ON SOME p(x)-KIRCHHOFF TYPE EQUATIONS WITH WEIGHTS

Abstract

Consider a class of p(x)-Kirchhoff type equations of the form $$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$ where p(x), $q(x){\in}C({\bar{\Omega}})$ with 1 < $p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$ < N, $M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$ is a continuous function that may be degenerate at zero, ${\lambda}$ is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function V (x) may change sign or not.

1. Introduction

In this paper, we are concerned with the following p(x) -Kirchhoff type equations

where Ω ⊂ ℝN is a smooth bounded domain with boundary ∂Ω, with 1 < p− := infΩp(x) ≤p+ := supΩp(x) < N, M : ℝ+ → ℝ+ is a continuous function, ƒ is a Carathéodory function having special structures, and λ is a paramter.

Since the first equation in (1.1) contains an integral over Ω, it is no longer a pointwise identity, and therefore it is often called nonlocal problem. This problem models several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density, see [4]. Problem (1.1) is related to the stationary version of the Kirchhoff equation

presented by Kirchhoff in 1883, see [19]. This equation is an extension of the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the string during the vibrations. The parameters in (1.2) have the following meanings: L is the length of the string, h is the area of the cross section, E is the Young modulus of thematerial, ρ is themass density, and P0 is the initial tension.

In recent years, elliptic problems involving p-Kirchhoff type operators have been studied in many papers, we refer to some interesting works [2,5,9,21,22,25,26], in which the authors have used different methods to get the existence of solutions for (1.1) in the case when p(x) = p is a constant. To our knowledge, the study of p(x)-Kirchhoff type problems was firstly done by G. Dai et al. in the papers [11,12]. It is not difficult to see that the p(x)-Laplacian possesses more complicated nonlinearities than p-Laplacian, for example it is inhomogeneous. The study of differential equations and variational problems involving p(x)-growth conditions is a consequence of their applications. Materials requiring such more advanced theory have been studied experimentally since the middle of last century. In [11], the authors established the existence of infinitely many distinct positive solutions for problem (1.1) in the special case M(t) = a + bt. In [12], the authors considered the problem in the case when M : ℝ+ → ℝ is a continuous and non-descreasing function, satisfying the well-known condition:

which plays an enssential role in the arguments, see further papers [2,3,6,10,21,22]. There have been some authors improving (M0) in the sense that the Kirchhoff function M may be degenerate at zero, see for example [7,8,9,15]. In this paper, we assume that the Kirchhoff function M satisfies the following hypotheses:

Motivated by the ideas in [7,8,9,15] and the results in [18,23] for the p (x)-Laplacian, i.e., M(t) ≡ 1, in this paper, we consider problem (1.1) with ƒ(x, u) = λV (x) |u|q(x)−2u in two cases when the weight function V(x) may change sign or not. The results in this work suplement or complement our earlier ones in [7], in which we studied the problem in the case when the concave and convex nonlinearities were combined and the weight function did not change sign.

First, we consider the case when the parameter λ = 1 and ƒ(x, u) = V(x)|u|q(x)−2u in which the weight function V(x) does not change sign. Problem (1.1) then becomes

More exactly, V : Ω → [0,+∞) belongs to L∞(Ω) and satisfies

and the function q is assumed to satisfy

Definition 1.1. A function is said to be a weak solution of problem (1.3) if and only if

for all v ∈ X.

Our main result concerning problem (1.3) is given by the following theorem.

Theorem 1.2. Assume that the conditions (M1)-(M2), (V1) and (Q1)-(Q2) are satisfied. Then there exists a positive constant ε0 such that problem (1.3) has at least two non-trivial non-negative weak solutions, provided that|V|L∞(Ω) < ε0.

It should be noticed that Theorem 1.2 is only true when q(x) is a non-constant function while p(x) may be a constant. If p(x) = p is a constant then it follows from (Q2) that α = β.

Next, we consider problem (1.1) in the case when ƒ(x, u) = λV (x)|u|q(x)−2u, in which V(x) is a sign changing weight function, that is,

More exactly, we study the existence of solutions for (1.4) under the hypotheses (M1), (M2) and

and the function is assumed to satisfy the following condition

As we shall see in Section 4, due to the hypothesis (Q3), we cannot use the mountain pass theorem [1] in order to get the solutions for problem (1.4) as in Theorem 1.2. We emphasize that this is the main different point between two problems (1.3) and (1.4).

Definition 1.3. A function is said to be a weak solution of problem (1.4) if and only if

for all v ∈ X.

Our main result concerning problem (1.4) in this case is given by the following theorem.

Theorem 1.4. Assume that the conditions (M1)-(M2), (V2) and (Q3) are satisfied. Then there exists a positive constant λ∗ such that for any λ ∈ (0, λ∗), problem (1.4) has at least one non-trivial non-negative weak solution, i.e., any λ ∈ (0, λ∗) is an eigenvalue of eigenvalue problem (1.4).

Our paper is organized as follows. In the next section, we shall recall some useful concepts and properties on the generalized Lebesgue-Sobolev spaces. Section 3 is devoted to the proof of Theorem 1.2 while we shall present the proof of Theorem 1.4 in Section 4.

 

2. Preliminaries

We recall in what follows some definitions and basic properties of the generalized Lebesgue-Sobolev spaces Lp(x) (Ω) and W1,p(x) (Ω) where Ω is an open subset of ℝN. In that context, we refer to the book of Musielak [24] and the papers of Kováčik and Rákosník [20] and Fan et al. [16,17]. Set

For any we define h+ = supx∈Ω h(x) and h− = infx∈Ω h(x). For any , we define the variable exponent Lebesgue space

We recall the following so-called Luxemburg norm on this space defined by the formula

Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p− ≤ p+ < ∞ and continuous functions are dense if p+ < ∞. The inclusion between Lebesgue spaces also generalizes naturally: if 0 < |Ω| < ∞ and p1, p2 are variable exponents so that p1(x) ≤ p2(x) a.e. x ∈ Ω then there exists a continuous embedding the conjugate space of Lp(x)(Ω), where For any u ∈ Lp(x)(Ω) and v ∈ Lp′(x)(Ω) the Holder inequality

holds true.

Moreover, if h1, h2 and h3 : are three Lipschitz continuous functions such that then for any u ∈ Lh1(x)(Ω), v ∈ Lh2(x)(Ω) and w ∈ Lh3(x)(Ω), the following inequality holds:

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the Lp(x)(Ω) space, which is the mapping ρp(x) : Lp(x)(Ω) → ℝ defined by

Proposition 2.1 ([17]). If u ∈ Lp(x)(Ω) and p+ < ∞ then the following relations hold

provided that |u| p(x) > 1 while

provided that |u| p(x) < 1 and

Proposition 2.2 ([18]). Let p and q be measurable functions such that p ∈ L∞(Ω) and 1 ≤ p(x)q(x) ≤ ∞ for a.e. x ∈ Ω. Let u ∈ Lq(x)(Ω), u ≠ 0. Then the following relations hold

provided that |u| p(x) ≤ 1 while

provided that |u| p(x) ≥ 1 In particular, if p(x) = p is a constant, then

In this paper, we assume that is the space of all the functions of which are logarithmic Höolder continuous, that is, there exists R > 0 such that for all x, y ∈ Ω with see [13,16]. We define the space under the norm

Proposition 2.3 ([17,18]). The space is a separable and Banach space. Moreover, if then the embedding is compact and continuous, where

 

3. Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2, which is essentially based on the mountain pass theorem [1] combined with the Ekeland variational principle [14].

Let us define the functional by the formula

where

where Then, the functional J associated with problem (1.1) is well defined and of C1 class on X. Moreover, we have

for all u, v ∈ X. Thus, weak solutions of problem (1.3) are exactly the ciritical points of the functional J. Due to the conditions (M1) and (Q1), we can show that J is weakly lower semi-continuous in X. The following lemma plays an essential role in our arguments.

Lemma 3.1. The following assertions hold:

Proof. We shall prove Lemma 3.1 in details for the case

the remaining case is similarly proved.

(i) Let us define the function We denote By the conditions (Q1) and (Q2),

which helps us to deduce that X is continuously embedded in Then there exists a positive constant c1 such that

From (3.4), there exist two positive constants c2, c3 such that

and

Using the hypothesis (M1), relations (3.5) and (3.6) give us

for all u ∈ X with ||u|| < 1. Since the function g : [0, 1] → ℝ defined by

is positive in a neighbourhood of the origin, it follows that there exists ρ1 ∈ (0, 1) such that g(ρ1) > 0. On the other hand, defining

we deduce that, for any |V| L∞(Ω) < ε0, there exists γ1 > 0 such that for any u ∈ X with ∥u∥ = ρ1 we have J(u) ≥ γ1. (ii) Let and there exist x1 ∈ Ω\BR (x0) and ε > 0 such that for any x ∈ Bε x1) ⊂ (Ω\BR(x0)) we have ψ1(x) > 0. For any t > 1, we have

Since we infer that limt→ ∞ J (tψ1) = −∞. (iii) Let φ1 ≥ 0 and there exist x2 ∈ Br(x0) and ε > 0 such that for any x ∈ Bε(x2) ⊂ Br(x0) we have φ1(x) > 0. Letting 0 < t < 1 we find

Obviously, we have J(tφ1) < 0 for any where

The proof of Lemma 3.1 is complete.

Lemma 3.2. The functional J satisfies the Palais-Smale condition in X.

Proof. Let {um} ⊂ X be such that

where X∗ is the dual space of X.

We shall prove that {um} is bounded in X. In order to do that, we assume by contradiction that passing if necessary to a subsequence, still denoted by {um}, we have ∥um∥ → ∞ as m → ∞. By (3.12) and (M1)-(M2), for m large enough and |V| L∞(Ω) < ε0, we have

Dividing the above inequality by ∥um∥αp− taking into account that (3.3) holds true and passing to the limit as m → ∞ we obtain a contradiction. It follows that {um} is bounded in X. Thus, there exists u1 ∈ X such that passing to a subsequence, still denoted by {um}, it converges weakly to u1 in X. Then {∥um − u∥} is bounded. By (3.3), the embedding from X to the space Lq(x)(Ω) is compact. Then, using the Hölder inequality, Propositions 2.1-2.3, we have

This fact and relation (3.12) yield

Since {um} is bounded in X, passing to a subsequence, if necessary, we may assume that

If t0 = 0 then {um} converges strongly to u = 0 in X and the proof is finished. If t0 > 0 then we deduce by the continuity of M that

Thus, by (M1), for sufficiently large m, we have

From (3.15), (3.16), it follows that

Thus, {um} converges strongly to u in X and the functional J satisfies the Palais-Smale condition.

Proof of Theorem 1.2. By Lemmas 3.1 and 3.2, all assumptions of the mountain pass theorem in [1] are satisfied. Then we deduce u1 as a non-trivial critical point of the functional J with J(u1) = and thus a non-trivial weak solution of problem (1.3).

We now prove that there exists a second weak solution u2 ∈ X such that u2≠ u1. Indeed, let ε0 as in the proof of Lemma 3.1(i) and assume that |V| L∞(Ω) < ε0. By Lemma 3.1(i), it follows that on the boundary of the ball centered at the origin and of radius ρ1 in X, denoted by Bρ1(0) = {u ∈ X : ∥u∥ < ρ1}, we have

On the other hand, by Lemma 3.1(ii), there exists φ1 ∈ X such that J(tφ1) < 0 for all t > 0 small enough. Moreover, from (3.7), the functional J is bouned from below on Bρ1 (0). It follows that

Applying the Ekeland variational principle in [14] to the functional it follows that there exists such that

By Lemma 3.1, we have

Let us choose ε > 0 such that

Then, J(uε) < infu∈∂Bρ1(0) J(u) and thus, uε ∈ Bρ1 (0).

Now, we define the functional It is clear that uε is a minimum point of I and thus

for all t > 0 small enough and all v ∈ Bρ1 (0). The above information shows that

Letting t → 0+, we deduce that

It should be noticed that −v also belongs to Bρ1 (0), so replacing v by −v, we get

or

which helps us to deduce that ∥J′(uε)∥X∗ ≤ ε. Therefore, there exists a sequence {um} ⊂ Bρ1(0) such that

From Lemma 3.2, the sequence {um} converges strongly to u2 as m → ∞. Moreover, since J ∈ C1(X,ℝ), by (3.17) it follows that Thus, u2 is a non-trivial weak solution of problem (1.2).

Finally, we point out the fact that u1 ≠ u2 since Moreover, since J(u) = J(|u|), problem (1.3) has at least two non-trivial nonnegative weak solutions. The proof of Theorem 1.2 is complete.

 

4. Proof of Theorem 1.4

In this section, assume that we are under the hypotheses of Theorem 1.4, we shall prove Theorem 1.4 using the Ekeland variational principle [14]. For each λ ∈ ℝ, define the functional

where

From (V2), (2.4) and (2.5), it is clear that for all u ∈ X,

On the other hand, by (V2) and (Q3), we have and thus the embeddings are continuous and compact. For these reasons, we can use the similar arguments as in [18, Proposition 2] in order to show that the functional Jλ is well-defined. Moreover, Jλ is of C1 class in X and

for all u, v ∈ X. Thus, weak solutions of problem (1.4) are exactly the ciritical points of the functional Jλ.

Lemma 4.1. For any ρ2 ∈ (0, 1), there exist λ∗ > 0 and γ2 > 0 such that for all u ∈ X with ∥u∥ = ρ2,

Proof. Since the embedding is continuous, there exists a positive constant c7 such that

Now, let us assume that where c7 is the positive constant from above. Then we have Using relations (2.2), (4.2), the condition (M1) and the Hölder inequality, we deduce that for any u ∈ X with ∥u∥ = ρ2 ∈ (0, 1) the following inequalities hold true

By (Q3) we have q− ≤ q+ < p− ≤ p+ < αp+. So, if we take

then for any λ ∈ (0, λ∗) and u ∈ X with ∥u∥ = ρ2, there exists γ2 > 0 such that Jλ(u) ≥ γ2 > 0. The proof of the Lemma 4.1 is complete.

Lemma 4.2. For any λ ∈ (0, λ∗), where λ∗ is given by (4.4), there exists ψ2 ∈ X such that ψ2 ≥ 0, ψ2 ≠ 0 and Jλ(tψ2) < 0 for all t > 0 smaller than a certain value depending on λ.

Proof. From (Q3) we have q(x) < βp(x) for all where Ω0 is given by (V2). In the sequel, we use the notation Let δ0 > 0 be such that there exists an open set Ω1 ⊂ Ω0 such that for all x ∈ Ω1. It follows that

Let such that supp(ψ2) ⊂ Ω1 ⊂ Ω0, ψ2 = 1 in a subset Then, using (M1) we have

Therefore

Finally, we shall point that

In fact, due to the choice of ψ2, if Using (2.3), we deduce that |∇ψ2| = 0 and consequently ψ2 = 0 in Ω, which is a contradiction. The proof of Lemma 4.2 is complete.

Proof of Theorem 1.4. Let λ∗ > 0 be defined by (4.4) and λ ∈ (0, λ∗). By Lemma 4.1, it follows that on the boundary of the ball centered at the origin and of radius ρ2 in X, denoted by Bρ2 (0), we have

On the other hand, by Lemma 4.2, there exists ψ2 ∈ X such that Jλ(tψ2) < 0 for all t > 0 small enough. Moreover, relation (4.3) implies that for any u ∈ Bρ2(0) we have

It follows that

Using the Ekeland variational principle [14] and the similar arguments as those used in the proof of Theorem 1.1, we can deduce that there exists a sequence {um} ⊂ Bρ2(0) such that

It is clear that {um} is bounded in X. Thus, there exists u ∈ X such that, up to a subsequence, {um} converges weakly to u in X. Since we deduce that X is compactly embedded in hence the sequence {um} converges strongly to u in Using the Hölder inequality, we have

Now, if

The compact embedding ensures that

Relation (4.6) yields

Using the above information, we also obtain relation (3.15) and thus, {um} converges strongly to some u in X. So, by (4.6), It is clear that Jλ(|u|) = Jλ(u). Therefore, u is a non-trivial non-negative weak solution of problem (1.4). Theorem 1.4 is completely proved.

Remark 4.3. We cannot use the mountain pass argument in the proof of Theorem 1.4 since the functional Jλ does not satisfy the geometry of the mountain pass theorem. More exactly, we cannot find a function φ2 ≥ 0 such that Jλ (tφ2) → −∞ as t → ∞ as in Lemma 3.1.