A class of p₁(x, ⋅) & p₂(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝ^N

1 Introduction

In this article, we study the existence and multiplicity of solutions for the following Kirchhoff-type equation:

for all x ∈ R N . M i ( i = 1 , 2 ) are continuous Kirchhoff-type functions in R N , λ is a real positive parameter, and the nonlinearity f is a Carathéodory function, whose hypothesis will be introduced later. ( − Δ ) p i ( x , ⋅ ) s ( x , ⋅ ) are called fractional p i ( x , ⋅ ) -Laplacian with variable s ( x , ⋅ ) -order, given p i ( x , ⋅ ) : R 2 N → ( 1 , + ∞ ) ( i = 1 , 2 ) and s ( x , ⋅ ) : R 2 N → ( 0 , 1 ) with N > s ( x , y ) p i ( x , y ) for all ( x , y ) ∈ R N × R N , which can be defined as follows:

for all x in R N , η ∈ C 0 ∞ ( R N ) and P.V. stands for the Cauchy principal value. Especially, when s ( x , ⋅ ) ≡ s and p i ( x , ⋅ ) ≡ p i , ( − Δ ) p i ( x , ⋅ ) s ( x , ⋅ ) in (1.1) reduces to the fractional p -Laplace operator ( − Δ ) p s , e.g., see [1] involving the fractional p -Laplacian problem without the Ambrosetti-Rabinowitz (A-R) condition and see [2] on the fractional p & q -Laplacian problem with critical Sobolev-hardy exponents.

Throughout this article, we assume that p i ( x , y ) ( i = 1 , 2 ) and s ( x , y ) are continuous functions and the hypotheses we impose on p i ( x , y ) and s ( x , y ) are as follows:

1. p i ( x , y ) are symmetric functions, i.e., p i ( x , y ) = p i ( y , x ) , 1 < p i − ≔ inf ( x , y ) ∈ R N × R N p i ( x , y ) ≤ p i + ≔ sup ( x , y ) ∈ R N × R N p i ( x , y ) < + ∞ . We denote p max ( x , y ) = max { p 1 ( x , y ) , p 2 ( x , y ) } , p min ( x , y ) = min { p 1 ( x , y ) , p 2 ( x , y ) } , and p i ¯ ( x ) = p i ( x , x ) .

2. s ( x , y ) is a symmetric function, i.e., s ( x , y ) = s ( y , x ) , 0 < s − ≔ inf ( x , y ) ∈ R N × R N s ( x , y ) ≤ s + ≔ sup ( x , y ) ∈ R N × R N s ( x , y ) < 1 , and s ¯ ( x ) = s ( x , x ) .

Kirchhoff in [3] introduced the following model, which came to be known as the Kirchhof-type equation:

where parameters ρ , p 0 , h , E , and L , with some specific physical meaning, are real positive constants. Particularly, Equation (1.3) is nonlocal fractional problem that contains a nonlocal coefficient p 0 h + E 2 L ∫ 0 L ∂ η ( x ) ∂ t 2 d x and can be used to model some physical systems in concrete real-world application, such as anomalous diffusion, ultra-relativistic of quantum mechanics, and water waves. Since then the literature on Kirchhof-type equations and Kirchhoff-type systems are quite large, and here we just list a few, e.g., see [4,5, 6,7] for further details.

We assume that M i : R 0 + → R + ( i = 1 , 2 ) are continuous functions, which the following conditions are satisfied:

1. There are some positive constants ϑ i ∈ [ 1 , p s ( x , ⋅ ) ∗ / p max + ) ( i = 1 , 2 ) and ϑ = max { ϑ 1 , ϑ 2 } such that

   t M i ( t ) ≤ ϑ i M ˜ i ( t ) , for any t ∈ R 0 + , where M ˜ i ( t ) = ∫ 0 t M i ( τ ) d τ .

2. There are some positive constants m i = m i ( τ ) > 0 ( i = 1 , 2 ) for all τ > 0 such that

   M i ( t ) ≥ m i , for any t > τ .

3. M i ( t ) is a decreasing function.

The evolution of the Laplace operator has been progressively deepened and has taken many forms so far. Many mathematical scholars have been devoted to the integer Laplace operators, fractional Laplace operators, and variable-order fractional Laplace operators. For some important results of these operators, we recommend the readers to refer to previous studies in [8,9,10, 11,12,13, 14,15,16, 17,18,19] and literature cited therein.

In the framework of variable exponents involving fractional p ( x , ⋅ ) -Laplace operator with variable s ( x , ⋅ ) -order, there have been some papers on this topic involving both with and without a Kirchhoff coefficient, for instance, see [20,21,22, 23,24,25]. Especially, an embedding theorem for variable exponential Sobolev spaces was first proved in [26]. In addition, with the help of variational methods, Zuo et al. in [27] studied a class of fractional p ( x , ⋅ ) -Kirchhoff-type problem with the presence of a single Laplace operator in the whole space R N .

Problem (1.1) comes from the following system:

where D η = ∣ ∇ η ∣ p − 2 + ∣ ∇ η ∣ q − 2 . System (1.4) had a wide range of applications in the field of physics and related sciences and had been paid much attention, for example, see [28,29,30, 31,32].

Since both p & q -Laplacian that is not homogeneous are involved, it is more difficult to get the corresponding estimates to compare to the case p = q > 1 ; therefore, we do need more careful analysis. The case on the whole space R N was studied in [33], and He and Li used the constraint minimization to study the subcritical growth problem:

where m , n > 0 , N ≥ 3 , and 1 < q < p < N , f ( x , η ) / η p − 1 tend to a positive constant l as v → ∞ .

Chaves et al. in [34] analyzed the existence of weak solution in D 1 , p ( R N ) ⋂ D 1 , q ( R N ) for the equation involving weight functions as follows:

where 1 < q < p < q ⋆ ≔ N q N − q , p < N . They proved that problem (1.5) possessed at least one weak solution even if the nonlinear term f did not satisfy the (A-R) condition.

There are few papers to consider the p ( x , ⋅ ) & q ( x , ⋅ ) -Laplacian problem. The case is on the bounded domain Ω , and Zuo et al. in [35] investigated a kind of the Choquard-type problems without a Kirchhoff coefficient:

where the operators ( − Δ ) p ( x , ⋅ ) s ( x , ⋅ ) & ( − Δ ) q ( x , ⋅ ) s ( x , ⋅ ) are two fractional Laplace operators with variable order s ( x , ⋅ ) : R 2 N → ( 0 , 1 ) and different variable exponents p ( x , ⋅ ) , q ( x , ⋅ ) : R 2 N → ( 1 , ∞ ) . The results of problem (1.6) are different from the single fractional Laplace operator.

While combining the p ( x , ⋅ ) & q ( x , ⋅ ) -Laplacian with Kirchhoff coefficients, Zhang in [36] devoted to the study of the following equations:

where M i ( i = 1 , 2 ) is a model of Kirchhoff coefficient on the bounded domain Ω . and ( − Δ p i ( x ) ) s is fractional Laplace operators with a constant order. On the basis of variational method and critical point theory, he proved the existence of solutions for problems (1.7) in an appropriate space of functions.

In the famous paper [37], Ambrosetti and Rabinowitz introduced the well-known (A-R) condition on the nonlinearity, that is, there exist some positive constants μ 0 such that

where F ( x , η ) = ∫ 0 η f ( x , t ) d t . As is known, the (A-R) condition plays a very important role in the application of the variational method, which is widely used to guarantee that the Palais-Smale sequences are bounded and the function I λ has a mountain pass geometry. However, some interesting nonlinearities do not satisfy the (A-R) condition, and an example of such a function (see [38]) is expressed as follows:

where 1 < p 0 < r 0 < p ∗ = N p 0 N − p 0 . Indeed, this function does not satisfy

where d 1 , d 2 > 0 . Hence, many researchers pay attention to find the new reasonable conditions instead of the (A-R) condition, see [24,39] and the references therein.

As far as we know, there is no result for Kirchhoff-type equation involving double fractional p 1 ( x , ⋅ ) & p 2 ( x , ⋅ ) -Laplace operators with a variable s ( x , ⋅ ) -order without the (A-R) condition in the whole space R N . Therefore, motivated by the previous and aforementioned cited works, we will investigate the existence and multiplicity of solutions for this kind of equation, which is different from the work of [15,34,35] that the equations of these problems involve the fractional p -Laplace operator with a constant order or do not contain Kirchhoff terms, and more general than (1.7), which authors considered a local version of the fractional operator, that is, with an integral set in Ω and not in the whole space R N . Our study extends previous results in some ways.

Throughout this article, C j ( j = 1 , 2 , … , N ) denote distinct positive constants, and i = 1 and 2. For any real-valued function H defined on a domain D , we denote:

The function a i : R N → R are continuous functions, which satisfy the following conditions:

1. a , a i ∈ L h ( x ) ( R N ) such that a ( x ) , a i ( x ) ≥ 0 , where h ∈ C + ( R N ) .

2. a , a i ∈ C ( R N × R ) such that a ( x ) , a i ( x ) ≥ 0 for all x ∈ R N and a , a i ≠ 0 .

The nonlinearity f : R N × R → R is a Carathéodory function, satisfying:

1. Let p i , q ∈ C + ( D ) and 1 < p max + < q − ≤ q ( x ) ≤ q + < ϑ < p s ∗ ( x ) , and there exist a 1 ( x ) and a 2 ( x ) , given by ( A 1 ) such that

   ∣ f ( x , t ) ∣ ≤ a 1 ( x ) + a 2 ( x ) ∣ t ∣ q ( x ) − 1 for all ( x , t ) ∈ R N × R .

2. lim ∣ t ∣ → ∞ F ( x , t ) ∣ t ∣ ϑ p max + = + ∞ , uniformly for all x ∈ R N , where F ( x , t ) = ∫ 0 t f ( x , s ) d s > 0 .

3. There exists β ( x ) ∈ L ∞ ( R N ) + such that lim sup ∣ t ∣ → 0 p max + F ( x , t ) ∣ t ∣ p max + ≤ β ( x ) , uniformly for all x ∈ R N .

4. There exists a constant τ ≥ 1 such that τ ϱ ( x , t ) ≥ ϱ ( x , ι t ) for all ( x , t ) ∈ R N × R and ι ∈ [ 0 , 1 ] , where ϱ ( x , t ) ≔ f ( x , t ) t − p max + F ( x , t ) ;

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

The paper is organized into five sections. Aside from Sections 1, 2 presents the main results, Section 3 presents some preliminary notions and results about fractional Lebesgue spaces and Sobolev spaces, Section 4 proves the compactness condition of Cerami sequence and Theorems 2.1–2.3, and Section 5 presents a conclusion.

2 The main results

We need to present the corresponding definition and variational framework before stating our main results.

for any φ ∈ X , where X will be introduced in Section 2 and

The problem (1.1) has a variational form with the function I λ : X → R , defined as follows:

for all η ∈ X and M i ˜ given in ( ℳ 1 ) . Moreover, the function I λ is well defined on the Sobolev spaces X and belongs to the class C 1 ( X , R ) , which the argument is similar to [21], and

for any η , φ ∈ X . Thus, under our assumptions, the existence and multiplicity of solutions for problem (1.1) is equivalent to the existence of critical points for the function I λ .

Now, we are ready to state three results of this paper.

3 Preliminary results