x , we have

(1) if | u | P < 1 , then | u | P p + ≤ ρ ( u ) ≤ | u | P p − ;

(2) if | u | P > 1 , then | u | P p − ≤ ρ ( u ) ≤ | u | P p + ;

(3) if 0 < t < 1 , then t p + P ( u ) ≤ P ( t u ) ≤ t p − P ( u ) ;

(4) if t > 1 , then t p − P ( u ) ≤ P ( t u ) ≤ t p + P ( u ) .

Lemma 2.4 111213

Let E = V + X , where E is a real Banach space and V is finite dimensional. Suppose I ∈ C 1 ( E , R ) is an even functional satisfying I ( 0 ) = 0 and

( I 1 ) there is a constant ρ > 0 such that I | ∂ B ρ ∩ X ≥ 0 ;

( I 2 ) there is a subspace W of E with dim V < dim W < ∞ and there is M > 0 such that max u ∈ W I ( u ) < M ;

( I 3 ) considering M>0 given by (I2), I satisfies (PS)_c for 0 ≤ c ≤ M .

Then I possesses at least dim W − dim V pairs of nontrivial critical points.

Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.

Theorem 2.1 Assume that f(x,t) is odd in t, satisfies (f∗) with p − < h + ≤ p + and the following assumptions:

( f 1 ) there exist η > p + and 1 < σ < p − , and a 1 , a 2 > 0 , such that 1 η f ( x , t ) t − F ( x , t ) ≥ − a 1 − a 2 | t | σ for every t ∈ R , a.e. in Ω.

( f 2 ) there is Ω 0 ⊂ Ω with | Ω 0 | > 0 such that lim inf | t | → ∞ F ( x , t ) / | t | p + = ∞ uniformly a.e. in Ω 0 .

Then for any given k ∈ N , the problem (1.1) possesses at least k pairs of nontrivial solutions.

In this section, we assume that N ≥ 1 and E = W 0 1 , P ( Ω ) , u ∈ E is called a weak solution of the problem (1.1) if

∫ Ω a ( | ∇ u | ) ∇ u ∇ ϕ d x = ∫ Ω f ( x , u ) ϕ d x , ∀ ϕ ∈ E .

Set

I ( u ) = ∫ Ω P ( | ∇ u | ) d x − ∫ Ω F ( x , u ) d x , ∀ u ∈ E

and we know that the critical points of I are just the weak solutions of the problem (1.1).

For E is a separable and reflexive Banach space, then there exist (see 9) { e n } n = 1 ∞ ⊂ E and { e n ∗ } n = 1 ∞ ⊂ E ∗ such that

e n ∗ ( e m ) = δ n , m = { 1 , if  n = m ; 0 , if  n ≠ m . and e n ∗ ( v ) = α n for  v = ∑ i = 1 ∞ α i e i ∈ E .

Now, we set V j = { u ∈ W 0 1 , P ( Ω ) : e i ∗ ( u ) = 0 , i > j } , X j = { u ∈ W 0 1 , P ( Ω ) : e i ∗ ( u ) = 0 , i ≤ j } , so

W 0 1 , P ( Ω ) = V j ⊕ X j .

Lemma 3.1 Given δ > 0 , there is j ∈ N such that for all u ∈ X j , | u | H ≤ δ ∥ u ∥ .

Proof We prove the lemma by contradiction. Suppose that there exist δ>0 and u j ∈ X j for every j∈N such that | u j | H ≥ δ ∥ u j ∥ . Taking v j = u j | u j | H , we have | v j | H = 1 for every j∈N and ∥ v j ∥ ≤ 1 δ . Hence, { v j } ⊂ W 0 1 , P ( Ω ) is a bounded sequence, and we may suppose, without loss of generality, that v j ⇀ v in W01,P(Ω). Furthermore, e n ∗ ( v ) = 0 for every n ∈ N since e n ∗ ( v j ) = 0 for all j ≥ n . This shows that v = 0 . On the other hand, by the compactness of embedding W01,P(Ω)↪LH(Ω), we conclude that | v | H = 1 . This proves the lemma. □

Lemma 3.2 Suppose f satisfies (f∗), then there exist j∈N and ρ , α > 0 such that

I | ∂ B ρ ∩ X j ≥ α .

Proof Now suppose that ∥ u ∥ > 1 . From (f∗), we know that

I ( u ) = ∫ Ω P ( | ∇ u | ) d x − ∫ Ω F ( x , u ) d x ≥ ∥ u ∥ p − − C 1 | u | H h + − C 2 .

Consequently, considering δ>0 to be chosen posteriorly by Lemma 3.1, we have for all u∈Xj and j sufficiently large,

I ( u ) ≥ ∥ u ∥ p − ( 1 − C 1 δ h + ∥ u ∥ h + − p − ) − C 2 .

Now, taking ∥ u ∥ = ρ ( δ ) = ( 1 2 C δ h + ) 1 h + − p − and noting that ρ ( δ ) → + ∞ , if δ → 0 , we can choose δ>0 such that 1 2 ρ p − > C 2 , ρ > 1 , and I ( u ) > 0 for every u∈Xj, ∥ u ∥ = ρ , the proof is complete. □

Lemma 3.3 Suppose f satisfies (f2). Then given m ∈ N , there exist a subspace W of W01,P(Ω) and a constant M m > 0 such that dim W = m and max u ∈ W I ( u ) < M m .

Proof Let x 0 ∈ Ω 0 and r 0 > 0 be such that B ( x 0 , r 0 ) ¯ ⊂ Ω , and 0 < | B ( x 0 , r 0 ) ¯ ∩ Ω 0 | < | Ω 0 | 2 . First, we take v 1 ∈ C 0 ∞ ( Ω ) with supp ( v 1 ) = B ( x 0 , r 0 ) ¯ . Considering Ω 1 = Ω 0 ∖ [ B ( x 0 , r 0 ) ¯ ∩ Ω 0 ] ⊂ Ω ˆ 0 = Ω ∖ B ( x 0 , r 0 ) ¯ , we have | Ω 1 | ≥ | Ω 0 | 2 > 0 . Let x 1 ∈ Ω 1 and r 1 > 0 be such that B ( x 1 , r 1 ) ¯ ⊂ Ω ˆ 0 , and 0 < | B ( x 1 , r 1 ) ¯ ∩ Ω 1 | < | Ω 1 | 2 . Next, we take v 2 ∈ C 0 ∞ ( Ω ) with supp ( v 2 ) = B ( x 1 , r 1 ) ¯ . After a finite number of steps, we get v 1 , v 2 , … , v m such that supp ( v i ) ∩ supp ( v j ) = ∅ , i ≠ j , and | supp ( v j ) ∩ Ω 0 | > 0 for all i , j ∈ { 1 , 2 , … , m } . Let W = span { v 1 , v 2 , … , v m } , by construction, dimW=m, and ∫ Ω | v | p + d x > 0 for every v ∈ W ∖ { 0 } .

Since max u ∈ W ∖ { 0 } I ( u ) = max t > 0 , v ∈ W ∩ ∂ B 1 ( 0 ) ( ∫ Ω P ( t | ∇ v | ) d x − ∫ Ω F ( x , t v ) d x ) , if t>1, then I ( t v ) ≤ t p + − ∫ Ω F ( x , t v ) d x = t p + ( 1 − 1 t p + ∫ Ω F ( x , t v ) d x ) . Now, it suffices to verify that

lim t → ∞ 1 t p + ∫ Ω F ( x , t v ) d x > 1 .

From the condition (f2), given L > 0 , there is C > 0 such that for every s ∈ R , a.e. x in Ω0,

F ( x , s ) ≥ L | s | p + − C .

Consequently, for v ∈ ∂ B 1 ( 0 ) ∩ W and t>1,

∫ Ω F ( x , t v ) d x ≥ L t p + ∫ Ω 0 | v | p + d x − C t h + ∫ Ω ∖ Ω 0 H ( v ) d x − C 2 ,

and

lim t → ∞ ∫ Ω f ( x , t v ) d x t p + ≥ L ∫ Ω 0 | v | p + d x − C ∫ Ω ∖ Ω 0 H ( v ) d x ≥ L r − C R ,

where r = min { ∫ Ω 0 | v | p + d x , v ∈ ∂ B 1 ( 0 ) ∩ W } and R = max { ∫ Ω ∖ Ω 0 H ( v ) d x , v ∈ ∂ B 1 ( 0 ) ∩ W } . Observing that W is finite dimensional and we have R < + ∞ , r > 0 , the inequality is obtained by taking L > 1 r ( 1 + C R ) ; the proof is complete. □

Lemma 3.4 Suppose f satisfies (f1), then I satisfies the (PS) condition.

Proof We suppose that ∥ u n ∥ > 1 ,

Noting that 1<σ<p−, η>p+, { u n } is bounded. By 9, Lemma 3.1, we know that I satisfies the (PS) condition. □

Proof of Theorem 2.1 First, we recall that W 0 1 , P ( Ω ) = V j ⊕ X j , where V j and X j are defined in (3.1). Invoking Lemma 3.2, we find j∈N, and I satisfies I1 with X = X j . Now, by Lemma 3.3, there is a subspace W of W01,P(Ω) with dim W = k + j = k + dim V j and such that I satisfies (I2). Since I(0)=0 and I is even, we may apply Lemma 2.4 to conclude that I possesses at least k pairs of nontrivial critical points. The proof is complete. □

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.