Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents

Changmu Chu; Yanling Xie; Dizhi Zhou

doi:10.1515/math-2022-0520

Article Open Access

Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents

Changmu Chu , Yanling Xie and Dizhi Zhou

Published/Copyright: January 31, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we study a class of new p(x)-Kirchhoff problem without satisfying the Ambrosetti-Rabinowitz type growth condition. Under some suitable superliner conditions, we introduce new methods to show the boundedness of Cerami sequences. By using the mountain pass lemma and the symmetric mountain pass lemma, we prove that the p(x)-Kirchhoff problem has a nontrivial weak solution and infinitely many solutions.

Keywords: p(x)-Kirchhoff problem; Cerami sequences; mountain pass lemma; infinitely many solutions

MSC 2010: 35A15; 35J60

1 Introduction and main results

This work deals with the following nonlocal p(x)-Kirchhoff problem:

(1.1) − a − b ∫ Ω ∣ ∇ u ∣ p ( x ) p ( x ) d x Δ p ( x ) u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω ,

where Ω ⊂ R N is a smooth bounded domain with boundary ∂ Ω , a , b > 0 are constants, p ∈ C ( Ω ¯ ) with 1 < p − = inf x ∈ Ω ¯ p ( x ) ≤ p ( x ) ≤ sup x ∈ Ω ¯ p ( x ) = p + < N , and f : Ω ¯ × R → R is a continuous function that satisfies some conditions which will be stated later on.

The study of Kirchhoff-type problems with variable exponent growth conditions has received more and more attention because it arises from various applications, we can refer to [1–4]. We often called problem (1.1) a nonlocal problem because an integral term appears on the left-hand side of the problem (1.1). Problem (1.1) is related to the stationary problem of a model introduced by Kirchhoff [5]. More precisely, Kirchhoff introduced a model given by the following equation:

(1.2) ρ ∂ 2 u ∂ t 2 − P 0 h + E 2 L ∫ 0 L ∂ u ∂ x 2 d x ∂ 2 u ∂ x 2 = 0 ,

which extends the famous D’Alembert wave equations for free vibration of elastic strings, where ρ , P 0 , h , E , and L are constants that represent some physical meanings.

Recently, for the case f ( x , u ) = λ ∣ u ∣ p ( x ) − 2 u + g ( x , u ) , the existence and multiplicity of solutions of problem (1.1) are obtained in [6] and [7]. The authors in [6] require g ( x , u ) to satisfy the Ambrosetti-Rabinowitz type growth condition ((AR) condition), that is,

(AR): there exist s A > 0 and θ > p + such that

0 ≤ f ( x , t ) t − θ F ( x , t ) , for all ∣ t ∣ ≥ s A , x ∈ Ω .

In fact, the (AR) condition guarantees the boundedness of the Palais-Smale sequence of the Euler-Lagrange functional [8,9]. However, it puts strict constraints on the growth of nonlinear term and eliminates many nonlinearities [7,10–12]. Therefore, the authors in [7] give a more weaker superlinear condition:

( f 0 ): there exists a positive constant C 0 > 0 such that

(1.3) G ( x , t ) ≤ G ( x , s ) + C 0

for any x ∈ Ω , 0 < t < s or s < t < 0 , where G ( x , t ) ≔ t f ( x , t ) − p + F ( x , t ) . It is worth mentioning that the proof of the main results in [6] and [7] actually require that p(x) belong to the modular Poincaré inequality

(1.4) ∫ Ω ∣ u ∣ p ( x ) d x ≤ C ∫ Ω ∣ ∇ u ∣ p ( x ) d x , ∀ u ∈ W 0 1 , p ( x ) ( Ω ) .

However, the modular Poincaré inequality is not always hold (for details, see [13]). To remove this constraint, we study problem (1.1) without the (AR) condition. Let us assume that f satisfies the following conditions:

( f 1 ): f ∈ C ( Ω ¯ × R , R ) with F ( x , t ) ≥ 0 , for all ( x , t ) ∈ Ω × R , where F ( x , t ) = ∫ 0 t f ( x , s ) d s ;

( f 2 ): there exists a function α ∈ C ( Ω ¯ ) , p + < α − ≤ α ( x ) < p ∗ ( x ) , for all x ∈ Ω , and a number Λ 0 > 0 such that, for each λ ∈ ( 0 , Λ 0 ) , there exists C > 0 such that

f ( x , t ) ≤ λ ∣ t ∣ p ( x ) − 1 + C ∣ t ∣ α ( x ) − 1 , for all ( x , t ) ∈ Ω × R ,

where p ∗ ( x ) = N p ( x ) N − p ( x ) ;

( f 3 ): lim t → + ∞ F ( x , t ) ∣ t ∣ 2 p − = + ∞ , for x ∈ Ω ¯ uniformly;

( f 4 ): there exist ϖ 0 > 0 and μ ∈ p + , 2 ( p − ) 2 p + such that

(1.5) − ϖ 0 ∣ t ∣ 2 p − ≤ 1 μ f ( x , t ) t − F ( x , t ) , for all ( x , t ) ∈ Ω × R ;

( f 5 ): f ( x , − t ) = − f ( x , t ) , for all ( x , t ) ∈ Ω × R .

Now our main results are as follows:

Theorem 1.1

Suppose that the function p ∈ C ( Ω ¯ ) satisfies

1 < p − < p ( x ) < p + < 2 p − < α − ≤ α ( x ) < p ∗ ( x ) .

Then there exists Λ 0 > 0 such that, for each λ ∈ ( 0 , Λ 0 ) , with ( f 1 )–( f 4 ) satisfied, problem (1.1) has a nontrivial weak solution in W 0 1 , p ( x ) ( Ω ) .

Theorem 1.2

Suppose that the function p ∈ C ( Ω ¯ ) satisfies

1 < p − < p ( x ) < p + < 2 p − < α − ≤ α ( x ) < p ∗ ( x ) .

Then there exists Λ 0 > 0 such that, for each λ ∈ ( 0 , Λ 0 ) , with ( f 1 )–( f 5 ) satisfied, problem (1.1) has infinitely many solutions in W 0 1 , p ( x ) ( Ω ) .

Remark 1.3

The function f ( x , t ) = ∣ t ∣ 2 p − − 2 t [ ln ( 1 + ∣ t ∣ ) ] α ( x ) does not satisfy the (AR) condition. However, it is easy to check that f ( x , t ) satisfies conditions ( f 1 )–( f 5 ).

Throughout this article, the letters C and C i ( i = 1 , 2 , … ) denote various positive constants. This article is organized as follows. In Section 2, we give some basic properties of the variable exponent Lebesgue space and Sobolev space. In Section 3, we prove the Cerami compactness condition. In Section 4, we prove our main results by the mountain pass lemma and the symmetric mountain pass lemma.

2 Preliminaries

To discuss problem (1.1), we need some necessary properties on the functional space L p ( x ) ( Ω ) and W 1 , p ( x ) ( Ω ) . Set C + ( Ω ¯ ) = { h ∈ C ( Ω ¯ ) : h ( x ) > 1 , ∀ x ∈ Ω ¯ } , S ( Ω ) = { u ∣ u : Ω → R is measurable function } .

For any p ∈ C + ( Ω ¯ ) , the variable exponent Lebesgue space L p ( x ) ( Ω ) is defined as follows:

L p ( x ) ( Ω ) = u ∈ S ( Ω ) : ∫ Ω ∣ u ∣ p ( x ) d x < ∞

with the norm

∣ u ∣ p ( x ) = inf λ > 0 : ∫ Ω u ( x ) λ p ( x ) d x ≤ 1 .

The variable exponent Sobolev space W 1 , p ( x ) ( Ω ) is defined as follows:

W 1 , p ( x ) ( Ω ) = { u ∈ L p ( x ) ( Ω ) : ∣ ∇ u ∣ ∈ L p ( x ) ( Ω ) }

with the norm

‖ u ‖ 1 , p ( x ) = ∣ u ∣ p ( x ) + ∣ ∇ u ∣ p ( x ) .

Then define W 0 1 , p ( x ) ( Ω ) as the closure of C 0 ∞ ( Ω ) in W 1 , p ( x ) ( Ω ) . The space L p ( x ) ( Ω ) , W 0 1 , p ( x ) ( Ω ) , and W 1 , p ( x ) ( Ω ) are separable and reflexive Banach spaces [14]. Moreover, there is a constant C > 0 , such that

(2.1) ∣ u ∣ p ( x ) ≤ C ∣ ∇ u ∣ p ( x ) ,

for all u ∈ W 0 1 , p ( x ) ( Ω ) . Therefore, ‖ u ‖ = ∣ ∇ u ∣ p ( x ) and ‖ u ‖ 1 , p ( x ) are equivalent norms on u ∈ W 0 1 , p ( x ) ( Ω ) .

Proposition 2.1

[9,14] For any u ∈ L p ( x ) ( Ω ) , v ∈ L p ′ ( x ) ( Ω ) with 1 p ( x ) + 1 p ′ ( x ) = 1 , we have

∫ Ω u v d x ≤ 1 p − + 1 p ′ − ∣ u ∣ p ( x ) ∣ v ∣ p ′ ( x ) .

Proposition 2.2

[9,14] If p , q ∈ C + ( Ω ¯ ) satisfies 1 ≤ q ( x ) < p ∗ ( x ) for all x ∈ Ω ¯ , then the imbedding from W 1 , p ( x ) ( Ω ) to L q ( x ) ( Ω ) is continuous and compact.

Proposition 2.3

[6,9] Let ρ ( u ) = ∫ Ω ∣ u ∣ p ( x ) d x for u ∈ L p ( x ) ( Ω ) and { u n } ⊂ L p ( x ) ( Ω ) , then the following properties hold:

∣ u ∣ p ( x ) < 1 ( = 1 ; > 1 ) ⇔ ρ ( u ) < 1 ( = 1 ; > 1 ) ;
∣ u ∣ p ( x ) < 1 ⇒ ∣ u ∣ p ( x ) p + ≤ ρ ( u ) ≤ ∣ u ∣ p ( x ) p − , ∣ u ∣ p ( x ) > 1 ⇒ ∣ u ∣ p ( x ) p − ≤ ρ ( u ) ≤ ∣ u ∣ p ( x ) p + ;
lim n → ∞ ∣ u n ∣ p ( x ) = 0 ⇔ lim n → ∞ ρ ( u n ) = 0 , lim n → ∞ ∣ u n − u ∣ = 0 ⇔ lim n → ∞ ρ ( u n − u ) = 0 .

Proposition 2.4

[6,9] Let A ( u ) = ∫ Ω 1 p ( x ) ∣ ∇ u ∣ p ( x ) d x , for all u ∈ W 0 1 , p ( x ) ( Ω ) , then A ( u ) ∈ C 1 ( W 0 1 , p ( x ) ( Ω ) , R ) , ⟨ A ′ ( u ) , v ⟩ = ∫ Ω ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ v d x , for all u , v ∈ W 0 1 , p ( x ) ( Ω ) and the following properties hold:

A is convex and sequentially weakly lower semi-continuous;
A ′ : W 0 1 , p ( x ) ( Ω ) → ( W 0 1 , p ( x ) ( Ω ) ) ∗ is a mapping of type ( S + ) , that is, if u n ⇀ u in W 0 1 , p ( x ) ( Ω ) and lim n → ∞ ¯ ⟨ A ′ ( u n ) , ( u n − u ) ⟩ ≤ 0 , then u n → u in W 0 1 , p ( x ) ( Ω ) ;
A ′ : W 0 1 , p ( x ) ( Ω ) → ( W 0 1 , p ( x ) ( Ω ) ) ∗ is a strictly monotone operator and homeomorphism.

Definition 2.5

We say that u ∈ W 0 1 , p ( x ) ( Ω ) is a weak solution of problem (1.1), if for every v ∈ W 0 1 , p ( x ) ( Ω ) , it satisfies the following:

a − b ∫ Ω ∣ ∇ u ∣ p ( x ) p ( x ) d x ∫ Ω ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ v d x − ∫ Ω f ( x , u ) v d x = 0 .

The energy functional J λ : W 0 1 , p ( x ) ( Ω ) → R associated to problem (1.1) is defined as follows:

(2.2) J λ ( u ) = a ∫ Ω ∣ ∇ u ∣ p ( x ) p ( x ) d x − b 2 ∫ Ω ∣ ∇ u ∣ p ( x ) p ( x ) d x 2 − ∫ Ω F ( x , u ) d x .

Note that J λ is a C 1 ( W 0 1 , p ( x ) ( Ω ) , R ) functional and

(2.3) ⟨ J λ ′ ( u ) , v ⟩ = a − b ∫ Ω ∣ ∇ u ∣ p ( x ) p ( x ) d x ∫ Ω ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ v d x − ∫ Ω f ( x , u ) v d x ,

for all u , v ∈ W 0 1 , p ( x ) ( Ω ) . Hence, the critical points of the functional J λ are weak solutions of problem (1.1).

3 The Cerami compactness condition

We recall now the definition of the Cerami compactness condition.

Definition 3.1

Let ( W 0 1 , p ( x ) ( Ω ) , ‖ . ‖ ) be a real Banach space, and J λ ∈ C 1 ( W 0 1 , p ( x ) ( Ω ) , R ) . Given c ∈ R , we say that J λ satisfies the Cerami condition at any level c ∈ R ( ( C e ) c condition, for short), if any sequence { u n } ⊂ W 0 1 , p ( x ) ( Ω ) satisfying

(3.1) J λ ( u n ) → c and ‖ J λ ′ ( u n ) ‖ ( 1 + ‖ u n ‖ ) → 0 , as n → ∞ ,

has a convergent subsequence.

Lemma 3.2

Assume that ( f 1 )–( f 4 ) hold, then J λ satisfies the ( C e ) c condition at any level c ≠ a 2 2 b .

Proof

First, we prove that { u n } is bounded in W 0 1 , p ( x ) ( Ω ) . Let { u n } ∈ W 0 1 , p ( x ) ( Ω ) be a ( C e ) c sequence for c ≠ a 2 2 b .

Arguing by contradiction. So we suppose that { u n } is unbounded in W 0 1 , p ( x ) ( Ω ) , that is, ‖ u n ‖ → + ∞ , as n → + ∞ . We define the sequence { w n } ⊂ W 0 1 , p ( x ) ( Ω ) by w n = u n ‖ u n ‖ . Clearly, ‖ w n ‖ = 1 for all n ∈ N . Passing to a subsequence, if necessary, we may assume that

(3.2) w n ⇀ w , weakly in W 0 1 , p ( x ) ( Ω ) , w n → w , strongly in L q ( x ) ( Ω ) , 1 ≤ q ( x ) < p ∗ ( x ) , w n ( x ) → w ( x ) , a.e . x ∈ Ω .

Let Ω 0 ≔ { x ∈ Ω : w ≠ 0 } . If x ∈ Ω 0 , then it follows from (3.2) that

w ( x ) = lim n → ∞ w n ( x ) = lim n → ∞ u n ‖ u n ‖ ≠ 0 .

This implies that

∣ u n ( x ) ∣ = ∣ w n ( x ) ∣ ‖ u n ‖ → + ∞ , a.e . x ∈ Ω 0 , as n → + ∞ .

By using ( f 3 ) and Fatou’s lemma, if ∣ Ω 0 ∣ > 0 , we have

(3.3) lim n → ∞ ∫ Ω ∣ F ( x , u n ( x ) ) ∣ ‖ u n ‖ 2 p − d x = lim n → ∞ ∫ Ω ∣ F ( x , u n ( x ) ) ∣ ∣ w n ( x ) ∣ 2 p − ∣ u n ( x ) ∣ 2 p − d x = + ∞ .

Since ‖ u n ‖ > 1 , for n sufficiently large, it follows (2.2), (3.1), and Proposition 2.3 that

∫ Ω F ( x , u n ) d x ≤ a p − ∫ Ω ∣ ∇ u n ∣ p ( x ) d x − b 2 ( p + ) 2 ∫ Ω ∣ ∇ u n ∣ p ( x ) d x 2 + C 1 ≤ a p − ‖ u n ‖ p + − b 2 ( p + ) 2 ‖ u n ‖ 2 p − + C 1 .

Notice that p + < 2 p − , we can conclude that

(3.4) lim n → ∞ ∫ Ω ∣ F ( x , u n ) ∣ ‖ u n ‖ 2 p − d x ≤ lim n → ∞ a p − ‖ u n ‖ p + − 2 p − − b 2 ( p + ) 2 + C 1 ‖ u n ‖ 2 p − = − b 2 ( p + ) 2 .

By (3.3) and (3.4), we obtain + ∞ ≤ − b 2 ( p + ) 2 , which is a contradiction. Hence,

∣ Ω 0 ∣ = 0 and w ( x ) = 0 a.e. on Ω .

It follows ( f 4 ) and w n → w = 0 in L 2 p − ( Ω ) that

C 2 + ‖ u n ‖ ≥ J λ ( u n ) − 1 μ ⟨ J λ ′ ( u n ) , u n ⟩ = a ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x − b 2 ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x 2 − ∫ Ω F ( x , u n ) d x − 1 μ a − b ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x ∫ Ω ∣ ∇ u n ∣ p ( x ) d x − ∫ Ω f ( x , u n ) u n d x ≥ a p + ∫ Ω ∣ ∇ u n ∣ p ( x ) d x − b 2 ( p − ) 2 ∫ Ω ∣ ∇ u n ∣ p ( x ) d x 2 − ∫ Ω F ( x , u n ) d x − a μ ∫ Ω ∣ ∇ u n ∣ p ( x ) d x + b μ p + ∫ Ω ∣ ∇ u n ∣ p ( x ) d x 2 + 1 μ ∫ Ω f ( x , u n ) u n d x = a p + − a μ ∫ Ω ∣ ∇ u n ∣ p ( x ) d x + b μ p + − b 2 ( p − ) 2 ∫ Ω ∣ ∇ u n ∣ p ( x ) d x 2 + ∫ Ω 1 μ f ( x , u n ) u n − F ( x , u n ) d x ≥ a p + − a μ ‖ u n ‖ p − + b μ p + − b 2 ( p − ) 2 ‖ u n ‖ 2 p − − ϖ 0 ∫ Ω ∣ u n ∣ 2 p − d x .

Dividing the aforementioned inequality by ‖ u n ‖ 2 p − , we can conclude that

C 2 ‖ u n ‖ 2 p − + ‖ u n ‖ 1 − 2 p − ≥ a 1 p + − 1 μ ‖ u n ‖ − p − + b 1 μ p + − 1 2 ( p − ) 2 − ϖ 0 ∫ Ω ∣ w n ∣ 2 p − d x .

Since μ ∈ p + , 2 p − 2 p + , and passing to the limit as n → ∞ , we obtain

0 ≥ b 1 μ p + − 1 2 ( p − ) 2 ,

which is a contradiction. So { u n } is bounded in W 0 1 , p ( x ) ( Ω ) . Therefore, to a subsequence, still denoted by { u n } , there exists u ∈ W 0 1 , p ( x ) ( Ω ) such that

(3.5) u n ⇀ u , weakly in W 0 1 , p ( x ) ( Ω ) , u n → u , strongly in L r ( x ) ( Ω ) , 1 ≤ r ( x ) < p ∗ ( x ) , u n ( x ) → u ( x ) , a.e . x ∈ Ω .

By (3.5), ( f 2 ), and the Hölder inequality and proposition 2.2, it follows that

∫ Ω f ( x , u n ) ( u n − u ) d x ≤ ∫ Ω λ ∣ u n ∣ p ( x ) − 1 ∣ u n − u ∣ d x + ∫ Ω C ∣ u n ∣ α ( x ) − 1 ∣ u n − u ∣ d x ≤ λ ∣ ∣ u n ∣ p ( x ) − 1 ∣ p ( x ) p ( x ) − 1 ∣ u n − u ∣ p ( x ) + C ∣ ∣ u n ∣ α ( x ) − 1 ∣ α ( x ) α ( x ) − 1 ∣ u n − u ∣ α ( x ) ≤ C 3 λ max { ‖ u n ‖ p + − 1 , ‖ u n ‖ p − − 1 } ∣ u n − u ∣ p ( x ) + C 4 max { ‖ u n ‖ α + − 1 , ‖ u n ‖ α − − 1 } ∣ u n − u ∣ α ( x ) → 0 , as n → ∞ ,

which means that

(3.6) lim n → ∞ ∫ Ω f ( x , u n ) ( u n − u ) d x = 0 .

By (3.1), we have

(3.7) ⟨ J λ ′ ( u n ) , u n − u ⟩ = a − b ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x ∫ Ω ∣ ∇ u n ∣ p ( x ) − 2 ∇ u n ∇ ( u n − u ) d x − ∫ Ω f ( x , u n ) ( u n − u ) d x → 0 , as n → ∞ .

Hence, from (3.6) and (3.7), we obtain

(3.8) a − b ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x ∫ Ω ∣ ∇ u n ∣ p ( x ) − 2 ∇ u n ∇ ( u n − u ) d x → 0 , as n → ∞ .

Since { u n } is bounded in W 0 1 , p ( x ) ( Ω ) , passing to a subsequence, if necessary, we assume that

∫ Ω 1 p ( x ) ∣ ∇ u n ∣ p ( x ) d x → t 0 ≥ 0 , as n → ∞ .

If t 0 = a b , then a − b ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x → 0 . For any v ∈ W 0 1 , p ( x ) ( Ω ) , by ( f 2 ), (3.5) and the Hölder inequality, it implies that ∫ Ω ( f ( x , u n ) − f ( x , u ) ) v d x → 0 , as n → ∞ . Since ⟨ J λ ′ ( u n ) , v ⟩ = ( a − b ∫ Ω 1 p ( x ) ∣ ∇ u n ∣ p ( x ) d x ) ∫ Ω ∣ ∇ u n ∣ p ( x ) − 2 ∇ u n ∇ v d x − ∫ Ω f ( x , u n ) v d x → 0 , we have ∫ Ω f ( x , u ) v d x → 0 , as n → ∞ .

By the fundamental lemma of the variational method, we obtain f ( x , u ( x ) ) = 0 , for a.e. x ∈ Ω . It follows that u = 0 . So

∫ Ω F ( x , u n ) d x → ∫ Ω F ( x , u ) d x = 0 .

Hence, for t 0 = a b , we see that

J λ ( u n ) = a ∫ Ω 1 p ( x ) ∣ ∇ u n ∣ p ( x ) d x − b 2 ∫ Ω 1 p ( x ) ∣ ∇ u n ∣ p ( x ) d x 2 − ∫ Ω F ( x , u n ) d x → a 2 2 b ,

and this is a contradiction since J ( u n ) → c ≠ a 2 2 b , then a − b ∫ Ω 1 p ( x ) ∣ ∇ u n ∣ p ( x ) d x → 0 is not true.

Hence,

a − b ∫ Ω ∣ ∇ u n ∣ p ( x ) p ( x ) d x is bounded .

By (3.8), we obtain ∫ Ω ∣ ∇ u n ∣ p ( x ) − 2 ∇ u n ( ∇ u n − ∇ u ) d x → 0 .

By proposition 2.4, we can conclude that u n → u as n → ∞ . So J λ satisfies the ( C e ) c condition.□

4 The proof of main results

To prove Theorems 1.1 and 1.2, we recall the mountain pass theorem and the symmetric mountain pass theorem.

Theorem 4.1

[7] Let X be a Banach space, and let I ∈ C 1 ( X , R ) . Assume that the following conditions hold:

I ( 0 ) = 0 , for all u ∈ X ;
I satisfies ( C e ) c condition, for all c > 0 ;
there exists constant ρ , δ > 0 such that I ( u ) ≥ δ for all u ∈ X with ‖ u ‖ = ρ ;
there exists a function e ∈ X such that ‖ e ‖ > ρ and I ( e ) < 0 ,

then the function I has a critical value c ≥ δ , i.e., there exists u ∈ X , such that I ′ ( u ) = 0 and I ( u ) = c .

Theorem 4.2

[15] Let X = Y ⊕ Z is an infinite-dimensional Banach space, where Y is finite-dimensional, and let I ∈ C 1 ( X , R ) . Assume that the following conditions hold:

I ( 0 ) = 0 , I ( u ) = I ( − u ) , for all u ∈ X ;
I satisfies ( C e ) c condition, for all c > 0 ;
there exists constant ρ , δ > 0 such that I ( u ) ≥ δ for all u ∈ X with ‖ u ‖ = ρ ;
for every finite-dimensional subspace X ˜ ⊂ X , there is R = R ( X ˜ ) > 0 such that I ( u ) < 0 on X ˜ ⧹ B R .

Then the function I has an unbounded sequence of critical values.

Now we prove that function J λ satisfies the mountain pass geometry.

Lemma 4.3

Assume that 1 < p − < p ( x ) < p + < 2 p − < α − ≤ α ( x ) < p ∗ ( x ) . If ( f 1 ) and ( f 2 ) hold, then there exists Λ 0 , ρ , δ > 0 such that, for each λ ∈ ( 0 , Λ 0 ) , J λ ( u ) ≥ δ > 0 , for any u ∈ W 0 1 , p ( x ) ( Ω ) with ∣ ∣ u ∣ ∣ = ρ .

Proof

Let ρ ∈ ( 0 , 1 ) and u ∈ W 0 1 , p ( x ) ( Ω ) such that ∣ ∣ u ∣ ∣ = ρ . By condition ( f 2 ), we have

(4.1) ∣ F ( x , t ) ∣ ≤ λ p ( x ) ∣ t ∣ p ( x ) + C α ( x ) ∣ t ∣ α ( x ) , for all ( x , t ) ∈ Ω × R .

Thus, by considering (2.1), (4.1), and Propositions 2.2 and 2.3, for all u ∈ W 0 1 , p ( x ) ( Ω ) with ∣ ∣ u ∣ ∣ = ρ , we obtain

(4.2) J λ ( u ) = a ∫ Ω 1 p ( x ) ∣ ∇ u ∣ p ( x ) d x − b 2 ∫ Ω 1 p ( x ) ∣ ∇ u ∣ p ( x ) d x 2 − ∫ Ω F ( x , u ) d x ≥ a ∫ Ω 1 p ( x ) ∣ ∇ u ∣ p ( x ) d x − b 2 ∫ Ω 1 p ( x ) ∣ ∇ u ∣ p ( x ) d x 2 − ∫ Ω λ p ( x ) ∣ u ∣ p ( x ) d x − ∫ Ω C α ( x ) ∣ u ∣ α ( x ) d x ≥ a p + ∫ Ω ∣ ∇ u ∣ p ( x ) d x − b 2 ( p − ) 2 ∫ Ω ∣ ∇ u ∣ p ( x ) d x 2 − λ p − ∫ Ω ∣ u ∣ p ( x ) d x − C α − ∫ Ω ∣ u ∣ α ( x ) d x

≥ a p + ∣ ∣ u ∣ ∣ p + − b 2 ( p − ) 2 ∣ ∣ u ∣ ∣ 2 p − − λ C 5 ∣ ∣ u ∣ ∣ p − − C 6 ∣ ∣ u ∣ ∣ α − = ρ p + a p + − b 2 ( p − ) 2 ρ 2 p − − p + − λ C 5 ρ p − − p + − C 6 ρ α − − p + .

Set

g ( s ) = a p + − b 2 ( p − ) 2 s 2 p − − p + − C 6 s α − − p + , ∀ s ≥ 0 .

Since p + < 2 p − < α − , we can obtain that there exists a constant ρ > 0 , such that

g ( ρ ) = max g ( s ) > 0 , ∀ s ≥ 0 .

So taking Λ 0 = 1 2 C 5 ρ p + − p − g ( ρ ) > 0 , we can infer from (4.2) that there exist ρ > 0 , δ > 0 and Λ ∈ ( 0 , Λ 0 ) , such that

(4.3) J λ ( u ) ≥ 1 2 ρ p + g ( ρ ) = δ > 0 , for all u ∈ W 0 1 , p ( x ) ( Ω ) with ∣ ∣ u ∣ ∣ = ρ

for all λ ∈ ( 0 , Λ ) . The proof is complete.□

Lemma 4.4

Assume that 1 < p − < p ( x ) < p + < 2 p − < α − ≤ α ( x ) < p ∗ ( x ) . If ( f 3 ) hold, then for every finite-dimensional subspace W ⊂ W 0 1 , p ( x ) ( Ω ) , there exists R = R ( W ) > 0 such that J λ ( u ) < 0 , for all u ∈ W , with ‖ u ‖ > R .

Proof

By condition ( f 3 ), we know that for all M > 0 , there exists C M > 0 such that

(4.4) F ( x , u ) ≥ M ∣ u ∣ 2 p − − C M , for all ( x , u ) ∈ Ω × R .

Let R = R ( W ) > 1 , for all u ∈ W with ‖ u ‖ > R . By (4.3), we have

J λ ( u ) = a ∫ Ω 1 p ( x ) ∣ ∇ ( u ) ∣ p ( x ) d x − b 2 ∫ Ω 1 p ( x ) ∣ ∇ ( u ) ∣ p ( x ) d x 2 − ∫ Ω F ( x , u ) d x ≤ a p − ∫ Ω ∣ ∇ u ∣ p ( x ) d x − b 2 ( p + ) 2 ∫ Ω ∣ ∇ u ∣ p ( x ) d x 2 − M ∫ Ω ∣ u ∣ 2 p − d x + C M ∣ Ω ∣ .

Consequently, all norms on the finite-dimensional space W are equivalent, so there is C W > 0 such that

∫ Ω ∣ u ∣ 2 p − d x ≥ C W ‖ u ‖ 2 p − .

Therefore, we obtain

J λ ( u ) ≤ a p − ‖ u ‖ p + − b 2 ( p + ) 2 ‖ u ‖ 2 p − − C W M ‖ u ‖ 2 p − + C M ∣ Ω ∣ ≤ a p − ‖ u ‖ p + − b 2 ( p + ) 2 + C W M ‖ u ‖ 2 p − + C M ∣ Ω ∣ .

Since 2 p − > p + , it follows that for some ‖ u ‖ > R large enough, we can obtain J λ ( u ) < 0 . Hence, the proof of Lemma 4.4 is complete.□

Proof of Theorem 1.1

By Lemmas 4.3 and 4.4 and the fact that J λ ( 0 ) = 0 , we see that the functional J λ has the mountain pass geometry. Define

Γ = { γ ∈ C ( [ 0 , 1 ] , W 0 1 , p ( x ) ( Ω ) ) ∣ γ ( 0 ) = 0 , γ ( 1 ) = w } , c λ = inf γ ∈ Γ max t ∈ [ 0 , 1 ] J λ ( γ ( t ) ) ,

where w ∈ W with ‖ w ‖ > R . Noting that max t > 0 a t − b 2 t 2 = a 2 2 b for all u ∈ W 0 1 , p ( x ) ( Ω ) , we have

J λ ( u ) < a ∫ Ω 1 p ( x ) ∣ ∇ ( u ) ∣ p ( x ) d x − b 2 ∫ Ω 1 p ( x ) ∣ ∇ ( u ) ∣ p ( x ) d x 2 ≤ a 2 2 b .

So, one has c λ < a 2 2 b . Letting { u n } be a ( C e ) c λ sequence of J λ , by Lemma 3.2, we obtain J λ , which satisfies the ( C e ) c λ condition. By theorem 4.1, we can conclude that problem (1.1) has a nontrivial weak solution. This completes the proof of Theorem 1.1.□

Proof of Theorem 1.2

It is clear that J λ ( 0 ) = 0 and by condition ( f 5 ), J λ ( u ) = J λ ( − u ) . By Lemma 3.2, Theorem 4.2, and Lemmas 4.3 and 4.4, J λ satisfies the symmetric mountain pass theorem. Therefore, problem (1.1) has infinitely many solutions. This completes the proof of Theorem 1.2.□

Acknowledgements

The authors thank the anonymous referee for the careful reading and some helpful comments.

Funding information: This article was supported by National Natural Science Foundation of China (No. 11861021).
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.

References

[1] Y. M. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383–1406. 10.1137/050624522Search in Google Scholar

[2] M. Růžička, Electrorheological fluids: Modeling and mathematical theory, Lecture Notes in Mathematics, Springer, Berlin, 2000. 10.1007/BFb0104029Search in Google Scholar

[3] S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 1, 19–36. 10.1007/s11565-006-0002-9Search in Google Scholar

[4] S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), no. 3, 515–545. 10.1016/S0362-546X(04)00393-1Search in Google Scholar

[5] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. Search in Google Scholar

[6] M. K. Hamdani, A. Harrabi, F. Mtiri, and D. Repovš, Existence and multiplicity results for a new p(x)-Kirchhoff problem, Nonlinear Anal. 190 (2020), 111598. 10.1016/j.na.2019.111598Search in Google Scholar

[7] B. L. Zhang, B. Ge, and X. F. Cao, Multiple solutions for a class of new p(x)-Kirchhoff problem without the Ambrosetti-Rabinowitz conditions, Mathematics 8 (2020), no. 11, 1–13. 10.3390/math8112068Search in Google Scholar

[8] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[9] X. L. Fan and Q. H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852. 10.1016/S0362-546X(02)00150-5Search in Google Scholar

[10] S. T. Chen and X. H. Tang. Existence and multiplicity of solutions for Dirichlet problem of p(x)-Laplacian type without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl. 501 (2021), no. 1, 123882. 10.1016/j.jmaa.2020.123882Search in Google Scholar

[11] Q. H. Zhang and C. S. Zhao, Existence of strong solutions of a p(x)-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl. 69 (2015), 1–12. 10.1016/j.camwa.2014.10.022Search in Google Scholar

[12] Z. Yucedag, Existence of solutions for p(x) Laplacian equations without Ambrosetti-Rabinowitz type condition, Bull. Malays. Math. Sci. Soc. 38 (2015), 1023–1033. 10.1007/s40840-014-0057-1Search in Google Scholar

[13] X. L. Fan, Q. H. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317. 10.1016/j.jmaa.2003.11.020Search in Google Scholar

[14] X. L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

[15] M. K. Hamdani, J. Zuo, N. T. Chung, and D. D. Repovš, Multiplicity of solutions for a class of fractional p(x,⋅)-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition, Bound. Value Probl. 2020 (2020), 150. 10.1186/s13661-020-01447-9Search in Google Scholar

Received: 2022-05-23

Revised: 2022-07-30

Accepted: 2022-10-07

Published Online: 2023-01-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0520

Keywords for this article

p(x)-Kirchhoff problem; Cerami sequences; mountain pass lemma; infinitely many solutions

Creative Commons

BY 4.0