A topological analysis of p(x)-harmonic functionals in one-dimensional nonlocal elliptic equations

1 Introduction

Suppose that a and b are L 1 ( 0 , + ∞ ) functions. Define the finite convolution of a and b at t ≥ 0, denoted (a∗b)(t), by

The finite convolution is an important nonlocal functional, being as it occurs in many frequently studied nonlocal operators, the most well known of which may be the fractional differential and integral operators. For example, if we set a ( t ) ≔ 1 Γ ( α ) t α − 1 , where 0 < α < 1, then (a∗u)(t) is the α-th order fractional integral of Riemann–Liouville type – see, for example [1], [2], [3], [4], [5], for details.

At the same time, an important functional is the p(x)-harmonic functional, which for u sufficiently regular is defined, in the one-dimensional setting, by

where p is a sufficiently regular function defined on [0, 1]. In our setting, p and u′ will be continuous on [0, 1]. In the higher dimensional setting, the functional (1.1) takes the form

and it plays a very important role in regularity theory. In part, this is due to minimisers of the functional being related to weak solutions of p(x)-Laplacian equations – see, for example, [6]. Such functionals were introduced by Zhikov [7], subsequently analysed by Coscia and Mingione [8], and since then studied by many authors such as Fey and Foss [9], Ragusa and Tachikawa [10], [11], and Usuba [12]. These problems have applications in electrorheological fluids and thermistors, amongst other areas.

Finally, regarding nonlocal differential equations, one of the most well-studied is the Kirchhoff-type equation, which is a parabolic PDE having the form

where, clasically, p = 2. When u _tt ≡ 0 so that steady-state solutions are sought, and the space dimension is one, then the parabolic PDE reduces to the elliptic one-dimensional nonlocal equation

In this paper we consider a combination of these three themes: finite convolutions, p(x)-harmonic functionals, and Kirchhoff-type nonlocal equations in dimension one. More precisely, we study, for λ > 0 a parameter, both the existence and nonexistence of at least one positive solution to

subject to the boundary data

where in (1.2), and, indeed, throughout the remainder of this paper, we use the notation

Observe that in the model case a(s) ≡ 1, problem (1.2) reduces to the model case

which is a one-dimensional Kirchhoff problem with a p(x)-harmonic functional nonlocal coefficient. In case p(s) ≡ p ₀ ≥ 1, then the model case further reduces to

which is the classical one-dimensional Kirchhoff problem. Thus, problem (1.2) and (1.3) is a natural generalisation. Furthermore, as intimated above, part of the reason for considering the convolutional structure in the nonlocal coefficient is that this permits the study of a wide range of nonlocal coefficients, including fractional-order operators as mentioned above.

One of the difficulties in managing the variable exponent case is that a number of inequalities that we would typically use, amongst these are Sobolev, Jensen, and Hölder, do not interact so nicely with non-constant exponents. In regularity theory, there is a standard strategy for circumventing this difficulty. Indeed, there one typically uses some assumed regularity about the exponent p (often, for example, some degree of Hölder continuity) to approximate quantities involving p(x) by switching from p(x) to either the supremum or infimum of p on a set of very small measure. For typical examples of this strategy, one can look at the arguments in any number of regularity papers involving p(x)-harmonic functionals [6], [8], [10], [11], [12] – see also Diening, et al. [13]. The problem is that in our setting we cannot simply use sets of arbitrarily small measure. Instead we must constantly work on the entire interval on which the differential equation is valid – i.e., 0 < t < 1. This means that we cannot directly import the strategies from regularity theory.

A different strategy, which we utilised in a recent paper [14], is to instead try to utilise a Harnack inequality. In [14] we considered a nonlocal functional coefficient of the form

and to control the variable exponent we calculated, for C > 0 a constant,

passing from u(t) to ‖u‖_∞ from below by means of a Harnack-type inequality. Then by controlling the size of ‖u‖_∞ we could pass to a constant exponent from below. Once again, this strategy is of no use in the p(x)-harmonic functional setting. The reason is that because the nonlocal functional (in the model case a(t) ≡ 1) has the form

we cannot expect a Harnack-like equality to hold for u′.

So, this is where a completely new idea is required. Indeed, one of the principal contributions of this present work is to develop a way to handle the variable exponent with neither the tools from regularity theory nor the Harnack-inequality approach as in [14]. The resolution is a careful estimation – see Lemma 2.6 – of the variable exponent quantity in terms of a constant exponent quantity, and, crucially, this estimation is global in nature, not merely local as is the custom in regularity theory. Moreover, due to the Dirichlet boundary data (1.3), we also make careful use of a version of Sobolev’s inequality. This was an idea we recently utilised in [15], though some careful adaptations are required in the variable exponent setting.

We have spent most of this introduction explaining the motivation for considering and the difficulties inherent in treating p(x)-harmonic functionals. We wish to conclude this section by highlighting the broader literature on nonlocal differential operators. Most of the existing papers in nonlocal differential equations have a form roughly similar to either the case (and its ODE equivalent)

or the case (and, again, its ODE equivalent)

Equation (1.4), in which the nonlocal element depends only on u itself, has been studied in papers by Alves and Covei [16], Corrêa [17], Corrêa, Menezes, and Ferreira [18], do Ó, Lorca, Sánchez, and Ubilla [19], Goodrich [20], Stańczy [21], Wang, Wang, and An [22], Yan and Ma [23], and Yan and Wang [24]. Similarly, equation (1.5), in which the nonlocal element depends on du (or u′), has been studied by Afrouzi, Chung, and Shakeri [25], Ambrosetti and Arcoya [26], Azzouz and Bensedik [27], Boulaaras [28], Boulaaras and Guefaifia [29], Delgado, Morales-Rodrigo, Santos Júnior, and Suárez [30], Graef, Heidarkhani, and Kong [31], Infante [32], [33], and Santos Júnior and Siciliano [34]. Mostly this research has investigated existence of solution (as we do in Section 2), though there are a few works on nonexistence (as we do in Section 3), including recent papers by both Shibata [35], [36], [37] and the author [38], [39], [40]. Additionally, nonlocal differential and integral operators have been studied extensively, and here we direct the interested reader to the following references [41], [42], [43], [44], [45], some of which detail applications to beam deformation, nuclear reactor modelling, and heat transfer problems.

In the analysis of problems such as (1.4) and (1.5) a very common assumption is that M(t) > 0 for all t ≥ 0, and, sometimes in addition, some sort of growth assumptions on M. For example, the results of [17], [18], [19], [21], [22] cannot be applied to the case in which M is either sign-changing or vanishing. Similarly, Ambrosetti and Arcoya [26] require positivity eventually, Santos and Siciliano [34] require positivity on a neighbourhood of zero, and Infante [46] disallows negativity. Moreover, none of these authors’ methods were applied to either the convolutional setting or the p(x)-harmonic setting. By contrast, our recently developed methodology [47], [48], [49], [50], together with Lizama [51], allows a precise localisation of the argument of M, thereby avoiding all manner of assumptions on M; these techniques have been developed further by both Hao and Wang [52] and Song and Hao [53]. Moreover, it is fully compatible with realising the nonlocal element as a finite convolution. The key to this is the judicious construction of a specialised order cone, together with specially constructed open subsets of the order cone – see Section 2 for details.

But none of our previous work allows for p(x)-harmonic functional coefficients. So, in this present work we demonstrate that all of these desirable aspects of our theory carry over to the p(x)-harmonic functional setting. Furthermore, the fundamental lemma we introduce in order to switch between variable and constant exponents (Lemma 2.6) may be useful in other studies related to p(x)-harmonic functionals as it demonstrates a simple but fundamental connection between the variable exponent case and the constant exponent case.

In addition, our results here provide both for an analysis of existence (in Section 2) as well as nonexistence (in Section 3) of solution. The existence results treat the convex regime p(x) > 1, whereas the nonexistence results treat both the convex regime p(x) > 1 as well as the concave regime 0 < p(x) ≤ 1. Moreover, along the way we provide, especially in Section 3, a number of what may be called “norm localisation” results, whereby results are provided that localise, principally, ‖u‖_∞ based on some knowledge of the value of a ∗ | u ′ | p ( ⋅ ) ( 1 ) . Whilst results of this type have appeared in many recent papers on nonlocal differential equations, the ones we provide here are, to the best of the our knowledge, new. Moreover, it is likely that these localisation results will be useful in future studies related to problem (1.2) and (1.3) and its relatives.

2 Existence theory for problem (1.2) and (1.3)

In this section we will begin by detailing both the function spaces in which we work as well as the notation which we will employ throughout. We will then introduce a sequence of lemmata, which will be utilised in the proof of the main existence theorem. A concluding example is presented in order to illustrate the general application of the existence theory.

So, to begin, we will work within the space C 1 [ 0,1 ] equipped with the norm ‖ u ‖ C 1 ≔ max ‖ u ‖ ∞ , ‖ u ′ ‖ ∞ , which subsequently makes the space a Banach space; here we use the notation

for any u ∈ C [ 0,1 ] . The notation 1 will henceforth denote the constant function 1 : R → { 1 } . In an entirely similar way, the notation 0 will denote the constant function 0 : R → { 0 } . In addition, by the notation f [ a , b ] × [ c , d ] M and f [ a , b ] × [ c , d ] m we mean the quantities

respectively, where f is any continuous real-valued function defined on the set [a, b] × [c, d] ⊆ [0, 1] × [0, + ∞).

Since we will prove existence of solution to (1.2) and (1.3) by means of topological fixed point theory, we will study the boundary value problem by studying an equivalent Hammerstein integral operator. As such, we will denote by G : [ 0,1 ] × [ 0,1 ] → R the function

which is the Green’s function associated to the operator Lu ≔ − u″ subject to the Dirichlet boundary conditions u(0) = 0 = u(1). Henceforth, by G : [ 0,1 ] → R we denote the function

We next mention the general assumptions imposed on the various functions appearing in the differential equation (1.2).

H1: The functions M : [ 0 , + ∞ ) → R , f: [0, 1] × [0, + ∞) → [0, + ∞), and a: (0, 1] → [0, + ∞) satisfy the following properties.

1. Each of M and f is continuous on their respective domains.

2. Both a ∈ L 1 ( 0,1 ] ; [ 0 , + ∞ ) and (a∗1)(1) ≠ 0.

3. There exist real numbers 0 < ρ ₁ < ρ ₂ such that M(t) > 0 for t ∈ ρ 1 , ρ 2 .

H2: The function p: [0, 1] → (1, + ∞) is continuous, and there exist real numbers p ⁻ and p ⁺ such that

for each t ∈ [0, 1].

As suggested toward the conclusion of Section 1, we will use a positive order cone – namely,

where the coercivity constant C ₀ > 0 in (1∗u)(1) ≥ C ₀‖u‖_∞ is defined by

In addition, for each ρ ≥ 0 by V ̂ ρ denote the set

Observe that V ̂ ρ is (relatively) open in K and, moreover, that

which is the crucial topological property inasmuch as asserting precise control over the argument of M in (1.3) is concerned. Finally, by T : V ̂ ρ 2 ̄ \ V ̂ ρ 1 → C 1 [ 0,1 ] we denote the Hammerstein-type operator

Then fixed points of T correspond to solutions of (1.2) and (1.3). Note that T is well defined on the domain V ̂ ρ 2 ̄ \ V ̂ ρ 1 on account of condition (H1.3).

We now present a sequence of preliminary lemmata, the first three of which are older results and the remainder of which are original. To this end, we first recall a one-dimensional version of Sobolev’s inequality. The particular statement given here can be found either in Brown, Hinton, and Schwabik [54] or in Talenti [55]. In the sequel, the functions to which we apply Lemma 2.3 will be of class C 1 [ 0,1 ] , which means they will automatically satisfy the regularity hypothesis of absolute continuity in the statement of Lemma 2.3. Also, for notational convenience in the sequel, the function κ : [ 1 , + ∞ ) × ( 1 , + ∞ ) → R will be defined by

We next state the topological fixed point results that we will use. Our results originate from a paper by Yan [56]. The unusual aspect of Yan’s results are that whilst they are in a sense standard fixed point index results, they are nonstandard inasmuch as they do not require the sets in question to be bounded. In particular, in our setting Lemma 2.4 is crucial because whilst V ̂ ρ , for a given ρ ≥ 0, cannot be expected, in general, to satisfy

it does satisfy

as evidenced by Lemma 2.8. And thanks to Yan’s results, the latter is sufficient for our purposes. In addition, we remark that the notation i(T, Ω) in Lemmata 2.4 and 2.5 refers to the fixed point index relative to an operator T on a set Ω ⊆ K .

Our first original lemma is a pointwise estimate that allows us to switch from variable exponent growth to constant exponent growth. This lemma states a fundamental pointwise relationship between a function taken to a variable exponent and the same function taken to a constant exponent. As intimated in Section 1, Lemma 2.6 is a sort-of global version of the local variable-to-constant exponent estimates that are used frequently in regularity theory – cf., [6], [8].

As a consequence of Lemma 2.6 we obtain the following corollary.

One of the primary consequences of Corollary 2.7, and thus of Lemma 2.6, is that we conclude that for each ρ ≥ 0 the set V ̂ ρ satisfies the boundedness condition

Thus, much as in [15], we will be able to use Lemmata 2.4 and 2.5 in spite of the fact that

can occur. The former observation is the content of the next lemma. Furthermore, as an upshot of the next lemma we deduce a localisation of u ∈ V ̂ ρ in terms of both ‖u‖_∞ and ‖ u ‖ C 1 .

Our final preliminary lemma records the fact, which is well known to specialists in this context, that T maps a solid annular subregion of K into itself. Since this fact has been proved elsewhere in analogous circumstances and since the inclusion of the variable exponent does not materially alter such proofs, we merely record this fact and omit the proof.

With the necessary lemmata dispatched we state and prove our existence theorem. Prior to stating the theorem, we introduce the notation

for any ρ > 0. This notation will be used in the statement and proof of Theorem 2.11.