In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving (p₁(x), p₂(x))-Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

1. Introduction

We study the existence and multiplicity of solutions of the nonlocal equation

where Ω ⊂ ℝ^N (N ≥ 3) is a smooth bounded domain,

such that 2 ≤ p_i(x) < N for any

, and i = 1, 2.

The problem (P) is related to the stationary version of a model, the so-called Kirchhoff equation, introduced by [1]. To be more precise, Kirchhoff established a model given by the equation

()

where ρ, P₀, h, E, and L are constants, which extends the classical D’Alambert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. A distinguish feature of the Kirchhoff equation (1) is that the equation contains a nonlocal coefficient

which depends on the average

of the kinetic energy (1/2)|∂u/∂x|² on [0, L], and hence the equation is no longer a pointwise identity. For Kirchhoff-type equations involving the p(x)-Laplacian operator, see, for example, [2–4].

The p(x)-Laplacian operator −Δ_p(x)u : = div (|∇u|^p(x)−2∇u) is a natural generalization of the p-Laplacian operator −Δ_pu : = −div (|∇u|^p−2∇u) where p > 1 is a real constant. The main difference between them is that p-Laplacian operator is (p − 1) homogenous, that is, Δ_p(μu) = μ^p−1Δ_pu for every μ > 0, but the p(x)-Laplacian operator, when p(x) is not constant, is not homogeneous. This causes many problems; some classical theories and methods, such as the Lagrange multiplier theorem and the theory of Sobolev spaces, are not applicable. For p(x)-Laplacian operator, we refer the readers to [5–9] and references there in. Moreover, the nonlinear problems involving the p(x)-Laplacian operator are extremely attractive because they can be used to model dynamical phenomenons which arise from the study of electrorheological fluids or elastic mechanics. Problems with variable exponent growth conditions also appear in the modelling of stationary thermorheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes of filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the p(x)-Laplacian can be found in [10–14] and the references therein.

In the present paper, by considering the joint effects of different (p₁(x), p₂(x))-Laplace operator

, we study the existence and multiplicity of solutions for a nonlocal problem, that is, problem (P) via Mountain-Pass theorem and Fountain theorem. As far as we know, there is no paper that deals with a nonlocal problem involving (p₁(x), p₂(x))-Laplace operator except [15] in which the authors consider problem (P) for the case M₁ ≡ 1 and M₂ ≡ 1. Therefore, our paper deals with more general results than those obtained in [15]. Moreover, if we choose the functions M₁ ≡ M₂ ≡ 1 in problem (P), we get the equation

()

which is the well-known anisotropic

-Laplacian problem (see, e.g., [16] and references therein) in the case N = 1,2, that is,

()

As mentioned above, the p(x)-Laplacian can be applied to describe the physical phenomenon with pointwise different properties which earliest arose from the nonlinear elasticity theory. In that context, the systems involving the (p₁(x), p₂(x))-Laplacian (or (p₁(x), …, p_N(x))-Laplacian) can be good candidates for modeling phenomena which ask for distinct behavior of partial differential derivatives in various directions. For a mathematical model of a real physical phenomenon, one can consider the mean curvature operator

()

It is obvious that problem (P) is a degenerate version of (4) when M₁ ≡ M₂ ≡ 1.

2. Preliminaries

We state some basic properties of the variable exponent Lebesgue-Sobolev spaces L^p(x)(Ω) and W^1,p(x)(Ω), where Ω ⊂ ℝ^N is a bounded domain (for details, see, e.g., [17–19]).

Set

()

For any

, denote

()

and define the variable exponent Lebesgue space by

()

We define a norm, the so-called Luxemburg norm, on L^p(x)(Ω) by the formula

()

and then (L^p(x)(Ω), |·|_p(x)) becomes a Banach space.

Define the variable exponent Sobolev space by

()

then it can be equipped with the norm

()

The space

is defined as the closure of

in W^1,p(x)(Ω) with respect to the norm ∥u∥_1,p(x). For u∈

, we can define an equivalent norm

()

since Poincaré inequality holds; that is, there exists a positive constant C > 0 such that

()

for all

Proposition 1 (see [18], [19].)The conjugate space of L^p(x)(Ω) is , where (1/p^′(x)) + (1/p(x)) = 1. For any u ∈ L^p(x)(Ω) and v∈, we have

()

Proposition 2 (see [18], [19].)Denote , for all u, u_n ∈ L^p(x)(Ω) (n = 1,2, …); one has

(i)
;
(ii)
lim _n→∞ | u_n|_p(x) = 0⇔lim _n→∞ρ(u_n) = 0; lim _n→∞ | u_n|_p(x) → ∞⇔lim _n→∞ρ(u_n) → ∞.

Proposition 3 (see [18], [19].)If u, u_n ∈ L^p(x)(Ω) (n = 1,2, …), then the following statements are equivalent:

(i)
lim _n→∞ | u_n − u|_p(x) = 0;
(ii)
lim _n→∞ρ(u_n − u) = 0;
(iii)
u_n → u in measure Ω and lim _n→∞ρ(u_n) = ρ(u).

Proposition 4 (see [18], [19].)( i ) If 1 < p⁻ ≤ p⁺ < ∞, then spaces L^p(x)(Ω), W^1,p(x)(Ω), and are separable and reflexive Banach spaces.

( ii ) If and q(x) < p^*(x) for any if p(x) < N and p^*(x) = +∞ if p(x) ≥ N), then the embedding W^1,p(x)(Ω)↪L^q(x)(Ω) is compact and continuous.

Definition 5. Let X be a Banach space and J : X → ℝ a C¹ functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short), if any sequence {u_n} in X such that {J(u_n)} is bounded and J^′(u_n) → 0 as n → ∞ admits a convergent subsequence.

3. Main Results and Proofs

Let us consider the functional

()

where

with its norm given by ∥u∥ : = ∥u∥_1,p(x) + ∥u∥_2,p(x), for all u ∈ X. It is obvious that (X, ∥·∥) is also a separable and reflexive Banach space.

By using standard arguments, it can be proved that J ∈ C¹(X, ℝ) (see [20]), and the (p₁(x), p₂(x))-Laplace operator is the derivative operator of J in the weak sense. Denote L : = J^′ : X → X^*; then

()

where 〈·, ·〉 is the dual pair between X and its dual X^*.

Let us denote

()

By the definition, it is not difficult to see that p_M(x),

. For

such that

for any

, we have

, and the imbedding is continuous and compact.

We say that u ∈ X is a weak solution of (P) if

()

for any φ ∈ X.

We associate to the problem (P) the energy functional, defined as I : X → ℝ:

()

where

(i = 1,2.) and

. We know that from (M₀) and (f₀) (see below) I is well defined and in a standard way we can prove that I ∈ C¹(X, ℝ) and that the critical points of I are solutions of (P).

Moreover, the derivative of I is given by

()

for any u, φ ∈ X.

Now, we are ready to set and prove the first main result of the present paper.

Theorem 6. Assume that the following assumptions hold:

(M₀) M₁, M₂ : ℝ⁺ → ℝ⁺ are continuous functions and satisfy the conditions
()
for all t > 0, where C₁ and C₂ are positive constants and α > 1;
(f₀) is a Carathéodory function and satisfies the growth condition
()
where C₃ and C₄ are positive constants and such that , for all .

Then problem (P) has a weak solution.

Proof. Let ∥u∥ > 1. By the assumptions (M₀) and (f₀), we have

()

where

and c = min {C₁, C₂}. So, I is coercive. Since I is sequentially weakly lower semicontinuous, I has a minimum point u in X and u is a weak solution of (P).

Theorem 7. Assume that the following assumptions hold:

(M₁) M₁, M₂ : ℝ⁺ → ℝ⁺ are continuous functions and satisfy the conditions
()
for all t > 0, where C₅, C₆, C₇, C₈, and α are positive constants such that C₅ ≤ C₆ ≤ C₇ ≤ C₈ and α > 1;
(f₁) is a Carathéodory function and satisfies the growth condition
()
(f₂) , t → 0 uniformly for ,
where C₉ and C₁₀ are positive constants and such that , for all ;
(f₃) ;
(AR) ∃K > 0, such that
()
Then problem (P) has at least one nontrivial weak solution.

To obtain the result of Theorem 7, we need to show Lemmas 8 and 9 hold.

Lemma 8. Suppose (M₁), (f₁), and (AR) hold. Then I satisfies (PS) condition.

Proof. Let us assume that there exists a sequence {u_n} in X such that

()

Then by the assumptions (26), (M₁), and (AR), we get

()

where

. Since

, we have λ > 0 for θ large enough. Therefore, {∥u_n∥} is bounded in X. Passing to a subsequence, if necessary, there exists u ∈ X such that u_n⇀u. Therefore, we have the embeddings

()

By (26), we have 〈I^′(u_n), u_n − u〉 → 0. Thus

()

From (f₁) and Proposition 1, it follows that

()

If we consider the relations given in (28), we get

()

Hence,

()

From (M₁), it follows that

()

Furthermore, since u_n⇀u in X, we have

()

From (33) and (34), we deduce that

()

Next, we apply the following well-known inequality

()

valid for all r_i ≥ 2 (i = 1,2). From the relations (35) and (36), we infer that

()

and, consequently, u_n → u in X. We are done.

Lemma 9. Suppose (M₁), (f₁), (f₂), and (AR) hold. Then the following statements hold:

(i)
there exist two positive real numbers γ and a such that I(u) ≥ a > 0, u ∈ X with ∥u∥ = γ;
(ii)
there exists u ∈ X such that ∥u∥ > γ, I(u) < 0.

Proof. (i) Let ∥u∥ < 1. Then by (M₁) and Proposition 2, we have

()

where c^* = min {C₅, C₇}. Since

for all

, we have the continuous embeddings

and

, and also there are positive constants C₁₁, C₁₂ and C₁₃ such that

()

Let ε > 0 be small enough such that

. By the assumptions (f₁) and (f₂), we have

, for all

Then, for ∥u∥ ≤ 1 it follows that

()

Therefore, there exists two positive real numbers γ and a such that I(u) ≥ a > 0, for all u ∈ X with ∥u∥ = γ.

(ii) From (AR) it follows that F(x, t) ≥ c|t|^θ, for all and |t| ≥ K. In the other hand, when |t| ≥ K from (M₁) we obtain that

()

Hence, for any fixed ω ∈ X∖{0} and t > 1 we have

()

which implies I(tω) → −∞(t → +∞).

Proof of Theorem 7. From Lemmas 8 and 9 and the fact that I(0) = 0, I satisfies the Mountain-Pass theorem (see [20, 21]). Therefore, I has at least one nontrivial critical point; that is, (P) has a nontrivial weak solution. The proof is complete.

In the following, we will prove the second main result of the present paper.

Theorem 10. Suppose (M₁), (AR), (f₁), (f₂), and (f₃) hold. Then I has a sequence of critical points {u_n} such that I(u_n)→+∞ and (P) has infinite many pairs of solutions.

Since X is a reflexive and separable Banach space, then there are {e_i} ⊂ X and

such that

()

For convenience, we write X_i = span {e_i},

, and

Lemma 11. If such that for any , denote

()

Then lim _k→∞δ_k = 0.

Since the proof of Lemma 11 is similar to that of Lemma 4.9 in [7], we omit it.

Proof of Theorem 10. By the assumptions (M₁), (AR), and (f₁), I satisfies (PS) condition and from (f₃) it is also an even functional. In the sequel, we will show that if k is large enough, then there exist ρ_k > r_k > 0 such that

(i)
;
(ii)
.

Therefore, to obtain the results of Theorem 10 it is enough to apply Fountain theorem (see [21]).

(i)
For any u ∈ Z_k with ∥u∥ big enough, we have
()

Set ∥u∥ = r_k =

. Because δ_k → 0 and

, we have

()

(ii)
From (AR), we have F(x, t) ≥ C₁₆|t|^θ − C₁₇. Because and dim Y_k = k, it is obvious that I(u) → −∞ as ∥u∥ → ∞ for u ∈ Y_k.

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Solutions of Nonlocal (p₁(x), p₂(x))-Laplacian Equations

Abstract

1. Introduction

2. Preliminaries

3. Main Results and Proofs

References

References

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Solutions of Nonlocal (p1(x), p2(x))-Laplacian Equations

Abstract

1. Introduction

2. Preliminaries

3. Main Results and Proofs

References

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Solutions of Nonlocal (p₁(x), p₂(x))-Laplacian Equations