The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces

Given a Musielak–Orlicz function φ(x,s):Ω×[0,∞)→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}$$\end{document} on a bounded regular domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb R}^n$$\end{document} and a continuous function M:[0,∞)→(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:[0,\infty )\rightarrow (0,\infty )$$\end{document}, we show that the eigenvalue problem for the elliptic Kirchhoff’s equation -M∫Ωφ(x,|∇u(x)|)dxdiv∂φ∂s(x,|∇u(x)|)∇u(x)|∇u(x)|=λ∂φ∂s(x,|u(x)|)u(x)|u(x)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-M\left( \int \limits _{\Omega }\varphi (x,|\nabla u(x)|)\textrm{d}x\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|} $$\end{document} has infinitely many solutions in the Sobolev space W01,φ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_0^{1,\varphi }(\Omega )$$\end{document}. No conditions on φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} are required beyond those that guarantee the compactness of the Sobolev embedding theorem.


Introduction
In 1883, G. Kirchhoff [13] noted that the vibration of an elastic, variable-length string is modeled by means of the following variant of the classical wave equation: where M : [0, ∞) −→ [0, ∞) is a suitable increasing function.Since then, a vast amount of literature was devoted to studying the solvability of various Kirchhoff-type equations [2].In higher dimensions, (1.1) takes up the form O. Méndez (B) Department of Mathematical Sciences, University of Texas at El Paso, 124 Bell Hall, 500W University Ave., El Paso, TX, USA E-mail: osvaldomendez2007@hotmail.com; osmendez@utep.edu has been extensively studied under different assumptions on M and f , see for example [1,19,20,26,27] and the references therein.
Of particular interest is the extension of (1.2) to equations involving the p-Laplacian [10,18]: If 1 < p < ∞ is a real number and M ≥ 0 is a continuous function, the p-Kirchhoff operator is defined as (1.4) which clearly generalizes the right-hand side of (1.1).In (1.4), W 1, p 0 ( ) stands for the closure of C ∞ 0 ( ) in the usual Sobolev space W 1, p(•) ( ).The reader is referred to Sect. 3 for the precise terminology to be used in this work.
From the physical point of view, this operator arose from the need of finding a mathematical model for the motion of a vibrating string under less stringent assumptions than those assumed for the classical derivation of the linear wave equation.Specifically, the linear wave equation is obtained under the assumption that the length of the vibrating string remains constant during the motion.By removing this assumption, the nonlinear operator K p comes into the play.
Various boundary value problems associated to the p-Kirchhoff operator (1.4) have been studied for example in [10,18,19].The emergence of the variable exponent Lebesgue spaces and the subsequent realization of their role in applications [5] sparked interest in the study of boundary value problems of the type for a variable exponent p = p(x).
The variability of the exponent opens a new class of highly non-trivial difficulties, mainly related to its modular nature, that is to its direct relation to the functional u −→ |u(x)| p(x) dx rather than with the norm as discussed in [2,21].A vast amount of literature exists on boundary value problems of the type (1.5), under the assumption of the variability of the exponent p(x).We refer the reader to some of the most significant from the point of view of the present work, such as [2][3][4][6][7][8]21,23].
In this article we observe that the treatment of a wide class of eigenvalue problem for Kirchhoff-type operators, including, but not limited to the variable-exponent case, can be unified by the consideration of Musielak-Orlicz spaces.With this objective in mind, we study the eigenvalue problem for a general Kirchhoff equation in this framework.In fact, given a suitable Musielak-Orlicz (M O)-function ϕ and an appropriate function M (we refer the reader to the next section for a detailed account of the notation and terminology), the generalized Kirchhoff operator is naturally given by We provide a characterization of the first eigenvalue for the operator (1.6) via a Musielak-Orlicz Sobolev embedding theorem that has been obtained in [16,Theorem 5.1].
The present work is organized as follows.In the next section we introduce the notation and terminology to be used in the exposition and present a brief survey on the literature.In Section 3 the definition and basic properties of the Musielak-Orlicz spaces needed in the sequel are given.Section 4 is a brief survey on the Sobolev embedding theorems in the context of Musielak-Orlicz spaces.In Section 5 we delve into some natural properties of the Musielak-Orlicz operators to be considered later and the functional analytic stage is set for the treatment of the eigenvalue problems developed in detail in Section 7. Section 8 contains applications to the variable exponent case, i.e., to the case ϕ(x, t) = t p(x) .

Known results
In the sequel, ⊂ R n will denote a bounded domain with a regular boundary (the cone condition will do) and M( ) will stand for the vector space of all real-valued, Borel-measurable functions defined on .The subset of M consisting of functions will be denoted by P( ).The Lebesgue measure of a subset A ⊂ R n will be denoted by |A|.
For p ∈ P( ), the following notation will be used throughout this work: p − := essinf p, p + := esssup p.
For p ∈ P( ), the eigenvalue problem was studied in [2] for M subject to or 1 < q(x) < p(x) < p * (x) (2.5) in .Here and 1 A stands for the characteristic function of the set A.
An anisotropic variant of (2.1) was considered in [23], whereas [4] deals with the following weighted version of (2.1): is studied in [12].Specifically, problem (2. For M(t) = a + bγ t γ −1 , the study of the solvability of a hyperbolic equation related to the operator K p can be found in [3].A polyharmonic version of (1.5) was studied in [6].
Associated to every Musielak-Orlicz function ϕ, the so-called Matuszewska index of ϕ (see [17]) generalizes the role of the exponent p in the classical Lebesgue spaces; in particular the exponent p is easily verified to be the Matuszewska index of the M O function given by As is observed in [16], sharp conditions (trivially satisfied by the exponent p for the Sobolev embedding stated in [11,15]) on the Matuszewska index of the M O function ϕ guarantee the compactness of the Sobolev embedding for a bounded domain ⊂ R n .Via the compactness of the Sobolev embedding, a natural characterization of the first eigenvalue of the Kirchhoff's operator can be given, and the results outlined above can be regarded as particular cases of our more general approach, which allows for less stringent conditions than the ones stated in the first part of this Section.

Musielak-Orlicz spaces
Throughout this paper ⊂ R n , n ≥ 1 will stand for a bounded, Lipschitz domain.A convex, left-continuous function with ϕ(0) = 0, lim x→∞ ϕ(x) = ∞ and lim x→0 + ϕ(x) = 0 will be said to be an Orlicz function.In particular, any Orlicz function is non-decreasing.The term generalized Orlicz function or Musielak-Orlicz (MO) function will refer to a function is an Orlicz function for each fixed x ∈ and is Lebesgue measurable for each fixed y ∈ R.
The Musielak-Orlicz space L ϕ ( ), [24,25], is the real-vector space X ϕ of all extended-real valued, Borelmeasurable functions u on for which ϕ(x, λ|u(x)|) dx < ∞ for some λ > 0, furnished with the norm The functional is a convex, left-continuous modular on X ϕ [9,11,24].It is well known [9] that L ϕ ( ) is a Banach space.Since it will be needed in the sequel, we define the complementary function ϕ * of ϕ as The complementary function ϕ * is itself a M O-function (see [9]) and Hölder's inequality holds, namely for the Musielak-Orlicz Sobolev space W 1,ϕ ( ) consisting of all functions in L ϕ ( ) whose distributional derivatives are in L ϕ ( ), is a Banach space when furnished with the norm where ∇ stands for the gradient operator and

Sobolev-type embeddings
The central idea of this Section is the Sobolev Embedding Theorem 4.7.In order to facilitate the flow of ideas we present a few definitions.The Matuszewska index of an Orlicz function ϕ was introduced by Matuszewska and Orlicz in [17].
Definition 4.1 For ϕ as above and each x ∈ , set The Matuszewska index of ϕ is defined to be Definition 4. 2 The limit (4.1) is said to be uniform if for each δ > 0 there exist s 0 > 1 and T > 1 such that, for all (x, t) ∈ × [T, ∞), one as 3) The following examples illustrate the above definition for some well known M O functions: Example 4.3 Let ⊆ R n be a bounded domain and has Matuszewska index equal to p(x).In this case, the convergence (4.2) is trivially uniform on and the limit (4.2) is clearly uniform.for any s ≥ S 0 .
Condition (4.6) will be referred to as the condition.
Proof Fix δ > 0, then for some T 0 > 1 one has for any t ≥ T 0 , by virtue of (4.2) uniformly in .By definition of M(x, t) and on account of the uniformity assumption of the infimum (4.1), there exists a positive number N for which, uniformly for t ≥ T 0 and x ∈ , it holds that In particular, for all s ≥ N : Proof It suffices to show that if (u j ) ρ ϕ -converges to 0 and converges a.e. to 0. then it converges to 0 in the topology of the norm.This will be automatically implied by the validity of the equality Since the second term in the right-hand side tends a.e. to 0 as n → ∞, it follows that We refer the reader to [16] for the proof of the following theorem.
and that there exists a function such that, uniformly in and for t > 0: Then the embedding Proof See [16].

5 2 -type Musielak-Orlicz functions
From now on we assume that ⊆ R n is a bounded domain satisfying the cone condition and ϕ is a M Ofunction satisfying the conditions of Theorem 4.7.Denote the conjugate of ϕ by ϕ * .
Theorem 5.1 Let ϕ be a M O function on ; assume that ϕ satisfies the conditions of Theorem 4.7; in particular, ϕ satisfies the condition 4.6, i.e., for some K > 0, S 0 > 1 it holds that ϕ(x, 2s) ≤ K ϕ(x, s) for all s ≥ S 0 , x ∈ . (5.1) Then, Proof It follows from (5.1) that for arbitrary v ∈ L ϕ ( ) (recall that ϕ is nonnegative and nondecreasing) it is a simple matter to verify that if it is thus clear that if (5.4) holds, then: Therefore, It follows from the preceding reasoning in conjunction with Poincaré inequality that if ρ ϕ (|∇u|) ≤ r , then, for some C > 0, Hence, Otherwise, on account of the iteration of inequality (5.3) In all, (5.6) and (5.7) yield (5.2).
An immediate consequence of the preceding theorem is the following functional-analytic result: Lemma 5.2 For ϕ as in Theorem 4.7 and r > 0, the modular ball and there is no loss of generality in assuming that ∇u j → ∇u a.e. in .Theorem 4.7 guarantees that (u j ) can be chosen so that u j → u a.e. in .Since it follows that u ∈ B r , i.e., B r is norm-closed (and convex) and hence weakly closed.

Differentiability properties
Aiming at a full description of the Fréchet derivative of the functionals to be introduced momentarily, a further assumption is imposed unto the M O function ϕ at this point, namely, it is from now on required that ϕ be an N function.More precisely: It is well known [9] that if ϕ is an N -function, it can be written as where φ(x, •) is the right t-derivative of ϕ.On the other hand, the conjugate function ϕ * can be written as 3) The proof of the following theorem can be found in [14,22]; we include it here in the interest of completeness.
123 Theorem 6.2 Let ϕ be an N -function; assume that is continuous a.e.x ∈ .Define the operator T ϕ t as Then, from the assumption it follows that the operator is continuous and bounded. Proof then on account of the assumption on ϕ t , w is a Carathéodory function and w(x, 0) = 0.If If ϕ * satisfies the 2 condition, the latter is equivalent to that is Therefore, it is enough to show that T ϕ t is continuous at 0 under the assumption that ϕ t (x, 0) = 0 a.e. in .Assume that T ϕ t is not continuous at 0; let r > 0 and let (u n ) be a sequence that converges to 0 in L ϕ ( ), for which Since norm convergence implies modular convergence, one can, without loss of generality assume that and hence that ∞ n=1 ϕ(x, |u n (x)|) dx < ∞.Due to the validity of the 2 condition for ϕ * , norm convergence and modular convergence are equivalent on L ϕ * ( ).It follows that there exists (r ) > 0 such that ρ ϕ * T ϕ t (u n ) ≥ (r ) for any n ∈ N. We next claim the existence of a sequence of real numbers ( k ), a sequence ( k ) of subsets of and a subsequence (u n k ) of (u n ) satisfying the following conditions: We assume that k , n k and k are given, then, by assumption, ϕ t (•, |u n k (•)|) ∈ L ϕ * ( ) and on account of the 2 condition one has Since the measure defined on the Borel σ algebra B of subsets of is absolutely continuous with respect to the Lebesgue measure, one can find k+1 such that any The assumption k ≤ 2 k+1 would contradict (iii).We now proceed to the construction of k+1 and n k+1 .
It is well known [14] that the strong convergence of (u n ) in L ϕ ( ) implies the convergence in measure of both (T ϕ t (u n )) and (T ϕ * (T ϕ t (u n ))).Consequently, there exists n k+1 ∈ N such that

By construction
It is clear that v ∈ L ϕ ( ).Next, observe that: which follows by (6.5) and condition (iv).This contradicts assumption (6.4).Hence, T ϕ t is continuous at 0.
We next prepare the ground for the next lemma, which deals with the differentiability properties needed in the sequel.Let M : R −→ [0, ∞) be continuous and write Consider the maps and Recall that a M O function ϕ is said to be locally integrable if for any t > 0 and any subset W ⊆ with μ(W) < ∞ one has W ϕ(x, t) dx < ∞.

Lemma 6.3
In the terminology of the preceding paragraph, let ϕ be an N function; suppose that ∂ ∂t ϕ(x, •) is continuous a.e.x.Assume that the complementary function ϕ * of ϕ satisfies the condition and is locally integrable.If the maps and are well defined, then the functionals (6.7) and (6.8) are Fréchet differentiable for u = 0.In this case, the derivatives of F and H at u = 0 are given, respectively by and by with the understanding that x |x| = 0 if x = 0.
Proof It suffices to show that under the stipulated conditions (6.13) holds; a straightforward application of the chain rule will yield the full statement of the Lemma.
Observe that for u(x) = 0, h ∈ L ϕ ( ) one has By hypothesis, which in conjunction with the above limit yields: On the other hand, one has by assumption, one has a.e.x ∈ : and by convexity, for 0 it is easily derived by way of application of Lebesgue's theorem that In all, We conclude that F is Gâteaux differentiable and that its derivative is equal to the right-hand side of (6.13).
The proof of the Gâteaux differentiability of G follows along the same lines.Hence, it suffices to prove that under the additional assumptions (6.9) and (6.11), the operators are continuous at u = 0.A standard functional-analytic result guarantees that in this case F and G are Fréchet differentiable and that the Fréchet and the Gâteaux derivatives coincide.To this end, consider a convergent sequence (u j ) ⊂ L ϕ ( ), say u j −→ u in L ϕ ( ) as j −→ ∞: it is well known that there is no loss of generality by assuming that u j converges to u almost everywhere in (see [9,11])).
Since ∂ϕ ∂t (x, |u(x)|) ∈ L ϕ * ( ) there exists λ 0 > 0 for which Notice that for h ϕ ≤ 1 one has Theorem 6.2 guarantees the continuity of the map (6.9); it is apparent from this fact in conjunction with Hölder's inequality (3.4) that the integral in (6.21) tends to 0 as j tends to infinity.As to the remaining integral, set, for j ∈ N: Recall that in the terminology of Lemma 4.4, for any s ≥ S 0 .For λ 0 as in (6.20) and any λ > 2λ 0 let k be defined by the inequalities Notice that |r j | ≤ 2. For S 0 as in Lemma 4.4, one readily obtains, for any positive integer m, using the monotonicity of ϕ * in the second variable The iteration of the preceding inequality yields A routine application of Lebesgue's dominated convergence quickly shows that A similar argument shows that (6.25) holds for 0 < λ ≤ 2λ 0 .It follows from the arbitrariness of λ that On account of Hölders inequality (3.4), the integral (6.22) is bounded by The proof of the continuity of L G follows along the same lines and will be skipped.This concludes the continuity argument and hence F and G are Fréchet differentiable.Proof For any r > 0 set s r = M−1 (r ); it is then clear from the above assumptions that and the latter set is weakly closed (Lemma 5.2).

Kirchhoff-type eigenvalue problem
We are now ready to prove the main result of this work.We next observe that Theorem 4.7 implies the existence of a sequence (u j ) ⊂ V s r with u j u 0 in W 1,ϕ 0 ( ) and u j −→ u 0 in L ϕ ( ) such that ρ ϕ (u) −→ S r = ρ ϕ (u 0 ) .(7.4) To see this, we notice that a.e. in , and that on account of convexity, for any n ∈ N: Select n large enough so that 2 u − u n ϕ < 1; for such n, it holds, by way of the convexity of ϕ(x, •), Denote the left-hand side and the right-hand side of (7.5) by v n and w n respectively.Then the following conditions hold: Since w − v ≥ 0 a.e in , Fatou's Lemma leads to: The two last statements yield As is apparent from the above, u 0 is a solution of the constrained maximization problem of the type max F (v) with G(v) = r. (7.8) It is a routine matter to show, via the Implicit Function Theorem, that the preceding statement implies that kerG (u 0 ) = kerF (u 0 ), (7.9) from which it is clear (since neither functional is null) that u 0 satisfies equation (7.1) for some λ > 0. This concludes the proof of the claim.

Applications
A particular instance of Theorem 7.1 deserves to be stressed, namely its implication in the consideration of variable exponent Lebesgue spaces.More precisely, if ⊂ R n is bounded then With the aid of the following compactness theorem, the techniques used in Sections 6 and 7, one can derive Theorems 8.3 and 8.4, particular cases of which were obtained in [2] and [4], respectively, via different methods.

Lemma 4 . 4
Let ⊂ R n be a bounded domain and ϕ an MO function as described above.If the Matuszewska index m is the restriction to of a continuous function m on the closure of , i.e., m : −→ R, (4.5) and the convergence to the limits (4.1) and (4.2) is uniform, then there exist C > 1, T 0 > 1 and S 0 > 1 such that uniformly in it holds ϕ(x, sT 0 ) ≤ Cϕ(x, s) (4.6)

Theorem 4 . 7
Let ⊂ R n be a bounded domain andϕ : × [0, ∞) −→ Ra Musielak-Orlicz function that satisfies condition (3.6) and for which the limits in (4.1) and (4.2) are uniform; assume that the Matuszewska index m is the restriction to of a continuous function m defined on the closure of , that

Corollary 4 . 8
For ϕ satisfying the conditions of Theorem 4.7, there exists a positive constant C depending only on n, , ϕ such that for any u ∈ W

Definition 6 . 1
An M O function is said to be an N -function iff it satisfies the condition lim t→0