Investigation of weak solutions for p(z)-Kirchhoff equations by Young measure techniques

Mouad Allalou; Abderrahmane Raji

doi:10.1515/msds-2024-0006

Article Open Access

Investigation of weak solutions for p(z)-Kirchhoff equations by Young measure techniques

Mouad Allalou and Abderrahmane Raji

Published/Copyright: December 13, 2024

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From the journal Nonautonomous Dynamical Systems Volume 11 Issue 1

Abstract

The present article deals with the existence of weak solutions to a class of $p ( z )$ -Kirchhoff-type problems. To address these problems, we employ a variational approach in conjunction with the theory of variable exponent Sobolev spaces, while imposing suitable assumptions on the source term. Furthermore, we utilize the theory of Young measures.

Keywords: weak solution; p(z)-Kirchhoff-type; nonlocal problems; variable exponents; Young measures

MSC 2010: 35J60; 35J25; 35D30

1 Introduction

In the realm of mathematical analysis and partial differential equations, the study of equations involving nonlinearities, variable exponents, and nonlocal effects presents fascinating challenges. One such equation of significant interest, which is represented by $( P )$ , is

$( P ) − ℳ ∫ Ω A ( z , ∇ u ) d z div a ( z , ∇ u ) + ∣ u ∣ p ( z ) − 2 u = f ( z , u ) in Ω , u = 0 on ∂ Ω ,$

where $Ω$ represents a bounded domain in $R m ( m ≥ 1 )$ with a smooth boundary condition $∂ Ω$ . This equation represents a fascinating confluence of multiple intricate mathematical phenomena. It involves variable exponents denoted as $p ( z )$ and introduces nonlocality through the operators and functions $ℳ : R + → R + , A : Ω × R m → R , a : Ω × R m → R m$ , and $f : Ω × R → R$ all of which are subject to specific conditions (further detailed below).

The issue at hand $( P )$ pertains to the stationary variant of a model known as the Kirchhoff equation, originally introduced by Kirchhoff [35]. To elaborate further, Kirchhoff formulated a model represented by the following equation:

(1)

$ρ ∂ 2 u ∂ t 2 − P 0 h + E 2 L ∫ 0 L ∂ u ∂ x 2 d x ∂ 2 u ∂ x 2 = 0 .$

Incorporating constants $ρ$ , $P 0$ , $h$ , $E$ , and $L$ , this equation extends the classical D’Alembert’s wave equation to account for variations in string length during vibrations.

This type of equation has been extensively investigated by numerous researchers over the past two decades. Their significance extends across various domains of mathematics, including calculus of variations and partial differential equations [1,23,25]. Furthermore, they have found applications in diverse fields of physical and engineering sciences. These applications encompass modeling electrorheological fluids [38], analyzing Non-Newtonian fluids [41], studying fluid flow in porous media [7], examining magnetostatics [18], addressing image restoration [15], and exploring capillarity phenomena [14]. For more details and additional references, please refer [6,8–10,17,19,32] and references therein.

Recent contributions by Francisco et al. [21] proved the existence and multiplicity of solutions of the $p ( z )$ -Kirchhoff equation, with an additional nonlocal term,

(2)

$− ℳ ∫ Ω 1 p ( z ) ∣ ∇ u ∣ p ( z ) d z Δ p ( z ) u = f ( z , u ) ∫ Ω F ( z , u ) d z r in Ω , u = 0 on ∂ Ω ,$

via Krasnoselskii genus.

In [22], the present authors studied the existence of solutions for some classes of $p ( z )$ -Kirchhoff equation, with critical exponent and an additional nonlocal term,

(3)

$− ℳ ∫ Ω 1 p ( z ) ∣ ∇ u ∣ p ( z ) d z Δ p ( z ) u = λ f ( z , u ) ∫ Ω F ( z , u ) d z r + ∣ u ∣ q ( z ) − 2 u in Ω , u = 0 on ∂ Ω$

in Sobolev spaces with variable exponent by using a version of the concentration compactness principle, Yucedag [39] has demonstrated that

(4)

$− ℳ ∫ Ω A ( z , ∇ u ) d z div a ( z , ∇ u ) = ∣ u ∣ p ( z ) − 2 u in Ω , a ( z , ∇ u ) ∂ u ∂ v = λ f ( z , u ) on ∂ Ω ,$

while studying the multiplicity of solutions for a nonlocal $p ( z )$ -Kirchhoff-type problem with Steklov boundary value in variable exponent Sobolev spaces. To explore topics closely related to equation $( P )$ , it is recommended that the reader consult the following references: [5,16,20,24,29,36,37,42], along with additional sources cited therein. For a comprehensive discussion on the steady case, where the Young measures theory is employed, you can also refer [2–4,12,27].

Motivated by these advancements, our study investigates $( P )$ through the existence of weak solution theory, employing the framework of Young measures theory for the first time. This theoretical approach promises new insights into the existence and properties of solutions for $( P )$ under conditions $( ℐ 0 )$ through $( ℐ 3 )$ .

In this work, we emphasize the motivation behind studying $( P )$ , highlighting its contributions to the field, particularly in comparison to existing results. By leveraging Young measures theory, we aim to provide a deeper understanding of the behavior of solutions to $( P )$ , shedding light on its unique aspects and implications.

A weak solution for $( P )$ is defined as a function $u ∈ W 0 1 , p ( z ) ( Ω )$ satisfying the following equation for all $ϕ ∈ W 0 1 , p ( z ) ( Ω )$ :

$ℳ ∫ Ω A ( z , ∇ u ) d z ∫ Ω a ( z , ∇ u ) ⋅ ∇ ϕ d z + ∫ Ω ∣ u ∣ p ( z ) − 2 u ⋅ ϕ d z = ∫ Ω f ( z , u ) ϕ d z .$

Our underlying assumptions are outlined as follows:

$a : Ω × R m → R m$ and $f : Ω × R → R$ are Carathéodory functions which implies their measurability with respect to $z ∈ Ω$ and continuity with respect to the other variables. Additionally, the mapping $ξ ↦ a ( z , ξ )$ is both a $C 1$ function and monotonic, i.e.,
$( a ( z , ξ ) − a ( z , ξ ′ ) ) ⋅ ( ξ − ξ ′ ) ≥ 0 ∀ ξ , ξ ′ ∈ R m .$
There exists a constant $c 0 > 0$ , such that
$∣ a ( z , ξ ) ∣ ≤ c 0 ( α 1 ( z ) + ∣ ξ ∣ p ( z ) − 1 ) , ∣ f ( z , w ) ∣ ≤ α 2 ( z ) + ∣ w ∣ q ( z ) , p ( z ) A ( z , ξ ) ≥ a ( z , ξ ) ⋅ ξ ≥ c 1 ∣ ξ ∣ p ( z ) − α 3 ( z )$
for all $w ∈ R$ and $0 ≤ q ( z ) < p ( z ) − 1$ .
$A : Ω × R m → R$ is a Carathéodory function within the context of $( ℐ 0 )$ . Furthermore, the mapping $ξ ↦ A ( z , ξ )$ is both convex and $C 1$ -function and it fulfills the relation $a ( z , ξ ) = ∇ ξ A ( z , ξ ) = ( ∂ A ⁄ ∂ ξ ) ( z , ξ )$ .
$ℳ : R + → R +$ is continuous and non-decreasing function and satisfies the following inequality:
$m 0 s β ( z ) − 1 ≤ ℳ ( s ) ≤ m 1 s β ( z ) − 1 ,$
for all $s > 0$ and $m 0 , m 1$ are real numbers such that $0 < m 0 ≤ m 1$ and $β ( z ) ≥ 1$ .

Theorem 1.1

Assume that $( ℐ 0 )$ – $( ℐ 3 )$ hold. Then, problem $( P )$ has a weak solution in $W 0 1 , p ( z ) ( Ω )$ .

The structure of this study is outlined as follows: In Section 2, we provide a concise overview of essential concepts related to Young measures and we present some necessary preliminary knowledge of variable exponent Lebesgue-Sobolev spaces. Section 3 is dedicated to the development of approximate solutions and establishing a priori estimates. Section 4 focuses on presenting convergence outcomes and delivering the proof of the primary theorem.

2 Preliminaries

2.1 Lebesgue and Sobolev spaces

The generalized Lebesgue-Sobolev spaces $L p ( ⋅ ) ( Ω )$ , $W 1 , p ( ⋅ ) ( Ω )$ , and $W 0 1 , p ( ⋅ ) ( Ω )$ are defined for an open set $Ω$ of $R m$ and an associated continuous real-valued function $p : Ω ¯ ⟶ [ 1 , + ∞ )$ such that $1 < p ( ⋅ ) < m$ . The minimum and maximum values of $p ( ⋅ )$ are denoted by $p −$ and $p +$ , respectively. Further information on these spaces may be found in the work of Fan and Zhao [31]. Denote the Lebesgue space with variable exponent $L p ( ⋅ ) ( Ω )$ as the set of all measurable functions $w : Ω ⟶ R$ for which the convex modular

$ρ p ( ⋅ ) ( w ) = ∫ Ω ∣ w ∣ p ( z ) d s$

is finite. If the exponent is bounded, i.e., if $p + < + ∞$ , then the expression

$‖ w ‖ L p ( ⋅ ) ( Ω ) = inf μ > 0 ; ∫ Ω w ( z ) μ p ( z ) d z ≤ 1$

defines a norm in $L p ( ⋅ ) ( Ω )$ called the Luxemburg norm. The space $( L p ( ⋅ ) ( Ω ) ; ‖ ⋅ ‖ p ( ⋅ ) )$ is a separable Banach space. Moreover, if $1 < p − ≤ p + < + ∞$ , then $L p ( ⋅ ) ( Ω )$ is uniformly convex, hence reflexive and its dual space is isomorphic to $L p ′ ( ⋅ ) ( Ω )$ , where $1 p ( z ) + 1 p ′ ( z ) = 1$ , for $z ∈ Ω$ . The following inequality will be used later:

$min { ‖ w ‖ L p ( ⋅ ) ( Ω ) p − , ‖ w ‖ L p ( ⋅ ) ( Ω ) p + } ≤ ∫ Ω ∣ w ( z ) ∣ p ( z ) d z ≤ max { ‖ w ‖ L p ( ⋅ ) ( Ω ) p − , ‖ w ‖ L p ( ⋅ ) ( Ω ) p + } .$

Finally, the Hölder-type inequality

$∫ Ω w ϑ d z ≤ 1 p − + 1 p + ‖ w ‖ p ( ⋅ ) ‖ ϑ ‖ p ′ ( ⋅ ) ,$

where $w ∈ L p ( ⋅ ) ( Ω )$ and $ϑ ∈ L p ′ ( ⋅ ) ( Ω )$ . Let

$W 1 , p ( ⋅ ) ( Ω ) = { w ∈ L p ( ⋅ ) ( Ω ) ; ∣ ∇ w ∣ ∈ L p ( ⋅ ) ( Ω ) } ,$

which is Banach space equipped with the following norm:

$‖ w ‖ 1 , p ( ⋅ ) = ‖ w ‖ p ( ⋅ ) + ‖ ∇ w ‖ p ( ⋅ ) .$

The $( W 1 , p ( ⋅ ) ( Ω ) ; ‖ ⋅ ‖ 1 , p ( ⋅ ) )$ Banach space is separable and reflexive. Manipulating the generalized Lebesgue and Sobolev spaces is essential when it comes to the modular $ρ p ( ⋅ )$ of the space $L p ( ⋅ ) ( Ω )$ . This yields the following results.

Proposition 1

[28,33] If $w n , w ∈ L p ( ⋅ ) ( Ω )$ and $p + < + ∞$ , the following properties hold true:

$‖ w ‖ p ( ⋅ ) > 1 ⇒ ‖ w ‖ p ( ⋅ ) p + < ρ p ( ⋅ ) ( w ) < ‖ w ‖ p ( ⋅ ) p −$ ,
$‖ w ‖ p ( ⋅ ) < 1 ⇒ ‖ w ‖ p ( ⋅ ) p − < ρ p ( ⋅ ) ( w ) < ‖ w ‖ p ( ⋅ ) p +$ ,
$‖ w ‖ p ( ⋅ ) p ( ⋅ ) < 1$ (respectively $= 1 ; > 1 ) ⇔ ρ p ( ⋅ ) ( w ) < 1$ (respectively $= 1 ; > 1$ ),
$∥ w n ∥ p ( ⋅ ) p ( ⋅ ) ⟶ 0$ (respectively $⟶ + ∞ ) ⇔ ρ p ( ⋅ ) ( w n ) < 1$ (respectively $⟶ + ∞$ ),
$ρ p ( ⋅ ) w ‖ w ‖ p ( ⋅ ) = 1$ . For a measurable function $w : Ω ⟶ R$ , we introduce the following notation:
$ρ 1 , p ( ⋅ ) = ∫ Ω ∣ w ∣ p ( z ) d z + ∫ Ω ∣ ∇ w ∣ p ( z ) d z .$

Proposition 2

[28] If $w ∈ W 1 , p ( ⋅ ) ( Ω )$ and $p + < + ∞$ , the following properties hold true:

$‖ w ‖ 1 , p ( ⋅ ) > 1 ⇒ ‖ w ‖ 1 , p ( ⋅ ) p + < ρ 1 , p ( ⋅ ) ( w ) < ‖ w ‖ 1 , p ( ⋅ ) p −$ ,
$‖ w ‖ 1 , p ( ⋅ ) < 1 ⇒ ‖ w ‖ 1 , p ( ⋅ ) p − < ρ 1 , p ( ⋅ ) ( w ) < ‖ w ‖ 1 , p ( ⋅ ) p +$ ,
$‖ w ‖ 1 , p ( ⋅ ) < 1$ (respectively $= 1 ; > 1$ ) $⇔ ρ 1 , p ( ⋅ ) ( w ) < 1$ (respectively $= 1 ; > 1$ ).

2.2 Fundamentals of Young measures

Weak convergence stands as a fundamental and powerful tool in contemporary nonlinear analysis, offering the same advantageous compactness properties found in finite-dimensional space convergence [30]. Nevertheless, it is worth highlighting that this concept does not exhibit the desired behavior when applied to nonlinear functionals and operators. To address and effectively navigate these challenges, the adoption of Young measures becomes crucial.

We denote by $C 0 ( R m )$ the closure of the space comprising continuous functions on $R m$ with compact support concerning the $‖ ⋅ ‖ ∞$ -norm. Its dual space can be identified as $ℳ ( R m )$ , the space encompassing signed Radon measures with finite mass. The duality pairing for $ν : Ω → ℳ ( R m )$ is defined as:

$⟨ ν , h ⟩ = ∫ R m h ( λ ) d ν ( λ ) .$

Lemma 2.1

[30]. Assume that the sequence ${ y k } k ≥ 1$ is bounded in $L ∞ ( Ω ; R m )$ . Then, there exist a subsequence still denoted ${ y k } k$ and a Borel probability measure $ν z$ on $R m$ for a.e. $z ∈ Ω$ , such that for almost each $h ∈ C ( R m )$ we have

$h ( y k ) → * h ¯ weakly i n L ∞ ( Ω ) ,$

where

$h ¯ ( z ) = ⟨ ν z , h ⟩ = ∫ R m h ( λ ) d ν z ( λ ) for a.e. z ∈ Ω .$

Definition 2.2

We call $ν = { ν z } z ∈ Ω$ , the family of Young measures associated with the subsequence ${ y k } k$ . It is shown in [13], that if for all $R > 0$

$limsup k → ∞ ∣ { z ∈ Ω ∩ B R ( 0 ) : ∣ y k ( z ) ∣ ≥ L } ∣ = 0 ,$

then for any measurable $Ω ′ ⊂ Ω$

$h ( z , y k ) → ⟨ ν z , φ ( z , ⋅ ) ⟩ = ∫ R m h ( z , λ ) d ν z ( λ ) weakly in L 1 ( Ω )$

for any Carathéodory function $h : Ω ′ × R m → R$ such that $h ( z , y k )$ is equiintegrable.

If we consider $y k = ∇ w k$ , where $w k : Ω → R$ , the above properties remain true, and the following lemma can be proved in a similar way as in [11, Lemma 4.1].

Lemma 2.3

[34] Let $( ∇ w k )$ be a bounded sequence in $L p ( z ) ( Ω ; R m )$ . Then, the Young measure $ν z$ generated by $∇ w k$ in $L p ( z ) ( Ω ; R m )$ satisfies:

$∥ ν z ∥ ℳ ( R m ) = 1$ for a.e. $z ∈ Ω$ , i.e., $ν z$ is a probability measure.
The weak $L 1$ -limit of $∇ w k$ is given by $⟨ ν z , i d ⟩ = ∫ R m λ ⋅ d ν z ( λ )$ .
$ν z$ satisfies $⟨ ν z , i d ⟩ = ∇ u ( z )$ for a.e. $z ∈ Ω$ .

We will need the following Fatou-type inequality.

Lemma 2.4

[26] Let $h : Ω × R m → R$ be a Carathéodory function and $w k : Ω → R$ a sequence of measurable functions such that $∇ w k$ generates the Young measure $ν z$ with $∥ ν z ∥ ℳ ( R m ) = 1$ for almost every $z ∈ Ω$ . Then,

$liminf k → ∞ ∫ Ω h ( z , ∇ w k ) d z ≥ ∫ Ω ∫ R m h ( z , λ ) d ν z ( λ ) d z ,$

provided that the negative part of $h ( z , ∇ w k )$ is equiintegrable.

3 Proof of existence of weak solutions to problem

Let us consider the functional $S ( u ) : W 0 1 , p ( z ) ( Ω ) → R$ given by

$ϕ ↦ ℳ ∫ Ω A ( z , ∇ u ) d z ∫ Ω a ( z , ∇ u ) ⋅ ∇ ϕ d z + ∫ Ω ∣ u ∣ p ( z ) − 2 u ⋅ ϕ d z − ∫ Ω f ( z , u ) ϕ d z$

for arbitrary $u ∈ W 0 1 , p ( z ) ( Ω )$ and $ϕ ∈ W 0 1 , p ( z ) ( Ω )$ . Note that for any $ξ ∈ R m$ we have

$A ( z , ξ ) = ∫ 0 1 d d t A ( z , t ξ ) d t = ∫ 0 1 a ( z , t ξ ) ⋅ ξ d t .$

Then,

$A ( z , ξ ) ≤ c 3 ( ∣ ξ ∣ + ∣ ξ ∣ p ( z ) ) for all z ∈ Ω and ξ ∈ R m .$

The above inequality implies

(5)

$A ( z , ∇ u ) ≤ c 3 ( ∣ ∇ u ∣ + ∣ ∇ u ∣ p ( z ) ) .$

Lemma 3.1

The functional $S ( u )$ is well-defined, linear, and bounded.

Proof

First,

$∣ J 1 ∣ ≡ ℳ ∫ Ω A ( z , ∇ u ) d z ∫ Ω a ( z , ∇ u ) ⋅ ∇ ϕ d z ≤ m 1 ∫ Ω A ( z , ∇ u ) d z β ( z ) − 1 ∫ Ω ∣ a ( z , ∇ u ) ⋅ ∇ ϕ ∣ d z ≤ m 1 ∫ Ω c 3 ( ∣ ∇ u ∣ + ∣ ∇ u ∣ p ( z ) ) d z β ( z ) − 1 ∫ Ω c 0 ( α 1 ( z ) + ∣ ∇ u ∣ p ( z ) − 1 ) ∣ ∇ ϕ ∣ d z ≤ m 1 c 0 ( c 3 ‖ ∇ u ‖ p ( z ) + c 3 ‖ ∇ u ‖ p ( z ) p ( z ) − 1 ) β ( z ) − 1 ( ∥ α 1 ∥ p ′ ( z ) + ‖ ∇ u ‖ p ( z ) p ( z ) − 1 ) ‖ ∇ ϕ ‖ p ( z ) .$

Using the Hölder inequality and the growth condition of $a$ in ( $I 1$ ), ( $I 3$ ), and (5), we can derive the following. Additionally, based on the growth condition of $f$ in ( $I 1$ ) and without loss of generality by assuming $q ( z ) = p ( z ) − 1$ , along with the Hölder inequality, we obtain that

$∣ J 2 ∣ ≡ ∫ Ω f ( z , u ) ϕ d z ≤ ∫ Ω ∣ f ( z , u ) ϕ ∣ d z ≤ ( ∥ α 2 ∥ p ′ ( z ) + ‖ u ‖ p * ( z ) p ( z ) − 1 ) ‖ ϕ ‖ p ( z ) ≤ λ ( ∥ α 2 ∥ p ′ ( z ) + λ p ( z ) − 1 ‖ ∇ u ‖ p ( z ) p ( z ) − 1 ) ‖ ∇ ϕ ‖ p ( z ) ,$

with $λ$ denoting the constant in Poincaré inequality, there exists a positive constant $λ$ such that

$‖ ϕ ‖ p ( z ) ≤ λ ‖ ∇ ϕ ‖ p ( z ) ∀ ϕ ∈ W 0 1 , p ( z ) ( Ω ) .$

On the other hand,

$∣ J 3 ∣ ≡ ∫ Ω ∣ u ∣ p ( z ) − 2 u ⋅ ϕ d z ≤ ∫ Ω ∣ u ∣ p ( z ) − 1 ∣ ϕ ∣ d z ≤ 1 p − + 1 p + ∣ ∣ u ∣ ∣ p ( z ) p ( z ) − 1 ‖ ϕ ‖ p ( z ) .$

As the estimations of $I i$ for $i = 1 , 2 , 3$ are finite, $S ( u )$ is well defined. Moreover, $S ( u )$ is linear and for all $ϕ ∈ W 0 1 , p ( z ) ( Ω )$ , inequality

$∣ ⟨ S ( u ) , ϕ ⟩ ∣ ≤ ∣ I 1 ∣ + ∣ I 2 ∣ + ∣ J 3 ∣ ≤ C ‖ ∇ ϕ ‖ p ( z )$

holds, indicating that $S ( u )$ is bounded.□

According to Lemma 3.1, we can define the operator

$S : W 0 1 , p ( z ) ( Ω ) → W − 1 , p ′ ( z ) ( Ω ) , u ↦ S ( u ) ,$

which satisfies the following.

Lemma 3.2

The restriction of $S$ to a finite dimensional linear subspace $W$ of $W 0 1 , p ( z ) ( Ω )$ is continuous.

Proof

Let $W$ be a finite linear subspace of $W 0 1 , p ( z ) ( Ω )$ . Suppose $( u j )$ is a sequence in $W$ that converges to $u$ in $W$ . First, $u k → u$ and $∇ u k → ∇ u$ almost everywhere. Second,

$∫ Ω ∣ u k − u ∣ p ( z ) d z → 0 and ∫ Ω ∣ ∇ u k − ∇ u ∣ p ( z ) d z → 0 ,$

since $u k → u$ strongly in $W$ . Hence, there exist $Q 1 , Q 2 ∈ L 1 ( Ω )$ such that $∣ u k − u ∣ p ( z ) ≤ Q 1$ and $∣ ∇ u k − ∇ u ∣ p ( z ) ≤ Q 2$ . We know that for $γ > 1$

$∣ t 1 + t 2 ∣ γ ≤ 2 γ − 1 ( ∣ t 1 ∣ γ + ∣ t 2 ∣ γ ) ,$

then

$∣ u k ∣ p ( z ) = ∣ u k − u + u ∣ p ( z ) ≤ 2 p + − 1 ( ∣ u k − u ∣ p ( z ) + ∣ u ∣ p ( z ) ) ≤ 2 p + − 1 ( Q 1 + ∣ u ∣ p ( z ) ) .$

Likewise to the demonstration of $∣ u k ∣ p ( z )$ , it follows that $∥ u k ∥ p ( z )$ and $∥ ∇ u k ∥ p ( z )$ are bounded by a constant $C$ . Thus, the continuity condition in $( I 0 )$ , $( I 2 )$ , and $( I 3 )$ permits to deduce that

$ℳ ∫ Ω A ( z , ∇ u k ) d z a ( z , ∇ u k ) ⋅ ∇ ϕ ( z ) → ℳ ∫ Ω A ( z , ∇ u ) d z a ( z , ∇ u ) ⋅ ∇ ϕ ( z )$

and

$f ( z , u k ) ϕ ( z ) → f ( z , u ) ϕ ( z )$

almost everywhere as $k → ∞$ . Indeed, if $Ω ′$ be a measurable subset of $Ω$ , and $ϕ ∈ W 0 1 , p ( z ) ( Ω )$ , then

$∫ Ω ′ ∣ a ( z , ∇ u k ) ⋅ ∇ ϕ ( z ) ∣ d z ≤ ∫ Ω c 0 ( α 1 ( z ) + ∣ ∇ u k ∣ p ( z ) − 1 ) ∣ ∇ ϕ ∣ d z ≤ c 0 ( ∥ α 1 ∥ p ′ ( z ) + ∥ ∇ u k ∥ p ( z ) p ( z ) − 1 ︸ ≤ C ) ∫ Ω ′ ∣ ∇ ϕ ∣ p ( z ) d z 1 p ( z )$

and (for simplicity, we can consider $q ( z ) = p ( z ) − 1$ without loss of generality)

$∫ Ω ′ ∣ f ( z , u k ) ϕ ( z ) ∣ d z ≤ ∫ Ω ′ ( α 2 ( z ) + ∣ u k ∣ p ( z ) − 1 ) ∣ φ ∣ d z ≤ λ ( ∥ α 2 ∥ p ′ ( z ) + ∣ ∣ u k ∣ p ( z ) p ( z ) − 1 ︸ ≤ C ) ∫ Ω ′ ∣ ∇ ϕ ∣ p ( z ) d z 1 p ( z ) ,$

By using the inequalities of Hölder’s and Poincaré, along with equation (5). Moreover,

$ℳ ∫ Ω ′ A ( z , ∇ u k ) d z ≤ m 0 ∫ Ω ′ A ( z , ∇ u k ) d z β ( z ) − 1 < ∞$

by $( I 3 )$ , (5), and the boundedness of $∥ ∇ u k ∥ p ( z )$ . By utilizing the Vitali theorem, we can establish the continuity of $S$ .□

Remark 3.3

In this section, we have only utilized the condition $q ( z ) ≤ p ( z ) − 1$ . Hence, Lemmas 3.1 and 3.2 remain valid as $q ( z ) = p ( z ) − 1$ .

Now, the problem $( P )$ is equivalent to find a solution $u ∈ W 0 1 , p ( z ) ( Ω )$ such that

$⟨ S ( u ) , ϕ ⟩ = 0 for all ϕ ∈ W 0 1 , p ( z ) ( Ω ) .$

In order to find such a solution, we apply a Galerkin in scheme. Since $W 0 1 , p ( z ) ( Ω )$ is separable, there exists a sequence $( W k )$ of finite dimensional subspaces such that $∪ k ≥ 1 W k$ is dense in $W 0 1 , p ( z ) ( Ω )$ . Let ${ x 1 , … , x r }$ be a basis of $W k$ where $dim W k = r$ . Let us define

$ℒ : R r → R r ( d i ) i = 1 , … , r → ( ⟨ S ( d i x i ) , x j ⟩ ) j = 1 , … , r . .$

Proposition 3

$ℒ$ is continuous and $ℒ ( d ) ⋅ d → ∞$ as $‖ d ‖ R r → ∞$ .

Proof

$ℒ$ is trivially continuous, by the continuity of $S$ restricted to $W k$ (refer Lemma 3.2 if necessary). Consider $d ∈ R r$ and $u = d i x i ∈ W k$ (with conventional summation). The condition $‖ d ‖ R r → ∞$ is equivalent to $‖ u ‖ 1 , p ( z ) → ∞$ , and we have

$ℒ ( d ) ⋅ d = ⟨ S ( u ) , u ⟩ .$

Note that

$I 1 ≡ ℳ ∫ Ω A ( z , ∇ u ) d z ∫ Ω a ( z , ∇ u ) ⋅ ∇ u d z ≥ m 0 ∫ Ω A ( z , ∇ u ) d z β ( z ) − 1 ∫ Ω a ( z , ∇ u ) ⋅ ∇ u d z ( by ( ℐ 3 ) ) ≥ m 0 p ( z ) β ( z ) − 1 ∫ Ω a ( z , ∇ u ) ⋅ ∇ u d z β ( z ) ( by ( ℐ 1 ) ) ≥ m 0 p ( z ) β ( z ) − 1 ∫ Ω c 1 ∣ ∇ u ∣ p ( z ) d z − ∫ Ω ′ α 3 ( z ) d z ,$

since $β ≥ 1$ . Finally, it follows from the growth condition $( I 1 )$ and equation (5), we have

$∣ I 2 ∣ ≡ ∫ Ω f ( z , u ) u d z ≤ ∫ Ω ∣ f ( z , u ) u ∣ d z ≤ ∫ Ω ( α 2 ( z ) + ∣ u ∣ q ( z ) ) ∣ u ∣ d z ≤ λ ∥ α 2 ∥ p ′ ( z ) ‖ ∇ u ‖ p ( z ) + λ q ( z ) + 1 ‖ ∇ u ‖ p ( z ) q ( z ) + 1 .$

Hence,

$⟨ S ( u ) , u ⟩ ≥ I 1 − I 2 ≥ c ‖ ∇ u ‖ p ( z ) p ( z ) − c ‖ α 3 ‖ p ′ ( z ) − λ ‖ α 2 ‖ ‖ p ′ ( z ) ‖ ∇ u ‖ p ( z ) − λ q ( z ) + 1 ‖ ∇ u ‖ p ( z ) q ( z ) + 1 → ∞$

as $‖ u ‖ 1 , p ( z ) → ∞$ , since $c > 0$ and $p ( z ) > max ( 1 , q ( z ) + 1 )$ .□

Proposition 4

For all $k ∈ N$ , there exists $u k ∈ W k$ such that

$⟨ S ( u k ) , ϕ ⟩ = 0 for a l l ϕ ∈ W k .$

Proof

By Proposition 3, there exists $R > 0$ such that for all $d ∈ ∂ B R ( 0 ) ⊂ R r$ we have $ℒ ( d ) ⋅ d > 0$ , and the, usual topological argument [40, Proposition 2.8], there exists $z ∈ B R ( 0 )$ such that $ℒ ( z ) = 0$ . Hence, for all $k ∈ N$ there exists $u k ∈ W k$ such that $⟨ S ( u k ) , φ ⟩ = 0$ for all $φ ∈ W k$ .□

Proposition 5

The constructed sequence $( u k )$ in Proposition 4 is uniformly bounded, i.e., there is a constant $R > 0$ such that $∥ u k ∥ 1 , p ( z ) ≤ R$ for all $k ∈ N$ .

Proof

Due to Proposition 3, there exists $R > 0$ with the property that $⟨ S ( u ) , u ⟩ > 1$ whenever $‖ u ‖ 1 , p ( z ) > R$ . Hence, for the sequence of Galerkin approximations, $u k ∈ W k$ , which satisfy $⟨ S ( u k ) , u k ⟩ = 0$ by Proposition 4, we obtain the uniform boundedness of $( u k )$ in $W 0 1 , p ( z ) ( Ω )$ .□

4 Proofs and propositions concerning convergence

In this section, we present general convergence findings pertaining to the functions denoted as $a ( ⋅ )$ , $A ( ⋅ )$ , and $f ( ⋅ )$ . Given that the sequence $( u k )$ remains within bounded limits in the space $W 0 1 , p ( z ) ( Ω )$ , as established in Propositions 3–5, we can infer, based on the assertions of Lemma 2.4, the existence of a Young measure denoted as $ν x$ . This measure is generated by $∇ u k$ within the space $L p ( z ) ( Ω ; R m )$ .

Lemma 4.1

The Young measure $ν z$ generated by $∇ u k$ satisfies

$( a ( z , λ ) − a ( z , ∇ u ) ) ⋅ ( λ − ∇ u ) = 0 on s u p p ν z ,$

where supp $ν z$ represents the support of $ν z$ for a.e. $z ∈ Ω$ .

Proof

Take into account the sequence

$e k ≔ ( a ( z , ∇ u k ) − a ( z , ∇ u ) ) ⋅ ( ∇ u k − ∇ u ) = a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) − a ( z , ∇ u ) ⋅ ( ∇ u k − ∇ u ) = e k , 1 + e k , 2 .$

Given the growth condition of $a$ in $( I 1 )$ and the weak convergence as described in Lemma 2.3, it follows that:

$liminf k → ∞ ∫ Ω e k , 2 d z = ∫ Ω a ( z , ∇ u ) ⋅ ( ∫ R n λ d ν z ( λ ) ︸ ≕ ∇ u ( z ) − ∇ u ) d z = 0 .$

Thus,

$e ≔ liminf k → ∞ ∫ Ω e k d z = liminf k → ∞ ∫ Ω e k , 1 d z .$

By the growth condition of $a$ in $( I 1 )$ , $( a ( z , ∇ u k ) ⋅ ∇ u )$ is equiintegrable. Let us fix an arbitrary measurable subset $Ω ′ ⊂ Ω$ . Then, the coercivity condition in $( I 1 )$ implies

(6)

$∫ Ω ′ ∣ min ( a ( z , ∇ u k ) ⋅ ∇ u k , 0 ) ∣ d z ≤ c 1 ∫ Ω ′ ∣ ∇ u k ∣ p ( z ) d z + ∫ Ω ′ ∣ α 3 ( z ) ∣ d z < ∞ .$

This shows the equiintegrability of $( a ( z , ∇ u k ) ⋅ ∇ u k )$ . Therefore, $( a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) )$ is also equiintegrable, and by virtue of Lemma 2.4, it yields

$∫ Ω ∫ R m a ( z , λ ) ⋅ ( λ − ∇ u ) d ν z ( λ ) d z ≤ liminf k → ∞ ∫ Ω a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) d z = e .$

Now, let us show that $e ≤ 0$ . By Proposition 4, we can write

$ℳ ∫ Ω A ( z , ∇ u k ) d z ∫ Ω a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) d z = ∫ Ω f ( z , u k ) ( u k − u ) d z − ∫ Ω ∣ u k ∣ p ( z ) − 2 u k . ( u k − u ) d z .$

According to $( I 3 )$ , it follows that

$m 0 ∫ Ω A ( z , ∇ u k ) d z β ( z ) − 1 ∫ Ω a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) d z ≤ ∫ Ω f ( z , u k ) ( u k − u ) d z − ∫ Ω ∣ u k ∣ p ( z ) − 2 u k . ( u k − u ) d z .$

Since $( u k )$ is bounded and $∫ Ω A ( z , ∇ u k ) d z ≤ C < ∞$ , then

$∫ Ω a ( z , ∇ u k ) ⋅ ( ∇ u k − ∇ u ) d z ≤ 1 m 0 ∫ Ω A ( z , ∇ u k ) d z 1 − β ( z ) ︸ ≤ C ∫ Ω f ( z , u k ) ( u k − u ) d z − ∫ Ω ∣ u k ∣ p ( z ) − 2 u k . ( u k − u ) d z ≤ C ∫ Ω f ( z , u k ) ( u k − u ) d z ≤ C ( ∥ d 2 ∥ p ′ ( z ) + ∥ u k ∥ p ( z ) p ( z ) − 1 ︸ ≤ C ) ∥ u k − u ∥ p ( z ) → 0 as k → ∞ .$

This is achieved by Hölder’s inequality and the fact that $u k → u$ in $W 0 1 , p ( z ) ( Ω ) ↪ L p ( z ) ( Ω )$ . Hence,

$∫ Ω ∫ R m a ( z , λ ) ⋅ ( λ − ∇ u ) d ν z ( λ ) d z ≤ 0 .$

In conclusion, we can infer from this and equation (6) that

$∫ Ω ∫ R m ( a ( z , λ ) − a ( z , ∇ u ) ) ⋅ ( λ − ∇ u ) d ν z ( λ ) d z ≤ 0 .$

The function $a$ being monotonic, the integral above evaluates to zero with respect to the product measure $d ν x ( λ ) ⊗ d z$ , meaning

$∫ Ω ∫ R m ( a ( z , λ ) − a ( z , ∇ u ) ) ⋅ ( λ − ∇ u ) d ν z ( λ ) ⊗ d z = 0 .$

Consequently,

□

$( a ( z , λ ) − a ( z , ∇ u ) ) ⋅ ( λ − ∇ u ) = 0 on supp ν z .$

Proposition 6

For a.e. $z ∈ Ω$ , the support of $ν z$ is in the set where A agrees with the supporting hyper-plane $L ≡ { ( λ , A ( z , ∇ u ) + a ( z , ∇ u ) ⋅ ( λ − ∇ u ) ) }$ , i.e.,

$supp ν z ⊂ K z = { λ ∈ R m : A ( z , λ ) = A ( z , ∇ u ) + a ( z , ∇ u ) ⋅ ( λ − ∇ u ) } .$

Proof

Consider $λ ∈ supp ν z$ . As stated in Lemma 4.1 implies for all $t ∈ [ 0 , 1 ]$ , we have

(7)

$( 1 − t ) ( a ( z , λ ) − a ( z , ∇ u ) ) ⋅ ( λ − ∇ u ) = 0 .$

From the monotonicity condition and (7), we obtain

(8)

$0 ≤ ( 1 − t ) ( a ( z , λ ) − a ( z , ∇ u + t ( λ − ∇ u ) ) ) ⋅ ( λ − ∇ u ) = ( 1 − t ) ( a ( z , ∇ u ) − a ( z , ∇ u + t ( λ − ∇ u ) ) ) ⋅ ( λ − ∇ u ) .$

Employing the condition of monotonicity, we can express

$( a ( z , ∇ u ) − a ( z , ∇ u + t ( λ − ∇ u ) ) ) ⋅ t ( ∇ u − λ ) ≥ 0 ,$

and since $t ∈ [ 0 , 1 ]$ , we deduce

(9)

$( a ( z , ∇ u ) − a ( z , ∇ u + t ( λ − ∇ u ) ) ) ⋅ ( 1 − t ) ( ∇ u − λ ) ≥ 0 .$

Combining (8) and (9), we find

$( a ( z , ∇ u ) − a ( z , ∇ u + t ( λ − ∇ u ) ) ) ⋅ ( λ − ∇ u ) = 0 .$

By virtue of $( I 2 )$ , it follows that

$A ( z , λ ) = A ( z , ∇ u ) + ∫ 0 1 a ( z , ∇ u + t ( λ − ∇ u ) ) ⋅ ( λ − ∇ u ) d z = A ( z , ∇ u ) + a ( z , ∇ u ) ⋅ ( λ − ∇ u ) .$

Hence, $λ ∈ K z$ , i.e., supp $ν z ⊂ K z$ for almost every $z ∈ Ω$ .□

Now, we can proceed to prove the main result.

Proof

Since $ξ ↦ A ( z , ξ )$ is convex, we can express

$A ( z , λ ) ≕ V ( λ ) ≥ A ( z , ∇ u ) + a ( z , ∇ u ) ⋅ ( λ − ∇ u ) ≕ R ( λ )$

for all $λ ∈ R m$ . Assuming that $λ ↦ V ( λ )$ is a $C 1$ -function (as per the hypothesis), we have for every $ξ ∈ R m , t ∈ R$

$V ( λ + t ξ ) − V ( λ ) t ≥ R ( λ + t ξ ) − R ( λ ) t for t > 0 , V ( λ + t ξ ) − V ( λ ) t ≤ R ( λ + τ ξ ) − R ( λ ) t for t < 0 .$

This implies that $∇ λ V = ∇ λ R$ , i.e.,

(10)

$a ( z , λ ) = a ( z , ∇ u ) for all λ ∈ K z ⊃ supp ν z .$

Since $a ( z , ∇ u k )$ is equiintegrable, its weak $L 1$ -limit is:

and by virtue of (10) and Lemma 2.3

$a ¯ ( z ) ≔ ∫ R m a ( z , λ ) d ν z ( λ ) ,$

(11)

$a ¯ ( z ) = ∫ R m a ( z , λ ) d ν z ( λ ) = ∫ supp ν z a ( z , λ ) d ν z ( λ ) = ∫ supp ν z a ( z , ∇ u ) d ν z ( λ ) = a ( z , ∇ u ) .$

Now, consider the Carathéodory function

$ℬ ( z , λ ) = ∣ a ( z , λ ) − a ¯ ( z ) ∣ , λ ∈ R m .$

Since $a ( z , ∇ u k )$ is equiintegrable, $ℬ k ( z ) ≔ ℬ ( z , ∇ u k )$ is equiintegrable and its weak $L 1$ -limit is given by

$ℬ k → ℬ ¯ in L 1 ( Ω ) ,$

where

$ℬ ¯ ( z ) = ∫ R m ∣ a ( z , λ ) − a ¯ ( z ) ∣ d ν z ( λ ) = ∫ supp ν z ∣ a ( z , λ ) − a ¯ ( z ) ∣ d ν z ( λ ) = 0$

by (9) and (10). Importantly, the convergence of $ℬ k$ is strong since $ℬ k ≥ 0$ .

Applying $( I 1 )$ , we have

$A ( z , ∇ u k ) ≥ 1 p ( z ) a ( z , ∇ u k ) ⋅ ∇ u k ≥ 1 p ( z ) ( c 1 ∣ ∇ u k ∣ p ( z ) − α 3 ( z ) ) ≥ − 1 p ( z ) α 3 ( z ) ,$

thus

$∫ Ω ′ ∣ min ( A ( z , ∇ u k ) , 0 ) ∣ d z ≤ 1 p ( z ) ∫ Ω ′ ∣ α 3 ( z ) ∣ d z < ∞ .$

Hence, $A ( z , ∇ u k )$ is bounded and equiintegrable, its weak $L 1$ -limit is $∫ R m A ( z , λ ) d ν z ( λ )$ . Utilizing Lemma 4.1, it follows that

$∫ R m A ( z , λ ) d ν z ( λ ) = ∫ supp ν z A ( z , λ ) d ν z ( λ ) = ∫ supp ν z ( A ( z , ∇ u ) + a ( z , ∇ u ) ⋅ ( λ − ∇ u ) ) d ν z ( λ ) = A ( z , ∇ u ) (by Eq. ( E 2 ) .$

The continuity of the function $ℳ$ in $( I 3 )$ along with $ℬ k → 0$ in $L 1 ( Ω )$ allows us to deduce

$lim k → ∞ ℳ ∫ Ω A ( z , ∇ u k ) d z ∫ Ω a ( z , ∇ u k ) ⋅ ∇ φ ( z ) d z = ℳ ∫ Ω A ( z , ∇ u ) d z ∫ Ω a ( z , ∇ u ) ⋅ ∇ ϕ ( z ) d z , ∀ ϕ ∈ ∪ k ≥ 1 W k .$

To complete the proof of Theorem 1.1, we only need to consider the term $∫ Ω f ( z , u k ) ϕ ( z ) d z$ . We know that $( u k )$ is bounded in $W 0 1 , p ( z ) ( Ω )$ according to Propositions 3–5, up to a subsequence, $u k → u$ in $L p ( z ) ( Ω )$ . For some $ε$ positive, we have

$∫ Ω ∣ u k − u ∣ p ( z ) d z ≥ ∫ { z ∈ Ω : ∣ u k − u ∣ ≥ ε } ∣ u k − u ∣ p ( z ) d z ≥ ε p ( z ) ∣ { z ∈ Ω : ∣ u k − u ∣ ≥ ε } ∣ ,$

which implies

$∣ { z ∈ Ω : ∣ u k − u ∣ ≥ ε } ∣ ≤ 1 ε p ( z ) ∫ Ω ∣ u k − u ∣ p ( z ) d z → 0 as k → ∞ ,$

thus $u k → u$ in measure and almost everywhere. The continuity of the function $f$ in $( I 0 )$ implies

$f ( z , u i ) ϕ ( z ) → f ( z , u ) ϕ ( z ) almost everywhere.$

Due to the growth condition in $( I 1 )$ and uniform bound in Propositions 3–5, $f ( z , u k ) ϕ ( z )$ is equiintegrable. Consequently,

$f ( z , u k ) ϕ ( z ) → f ( z , u ) ϕ ( z ) in L 1 ( Ω )$

by the Vitali convergence theorem. Hence,

$lim k → ∞ ∫ Ω f ( z , u k ) ϕ ( z ) d z = ∫ Ω f ( z , u ) ϕ ( z ) d z ∀ ϕ ∈ ∪ k ≥ 1 W k .$

Since $∪ k ≥ 1 W k$ is dense in $W 0 1 , p ( z ) ( Ω )$ , it follows that $u$ is a weak solution of $( P )$ as desired.□

5 Conclusion

In this study, we have explored the existence of weak solutions for a class of equations represented by $( P )$ in the context of mathematical analysis and partial differential equations. This equation encompasses nonlinearities, variable exponents $p ( z )$ , and nonlocal effects through operators and functions $ℳ$ , $A$ , $a$ , and $f$ . Our approach employs Young measure techniques under mild monotonicity assumptions, enriching the theory in this domain. The findings extend the understanding of Young measures and their application in spaces such as variable exponents and Orlicz spaces.

Acknowledgement

The authors like to express their sincere thanks to the referee(s) for this paper.

Funding information: The authors state no funding involved.
Author contributions: All authors accepted the responsibility for the content of the manuscript and consented to its submission reviewed all the results, and approved the final version of the manuscript. The authors’ contributions are equal.
Conflict of interest: The authors state no conflict of interest.
Ethics and consent to participate declarations: Not applicable.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-10-19

Revised: 2024-08-11

Accepted: 2024-10-15

Published Online: 2024-12-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

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DOI: doi.org/10.1515/msds-2024-0006

Keywords for this article

weak solution; p(z)-Kirchhoff-type; nonlocal problems; variable exponents; Young measures

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