Singular Double Phase Kirchhoff Type Problem with a General Nonlocal Integrodifferential Operator

Abstract

In this study, we study a singular double-phase Kirchhoff problem involving a fractional nonlocal integrodifferential operator. More precisely, we reformulate the studied problem into an equivalent integral equation and derive the corresponding energy functional. By combining a variational method with monotonicity techniques, we establish that this functional admits a minimum point in an appropriate Sobolev space. However, due to the presence of the singular term, this minimum does not necessarily correspond to a critical point of the function. For this, we use the implicit function theorem to prove that this minimum corresponds to a weak solution for such a problem. To validate our main results, an example is presented.

1. Introduction

In this study, we investigate the following double-phase Kirchhoff problem containing a singular term on the bounded domain $\Lambda \subset {\mathbb{R}}^N$ with $N\ge 2$:

$$\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^2{G}_i\left(L_{A_i}\left(\chi \right)\right){\mathcal{L}}_{A_i}^{{\kappa }_i}\chi +{\left|\chi \right|}^{r\left(z\right)-2}\chi =\mu \left(z\right){\chi }^{-m\left(z\right)}+\lambda f(z,\chi ),\mathrm{in} \Lambda ,}\hfill \\ \\ \chi =0,\mathrm{on} \partial \Lambda ,\hfill \end{array}\right.$$

(1)

where $\partial \Lambda$ is the boundary of $\Lambda$, $\lambda \in \mathbb{R}$ is a specified parameter used to manipulate the mountain pass geometry related to the functional energy, $m\in C(\Lambda ,(0,1\left)\right)$, we adopt the following notations:

Notations: Throughout this paper, the following apply:

• The letter i will denote the integers 1 or 2.
• ${\kappa }_i:{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}_{+}$ is a measurable function.
• $G_i$ is a continuous function.
• $A_i:\mathbb{R}\to \mathbb{R}$ is a continuous, odd and increasing function, and ${\widehat{A}}_i$ is the function defined by

  $${\widehat{A}}_i\left(y\right)={\int }_0^y{A}_i\left(\zeta \right)d\zeta .$$
• $L_{A_i}\left(\chi \right)$ is defined by

  $$L_{A_i}\left(\chi \right)={\int }_{\Lambda \times \Lambda }{\widehat{A}}_i(\chi \left(z\right)-\chi \left(t\right)){\kappa }_i(z,t)dzdt.$$
• The operator ${\mathcal{L}}_{A_i}^{{\kappa }_i}$ is defined by

  $${\mathcal{L}}_{A_i}^{{\kappa }_i}u\left(x\right)=p.v.{\int }_{{\mathbb{R}}^N}{A}_i(u\left(x\right)-u\left(y\right)){\kappa }_i(x,y)dy,$$

  where $p.v.$ is a normalized constant.

We note that the operator ${\mathcal{L}}_{A_i}^{{\kappa }_i}$ generalizes several operators in the literature. In particular, if $A\left(z\right)={\left|z\right|}^{p\left(z\right)-2}z$ and $\kappa (z,t)={\displaystyle \frac{1}{{|z-t|}^{N+sp\left(z\right)}}},$ then ${\mathcal{L}}_A^{\kappa }$ becomes the well-known ${(-\Delta )}_{p(z,.)}^s$. This operator is given by

$${(-\Delta )}_{p(z,.)}^s\zeta \left(z\right)=p.v.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\zeta \left(z\right)-\zeta \left(y\right)|}^{p(z,y)-2}(\zeta \left(z\right)-\zeta \left(y\right))}{{|z-y|}^{N+sp(z,y)}}}dy.$$

Also, it is noted that operators like ${\mathcal{L}}_{A_i}^{{\kappa }_i}$ combine differentiation and integration into a single mathematical object, allowing it to capture both local and nonlocal behavior of a function. In many applications, the differential term reflects instantaneous interactions, while the integral term encodes cumulative effects. This makes such operators especially useful in several fields like fluid mechanics, neurology, and population dynamics (see [1]), in some closure models, the nonlocal effects of turbulence are represented by integrodifferential operators (see [2]). They explain how past behavior determines the current state, which is important for materials having mechanical and thermal memory (see [3]).

Very recently, many authors used several methods to study problems when the operator ${(-\Delta )}_{p(\zeta ,.)}^s$ appears; for instance, we cite the papers of Bahrouni et al. [4,5], Cabre and Tan [6], Caffarelli and Silvester [7], Coclite and Palmieri [8], and Servadei and Valdinoci [9,10].

Compared with the above works, our paper contains a Kirchhoff term. We note that the Kirchhoff equation usually refers to a class of nonlinear partial differential equations that generalize the classical wave equation by including a dependence on the integral of the gradient, reflecting nonlocal effects in the string or membrane tension. It originates from the work of G. Kirchhoff [11], who studied the vibration of elastic strings with variable tension. Precisely, Kirchhoff considered the following equation:

$$\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{{\rho }_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^{2}dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,$$

where $\rho ,{\rho }_0,L,h,E$ are constants.

In our study, we enrich the equation by adding a singular term. We note that the presence of the singularity implies more difficulty in the manipulation of such problems, since the functional energy is not of class $C^1$. Singular problems are studied by several authors. A. Fiscella et al. [12] studied the following problem:

$$\left\{\begin{array}{c}M\left(\int {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\zeta \left(z\right)-\zeta \left(y\right)|}^2}{{|z-y|}^{N+2s}}}dzdy\right){(-\Delta )}^s\zeta =\lambda f\left(z\right){\zeta }^{-\gamma \left(z\right)}+{\left|\zeta \right|}^{2^{*}-1},\mathrm{in} \Lambda ,\hfill \\ \\ \zeta >0,\mathrm{on} \partial \Lambda ,\hfill \\ \\ \zeta =0,\mathrm{on} {\mathbb{R}}^N\setminus \Lambda ,\hfill \end{array}\right.$$

(2)

where $2^{*}={\displaystyle \frac{2N}{N-2s}}$, $\gamma \in C(\mathbb{R},(0,1\left)\right)$. Using Nehari manifold techniques, the existence of two positive solutions is proved for small values of $\lambda$.

The investigation of double-phase singular problems, or singular problems in general, necessitates a comprehensive grasp of nonlinear analysis, the characteristics of singularities, boundary value problems, and the theory of regularity. This area of study is inherently interdisciplinary, integrating methods from both pure and applied mathematics to address intricate and demanding challenges encountered in a range of scientific and engineering contexts; see [13,14,15,16,17,18,19] and references therein.

Chammem et al. [20], investigated the following double-phase fractional $p(z,.)$-Laplacian problem with singular term:

$$\left\{\begin{array}{c}{\displaystyle {(-\Delta )}_{p_1(z,.)}^s\phi +{(-\Delta )}_{p_2(z,.)}^s\phi +{\left|\phi \right|}^{q\left(z\right)-2}\phi =g\left(z\right){\phi }^{-m\left(z\right)}+\lambda f(z,u),\mathrm{in} \Lambda ,}\hfill \\ \\ \phi =0,\mathrm{on} \partial \Lambda ,\hfill \end{array}\right.$$

(3)

where $p_i\in C(\mathbb{R}\times \mathbb{R},(1,\infty )),q\in C(\mathbb{R},(1,\infty ))$, $m\in C(\mathbb{R},(0,1\left)\right)$, and $\lambda \ge 0$. More precisely, the authors used the variational method to prove that Equation (3) admits a nontrivial solution.

Several other researchers have also tackled singular term problems using variations of the Nehari method and employing different operators, the nonlocal integrodifferential operator and the fractional Laplacian, to prove important existence results. Recently, E. Azroul et al. [21] used the variational tool based on the Nehari manifold approach and the fibering maps analysis to present existence results to the following fractional problem involving the integrodifferential operator

$$\left\{\begin{array}{cc}{T}_{\mathbb{K}}^{p(s,.)}\phi +{\left|\phi \right|}^{\overline{p}\left(s\right)-2}\phi =\lambda {\displaystyle \frac{a\left(s\right)}{{\left|\phi \right|}^{\gamma }}}+g\left(s\right){\left|\phi \right|}^{r\left(s\right)-2}\phi \hfill & \mathrm{in} \Lambda ,\hfill \\ \\ \phi =0,\hfill & \mathrm{in} {\mathbb{R}}^N\setminus \Lambda ,\hfill \end{array}\right.$$

(4)

where $\overline{p}\left(s\right)=p(s,s)$ with $p\in C(\mathbb{R}\times \mathbb{R},(1,+\infty \left)\right)$, $\gamma \in C(\mathbb{R},(0,1\left)\right)$, $\lambda >0$, and the operator $T_{\mathbb{K}}^{p(s,.)}$ is defined by

$$T_K^{p(s,.)}\left(\chi \right)={\int }_{\Lambda \times \Lambda }{|\chi \left(s\right)-\chi \left(t\right)|}^{p(s,t)}K(s,t)dsdt,$$

and the function $K:{\mathbb{R}}^N\times {\mathbb{R}}^N⟶(0,\infty )$ assumed to satisfy suitable conditions. For interested readers, other studies related to singular or Kirchhoff-type problems can be found in [22,23,24,25].

We note that our problem is a natural generalization of other problems in the literature; in fact, in some special cases, the operator ${\mathcal{L}}_{A_i}^{{\kappa }_i}$ is reduced to the fractional Laplacian operator ${(-\Delta )}^s$, to the fractional $p(x,.)$-Laplacian operator, and to the integrodifferential operator $T_{\mathbb{K}}^{p(x,.)}$ (see [12,20,21,26]). Moreover, the operator does not possess a simple kernel growth behavior. Its variable-exponent structure prevents the use of classical tools such as homogeneous Sobolev embeddings. On the other hand, the existence of the double-phase structure requires delicate control of oscillatory behavior in the Gagliardo seminorm. This interaction has no analogue in the classical integer-order case. Also, the combination of the Kirchhoff coefficient with the singular term causes the energy functional to be non-$C^1$; thereby, the direct variational technique is not valid for our study.

2. Preliminaries

In this section, we recall some definitions and key properties of the variable exponent Lebesgue spaces and the fractional Sobolev spaces. For further details and comprehensive discussions on these spaces, we refer the reader to the relevant literature [27,28,29,30,31,32].

We consider the set

$${\mathcal{F}}_{+}(\Lambda )=\{\zeta \in C(\Lambda ),\zeta \left(r\right)>1,\forall r\in \Lambda \}.$$

For any $r\in {\mathcal{F}}_{+}(\Lambda )$, we denote by

$$r^{-}=\underset{\overline{\Lambda }}{inf}r\left(z\right),r^{+}=\underset{\overline{\Lambda }}{sup}r\left(z\right),$$

and $M(\Lambda ,\mathbb{R})$ the space consisting by measurable functions. Now, we define the following space:

$$L^{r\left(z\right)}(\Lambda )=\{Z\in M(\Lambda ,\mathbb{R}),{\int }_{\Lambda }{\left|Z\left(\zeta \right)\right|}^{r\left(z\right)}d\zeta <\infty \}.$$

This space provides a flexible framework for modeling nonhomogeneous phenomena, appearing in applications such as fluid dynamics, image processing, and electrorheological fluids. So, it is natural to equip this space with a norm given by

$${\left|\phi \right|}_{r\left(z\right)}=inf\left\{\xi >0:{\int }_{\Lambda }{|{\displaystyle \frac{\phi \left(\zeta \right)}{\xi }}|}^{r\left(z\right)}d\zeta \le 1\right\}.$$

Equipped with the last norm, $L^{r\left(z\right)}(\Lambda )$ becomes a Banach space. If, in addition, r satisfies $1<r^{-}\le r^{+}<\infty$, then, it becomes reflexive and separable.

Next, we begin by recalling three lemmas. The first lemma concerns the Hölder inequality.

Lemma 1 

([5,20]). Let $\phi \in L^{r\left(z\right)}(\Lambda )$ and $\varphi \in L^{r\left(z\right)}(\Lambda )$ with ${\displaystyle \frac{1}{r\left(z\right)}}+{\displaystyle \frac{1}{r^{\prime }\left(z\right)}}=1,$ then we have

$$\left|{\int }_{\Lambda }\phi \left(z\right)\varphi \left(z\right)dz\right|\le \left({\displaystyle \frac{1}{r^{-}}}+{\displaystyle \frac{1}{{\left(r^{\prime }\right)}^{-}}}\right){\left|\phi \right|}_{r\left(z\right)}{\left|\varphi \right|}_{r^{\prime }\left(z\right)}.$$

Put

$${\rho }_{r\left(z\right)}\left(\phi \right)={\int }_{\Lambda }{\left|\phi \left(z\right)\right|}^{r\left(z\right)}dz.$$

We note that the last function is fundamental in defining, and analyzing variable exponent Lebesgue spaces. It determines the Luxemburg norm, characterizes convergence, and reflects the local behavior of functions according to the variable exponent $r\left(z\right)$. Moreover, it provides a natural framework for studying inequalities, variational problems, and the stability of solutions in spaces with nonuniform growth conditions. In the next lemma, we recall essential properties related to this function.

Lemma 2 

([5,20]). For all $\phi \in L^{r\left(z\right)}(\Lambda )$, we have,

1. 

$${\left|\phi \right|}_{r\left(z\right)}<1\iff {\rho }_{r\left(z\right)}\left(\phi \right)<1$$

moreover, the last inequality holds if we replace < with = or with >.

2. 

$$min\left({\left|\phi \right|}_{r\left(z\right)}^{r^{-}},{\left|\phi \right|}_{r\left(z\right)}^{r^{+}}\right)\le {\rho }_{r\left(z\right)}\left(\phi \right)\le max\left({\left|\phi \right|}_{r\left(z\right)}^{r^{-}},{\left|\phi \right|}_{r\left(z\right)}^{r^{+}}\right).$$

Lemma 3 

([5,20]). Let r and p are measurable functions such that $p\in L^{\infty }\left({\mathbb{R}}^N\right)$ such that $1\le p\left(z\right)r\left(z\right)\le \infty$ for all $z\in {\mathbb{R}}^N$. Let $\phi \in L^{r\left(z\right)}\left({\mathbb{R}}^N\right),\phi \ne 0$.Then, we have

$$min\left({\left|\phi \right|}_{p\left(z\right)r\left(z\right)}^{r^{-}},{\left|\phi \right|}_{p\left(z\right)r\left(z\right)}^{r^{+}}\right)\le {\left|\right|\phi |}^{p\left(z\right)}{|}_{r\left(z\right)}\le max\left({\left|\phi \right|}_{p\left(z\right)r\left(z\right)}^{r^{-}},{\left|\phi \right|}_{p\left(z\right)r\left(z\right)}^{r^{+}}\right).$$

Hereafter, we assume that $0<s<1$, $r\in C\left(\overline{\Lambda }\right)$ and p is symmetric in $C(\overline{\Lambda }\times \overline{\Lambda })$, such that the following equations hold:

$$1<p^{-}=\underset{\overline{\Lambda }\times \overline{\Lambda }}{inf}p(z,\zeta )\le p(z,\zeta )\le p^{+}=\underset{\overline{\Lambda }\times \overline{\Lambda }}{sup}p(z,\zeta )<\infty ,$$

(5)

and

$$1<r^{-}\le r\left(z\right)\le r^{+}<\infty .$$

(6)

Under hypotheses (5) and (6), we can define the following space:

$$W^{s,r\left(z\right),p(z,\zeta )}(\Lambda )=\left\{g\in L^{r\left(z\right)}(\Lambda ):H(g,\tau )<\infty ,\mathrm{for} \mathrm{some}\tau >0\right\},$$

where

$$H(g,\tau )={\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|g\left(z\right)-g\left(\zeta \right)|}^{p(z,\zeta )}}{{\tau }^{p(z,\zeta )}{|z-\zeta |}^{N+sp(z,\zeta )}}}dzd\zeta .$$

Now, we introduce the following Gagliardo semi-norm:

$${\left[g\right]}_{s,p(z,\zeta )}=inf\left\{\tau >0:H(g,\tau )\le 1\right\},$$

and the following norm

$${\left|\right|g\left|\right|}_{W^{s,r\left(z\right),p(z,\zeta )}(\Lambda )}={\left|\right|g\left|\right|}_{L^{r\left(z\right)}(\Lambda )}+{\left[g\right]}_{s,p(z,\zeta )}.$$

Endowed with the last norm, $X=W^{s,r\left(z\right),p(z,\zeta )}(\Lambda )$ is a separable Banach space, which is also a reflexive space. In this space and according to this norm, the closure of the space $C_0^{\infty }(\Lambda )$ will be denoted by E; it has the same properties as the space $W^{s,r\left(z\right),p(z,\zeta )}(\Lambda )$, moreover, it can be equipped with the norm

$$\left|\right|g\left|\right|={\left[g\right]}_{s,p(z,\zeta )}.$$

Next, we put

$$p(z,\zeta )=max\left(p_1(z,\zeta ),p_2(z,\zeta )\right).$$

Lemma 4 

([5,20]). The following inequalities hold:

$$min\left({\left[g\right]}_{s,p(z,\zeta )}^{p^{-}},{\left[g\right]}_{s,p(z,\zeta )}^{p^{+}}\right)\le {\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|g\left(z\right)-g\left(\zeta \right)|}^{p(z,\zeta )}}{{|z-\zeta |}^{N+sp(\zeta ,\zeta )}}}dzd\zeta \le min\left({\left[g\right]}_{s,p(z,\zeta )}^{p^{-}},{\left[g\right]}_{s,p(z,\zeta )}^{p^{+}}\right).$$

Theorem 1 

([33,34]). Let $h:\overline{\Lambda }\to (1,\infty )$ be a continuous function such that, for each $(z,\zeta )\in \overline{\Lambda }\times \overline{\Lambda }$, we have $sp(z,\zeta )<N$, and

$$1<h^{-}\le h\left(z\right)<p^{*}\left(z\right):=p^{*}\left(z\right)={\displaystyle \frac{Np(z,z)}{N-sp(z,z)}}.$$

Then, there exists a constant $k>0$, such that for any $\phi \in E$, we have

$${\left|\phi \right|}_{L^{h\left(z\right)}}\le {k\left|\right|\phi \left|\right|}_E.$$

This means that the space E is continuously and compactly embedded in $L^{t\left(\zeta \right)}(\Lambda )$.

Finally, let $\phi \in X$, we define the following functional:

$$\rho \left(\phi \right)={\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p(z,\zeta )}}{{|z-\zeta |}^{N+sp(z,\zeta )}}}dzd\zeta +{\int }_{\Lambda }{\left|\phi \left(z\right)\right|}^{r\left(z\right)}dz,$$

and

$${\left|\right|\phi \left|\right|}_{\rho }=inf\left\{\zeta >0:\rho \left({\displaystyle \frac{\phi }{\zeta }}\right)\le 1\right\}.$$

Then, $\left|\right|·{\left|\right|}_{\rho }$ is a norm which is equivalent to the norm of the space X.

3. Existence Result for $\mathbf{\lambda }\mathbf{\ge }\mathbf{0}$

In this section, we present and prove the first main result of this work that concerns nonnegative values of the parameter $\lambda$. For this, we assume that the measurable function $A:\mathbb{R}\to \mathbb{R}$, is such that for any $(z,\zeta )\in {\mathbb{R}}^N\times {\mathbb{R}}^N$, we have

(A₁)

There exist $c_{A_i}>0$ and $C_{A_i}>0$, such that for each $t\in \mathbb{R}$, we have

$$A_i\left(t\right)t\ge c_{A_i}{\left|t\right|}^{p_i(z,\zeta )}\mathrm{and}{A}_i\left(t\right)\le C_{A_i}{\left|t\right|}^{p_i(z,\zeta )-2}t.$$

(A₂)

$\kappa :{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}_{+}$ be a measurable function, such that

$${\displaystyle \frac{a_{1i}}{{|z-\zeta |}^{N+s{p}_i(z,\zeta )}}}\le {\kappa }_i(z,\zeta )\le {\displaystyle \frac{a_{2i}}{{|z-\zeta |}^{N+s{p}_i(z,\zeta )}}},z\ne \zeta ,$$

for some positive constants $a_{1i}$ and $a_{2i}$.

Moreover, we assume that for all $(z,\zeta )\in \overline{\Lambda }\times \overline{\Lambda }$, we have

(H₀)

$G_i:\mathbb{R}\to [0,\infty )$ is a continuous, and there exist ${\theta }_i>1$, and $1<{\beta }_2\le {\beta }_1$, such that for any $t\ge 0$, one has

$${\displaystyle \frac{1}{{\theta }_i}}{t}^{{\beta }_i-1}\le G_i\left(t\right)\le {\theta }_i{t}^{{\beta }_i-1},{\beta }_{i}p(\zeta ,\zeta )<{\displaystyle \frac{N}{s}},$$

and

$$p^{*}\left(\zeta \right)>r\left(\zeta \right)\ge {\beta }_{1}p(\zeta ,\zeta )\ge {\beta }_{2}p(\zeta ,\zeta ),\mathrm{and}sp(z,\zeta )<N,$$

(H₁)

There exists $h\in C\left(\overline{\Lambda }\right)$, such that $1<h\left(z\right)<p^{*}\left(z\right)$, and

$$\mu \in L^{{\displaystyle \frac{h\left(z\right)}{h\left(z\right)+m\left(z\right)-1}}}(\Lambda ).$$

(H₂)

There exist $S,v\in C\left(\overline{\Lambda }\right)$, $\sigma \in L^{S\left(z\right)}(\Lambda )$, and $c>0$, such that

$$f(z,\zeta )\le {c\sigma \left(z\right)\left|\zeta \right|}^{v\left(z\right)-2}\zeta ,$$

and

$$1<v\left(z\right)<p_i(z,z)<{\displaystyle \frac{N}{s}}<S\left(z\right).$$

(H₃)

There exists ${\Lambda }_1\subset \subset \Lambda$ with $|{\Lambda }_1|>0$, and $f\ge 0$ in ${\Lambda }_1$.

Remark 1. 

The first parts of hypotheses $\left(H_0\right)$ and $\left(H_2\right)$ are used to prove some estimations for the energy functional, in particular, the coercivity. The second parts of these hypotheses are used in the embedding results. hypothesis $\left(H_1\right)$ is used to apply the Hölder inequality to prove that the energy functional has negative values, and hypothesis $\left(H_3\right)$ is used in the manipulation of the singular term, especially, to apply the Vitali theorem.

Now, let’s define what a weak solution to our problem is.

Definition 1. 

A function $\phi \in E$ is a weak solution of (1) if, for every $\varphi \in E$, one has

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left(\phi \right)\right){\int }_{\Lambda \times \Lambda }{A}_i(\phi \left(z\right)-\phi \left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(\zeta ,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|\phi \left(\zeta \right)\right|}^{r\left(\zeta \right)-2}\phi \left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|\phi \right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,\phi \left(\zeta \right))\varphi \left(\zeta \right)d\zeta =0.\end{array}$$

Now, let ${\mathcal{I}}_{\lambda }:E\to \mathbb{R}$ be a functional defined by

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\phi \right)& =& \widehat{G}\left(\phi \right)+{\int }_{\Lambda }{\displaystyle \frac{{\left|\phi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}}d\zeta -{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\left|\phi \right|}^{1-m\left(\zeta \right)}d\zeta -\lambda {\int }_{\Lambda }F(\zeta ,\phi \left(\zeta \right))d\zeta ,\hfill \end{array}$$

where ${\displaystyle \widehat{G}\left(\phi \right)=\sum _{i=1}^2\widehat{G_i}\left(L_{p_i}\left(\phi \right)\right)}$, $\widehat{G_i}\left(t\right)={\int }_0^t{G}_i\left(\tau \right)d\tau$ and $F(\zeta ,t)={\int }_0^{t}f(\zeta ,\xi )d\xi$.

The functional ${\mathcal{I}}_{\lambda }$ is well defined. Moreover, a simple calculation shows that the derivative of this functional contains ${\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{{\left|\phi \right|}^{m\left(\zeta \right)}}}d\zeta$ which is not defined at zero. So, ${\mathcal{I}}_{\lambda }$ is not differentiable.

The following is the first outcome of this work.

Theorem 2. 

Assume that the hypotheses (A₁)–(A₂) and (H₀)–(H₃) Hold. Then, for each $\lambda \ge 0,$ the problem (1) has a nontrivial solution.

Four lemmas relating to the min–max approach must be demonstrated in order to support our first main result. In the first one, we prove a lower bound for the functional $\widehat{G}$.

Lemma 5. 

Let $\phi \in X$ with $\parallel \phi \parallel >1$. Then, we have

$$\widehat{G}\left(\phi \right)\ge {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}.$$

Proof. 

Let $\phi \in X$ such that $\parallel \phi \parallel >1$. Then, from hypothesis $\left(H_0\right)$, we have

$${\widehat{G}}_i\left(t\right)\ge {\displaystyle \frac{1}{{\theta }_i{\beta }_i}}{t}^{{\beta }_i},\mathrm{for}\mathrm{all}t\in [0,\infty ).$$

Consequently, by (A₁)–(A₂) and Lemma 4 we have

$$\begin{array}{ccc}& & \widehat{G}\left(\phi \right)=\widehat{G_1}\left({\int }_{\Lambda \times \Lambda }{\widehat{A}}_1(\phi \left(z\right)-\phi \left(\zeta \right)){\kappa }_1(z,\zeta )dzd\zeta \right)+\widehat{G_2}\left({\int }_{\Lambda \times \Lambda }{\widehat{A}}_2(\phi \left(z\right)-\phi \left(\zeta \right)){\kappa }_2(z,\zeta )dzd\zeta \right)\hfill \\ & \ge & {\displaystyle \frac{1}{{\theta }_1{\beta }_1}}{\left({\int }_{\Lambda \times \Lambda }{\widehat{A}}_1(\phi \left(z\right)-\phi \left(\zeta \right)){\kappa }_1(z,\zeta )dzd\zeta \right)}^{{\beta }_1}+{\displaystyle \frac{1}{{\theta }_2{\beta }_2}}{\left({\int }_{\Lambda \times \Lambda }{\widehat{A}}_2(\phi \left(z\right)-\phi \left(\zeta \right)){\kappa }_2(z,\zeta )dzd\zeta \right)}^{{\beta }_2}\hfill \\ & \ge & {\displaystyle \frac{1}{{\theta }_1{\beta }_1}}{\left({\int }_{\Lambda }{a}_{11}{c}_{A_1}{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p_1(z,\zeta )}}{p_1(z,\zeta ){|z-\zeta |}^{N+s{p}_1(z,\zeta )}}}dzd\zeta \right)}^{{\beta }_1}+{\displaystyle \frac{1}{{\theta }_2{\beta }_2}}{\left(a_{12}{c}_{A_2}{\int }_{\Lambda }{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p_2(z,\zeta )}}{p_2(z,\zeta ){|z-\zeta |}^{N+s{p}_2(z,\zeta )}}}dzd\zeta \right)}^{{\beta }_2}\hfill \\ & \ge & {\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1{\beta }_1{\left(p_1^{+}\right)}^{{\beta }_1}}}{\left({\int }_{\Lambda }{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p_1(z,\zeta )}}{{|z-\zeta |}^{N+s{p}_1(z,\zeta )}}}dzd\zeta \right)}^{{\beta }_1}+{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2{\beta }_2{\left(p_2^{+}\right)}^{{\beta }_2}}}{\left({\int }_{\Lambda }{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p_2(z,\zeta )}}{{|z-\zeta |}^{N+s{p}_2(z,\zeta )}}}dzd\zeta \right)}^{{\beta }_2}\hfill \\ & \ge & min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1{\beta }_1{\left(p_1^{+}\right)}^{{\beta }_1}}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2{\beta }_2{\left(p_2^{+}\right)}^{{\beta }_2}}}){\left({\int }_{\Lambda }{\displaystyle \frac{{|\phi \left(z\right)-\phi \left(\zeta \right)|}^{p(z,\zeta )}}{{|z-\zeta |}^{N+sp(\zeta ,\zeta )}}}dzd\zeta \right)}^{{\beta }_2}\hfill \\ & \ge & {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}.\hfill \end{array}$$

□

After proving the lower bound for the functional $\widehat{G}$, we prove in the next lemma the coercivity of the functional ${\mathcal{I}}_{\lambda }.$

Lemma 6. 

Assume that hypotheses (A₁)–(A₂) and (H₀)–(H₃) hold. If $\lambda \ge 0$, then ${\mathcal{I}}_{\lambda }$ is coercive in E.

Proof. 

let $\phi \in E$ be such that $\left|\right|\phi \left|\right|>1$. Then, from Lemma 5 we obtain

$$\begin{array}{c}\hfill \widehat{G}\left(\phi \right)+{\int }_{\Lambda }{\displaystyle \frac{{\left|\phi \left(z\right)\right|}^{r\left(z\right)}}{r\left(z\right)}}dz\ge {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}.\end{array}$$

(7)

On the other hand, from Lemma 1, we have

$${\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\phi }^{1-m\left(\zeta \right)}d\zeta \le {\displaystyle \frac{1}{1-m^{+}}}{\int }_{\Lambda }\mu \left(\zeta \right){\phi }^{1-m\left(\zeta \right)}d\zeta \le {\displaystyle \frac{1}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}{\left|\right|\phi |}^{1-m\left(\zeta \right)}{|}_{{\displaystyle \frac{t\left(\zeta \right)}{1-m\left(\zeta \right)}}}.$$

So, from hypothesis $\left(H_1\right)$, Lemma 2, Theorem 1, and the fact that $1<t\left(\zeta \right)<p^{*}\left(\zeta \right)$, we obtain

$$\begin{array}{ccc}\hfill {\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\phi }^{1-m\left(\zeta \right)}d\zeta & \le & {\displaystyle \frac{1}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}{(|\phi |}_{t\left(\zeta \right)}^{1-m^{+}}+{\left|\phi \right|}_{t\left(\zeta \right)}^{1-m^{-}})\hfill \\ & \le & {\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}{\left(\right|\left|\phi \right||}^{1-m^{+}}+{\left|\right|\phi \left|\right|}^{1-m^{-}}).\hfill \end{array}$$

(8)

Now, from $\left(H_3\right)$, Lemmas 1 and 4, we obtain

$$\begin{array}{ccc}\hfill {\int }_{\Lambda }F(\zeta ,\phi \left(\zeta \right))d\zeta & \le & c{\int }_{\Lambda }\sigma \left(\zeta \right){\left|\phi \left(\zeta \right)\right|}^{v\left(\zeta \right)}d\zeta \hfill \\ & \le & {c\left|\sigma \right|}_{S\left(\zeta \right)}{\left|\right|\phi |}^{v\left(\zeta \right)}{|}_{S^{\prime }\left(\zeta \right)}\hfill \\ & \le & {c\left|\sigma \right|}_{S\left(\zeta \right)}\left({\left|\phi \right|}_{S^{\prime }\left(\zeta \right)v\left(\zeta \right)}^{v^{+}}+{\left|\phi \right|}_{S^{\prime }\left(\zeta \right)v\left(\zeta \right)}^{v^{-}}\right).\hfill \end{array}$$

We note that from $\left(H_3\right)$, we deduce that $S^{\prime }\left(\zeta \right)v\left(\zeta \right)<p^{*}\left(\zeta \right).$ So, from Theorem 1, we get

$${\int }_{\Lambda }F(\zeta ,\phi \left(\zeta \right))d\zeta \le c{\left|\sigma \right|}_{S\left(\zeta \right)}\left({\left|\right|\phi \left|\right|}^{v^{+}}+{\left|\right|\phi \left|\right|}^{v^{-}}\right).$$

(9)

Finally, by combining Equations (7), (8) with Equation (9), we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\varphi \right)& \ge & {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}-{\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}{(|\left|\phi \right||}^{1-m^{+}}+{\left|\right|\phi \left|\right|}^{1-m^{-}})\hfill \\ & -& {c\lambda \left|\sigma \right|}_{S\left(\zeta \right)}{(|\left|\phi \right||}^{v^{+}}+{\left|\right|\phi \left|\right|}^{v^{-}}).\hfill \end{array}$$

Since we have $1-m^{-}<v^{+}<{\beta }_2{p}^{-},$ then, we obtain ${\mathcal{I}}_{\lambda }\left(\phi \right)\to +\infty$ as $\left|\right|\phi \left|\right|\to \infty$. That is ${\mathcal{I}}_{\lambda }$ is coercive on E. □

Lemma 7. 

Assume that the hypotheses $\left(A_1\right)$–$\left(A_3\right)$ and $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, for all $\lambda \ge 0$, there exists a positive function $\varphi \in E$ such that, for each small $t>0$, we have

$${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0.$$

Proof. 

let $0<t<1$, $\varphi \in C_0^{\infty }(\Lambda )$ such that $supp\left(\varphi \right)\subset {\Lambda }_1\subset \subset \Lambda ,$$\varphi =1$ in a subset ${\Lambda }^{\prime }\subset supp\left(\varphi \right)$ and $\varphi \in (0,1)$ in $\Lambda$. Then, if we combine hypotheses $\left(A_1\right)$ and $\left(A_2\right)$ with Lemma 4, we obtain

$$\begin{array}{c}\hfill L_{p_i}\left(\varphi \right)\le {\displaystyle \frac{a_{2i}{C}_{A_i}}{p_i^{-}}}{max\left(\right|\left|\varphi \right||}^{p_i^{-}}{,\left|\right|\varphi \left|\right|}^{p_i^{+}}).\end{array}$$

(10)

Again, from $\left(A_1\right)$ and $\left(A_2\right)$, we get

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{i=1}^2\widehat{G_i}\left(L_{p_i}\left(t\varphi \right)\right)+{\int }_{\Lambda }\frac{t^{r\left(\zeta \right)}{\left|\varphi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}d\zeta }& \le & {\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\left(L_{p_1}\left(t\varphi \right)\right)}^{{\beta }_1}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\left(L_{p_2}\left(t\varphi \right)\right)}^{{\beta }_2}\hfill \\ & & +{\displaystyle \frac{1}{r^{-}}}{t}^{r^{-}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta .\hfill \end{array}$$

(11)

To finish the proof, we have to discuss two cases:

• Case 1: In this case, we assume that $\left|\right|\varphi \left|\right|\le 1$, so from Equations (10) and (11), we get

$$\begin{array}{ccc}& & \widehat{G_1}\left(L_{p_1}\left(t\varphi \right)\right)+\widehat{G_2}\left(L_{p_2}\left(t\varphi \right)\right)+{\int }_{{\Lambda }_1}{\displaystyle \frac{t^{r\left(\zeta \right)}{\left|\varphi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}}d\zeta \hfill \\ & \le & {\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}{t}^{p_1^{-}{\beta }_1}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}{t}^{p_2^{-}{\beta }_2}+{\displaystyle \frac{1}{r^{-}}}{t}^{r^{-}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta \hfill \\ & \le & t^{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})}\left({\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta \right).\hfill \end{array}$$

Since, $min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})>1-m^{-},$ and $\lambda \ge 0$, then using hypothesis $\left(H_3\right)$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(t\varphi \right)& \le & t^{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})}\left({\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta \right)\hfill \\ & -& t^{1-m^{-}}{\int }_{{\Lambda }_1}{\displaystyle \frac{\mu \left(\zeta \right)\varphi {\left(\zeta \right)}^{1-m\left(\zeta \right)}}{1-m\left(\zeta \right)}}d\zeta ,\hfill \\ & \le & t^{1-m^{-}}\left(t^{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})-1+m^{-}}A-B\right),\hfill \end{array}$$

where

$$A={\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta ,$$

and

$$B={\int }_{{\Lambda }_1}{\displaystyle \frac{\mu \left(\zeta \right)\varphi {\left(\zeta \right)}^{1-m\left(\zeta \right)}}{1-m\left(\zeta \right)}}d\zeta .$$

Finally, from the last inequality, if we take

$$0<t<min\left(1,{\left({\displaystyle \frac{B}{A}}\right)}^{{\displaystyle \frac{1}{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})}}}\right),$$

Then, we obtain ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0$.

• Case 2: In this case, we assume that $\left|\right|\varphi \left|\right|>1$. So, if we proceed as in case 1, we can prove that ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0,$ provided that

$$0<t<min\left(1,{\left({\displaystyle \frac{B}{C}}\right)}^{{\displaystyle \frac{1}{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})}}}\right),$$

where B is given in the first case, and

$$C={\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{+}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{+}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{+}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta .$$

□

Since the functional is coercive, then we can define $l={inf}_{\varphi \in E}{\mathcal{I}}_{\lambda }\left(\varphi \right)$. Moreover, in the following lemma, we show that this minimum is achieved.

Lemma 8. 

Assume that hypotheses $\left(H_1\right)$–$\left(H_4\right)$ hold. If in addition $\lambda \ge 0$, then there exists ${\psi }_1\in E$ such that, ${\mathcal{I}}_{\lambda }\left({\psi }_1\right)=l<0$.

Proof. 

Let $\left\{{\phi }_n\right\}$ be a sequence in E such that

$${\mathcal{I}}_{\lambda }\left({\phi }_n\right)\to l.$$

From the fact that ${\mathcal{I}}_{\lambda }$ is coercive, we can easily prove that $\left({\phi }_n\right)$ is bounded in E. Moreover, from the fact that the space E is reflexive, we can find a subsequence, denoted also by $\left\{{\phi }_n\right\}$ and ${\psi }_1\in E,$ such that

$$\left\{\begin{array}{c}{\phi }_n\rightharpoonup {\psi }_{1}weaklyinE\hfill \\ {\phi }_n\to {\psi }_{1}stronglyin{L}^{r\left(z\right)}(\Lambda ),1\le r\left(\zeta \right)<p^{*}\left(\zeta \right)\hfill \\ {\phi }_n\to {\psi }_{1}a.ein\Lambda .\hfill \end{array}\right.$$

Since $\left({\phi }_n\right)$ is bounded in E, from Theorem 1, we concluded that $\left\{{\phi }_n\right\}$ is also bounded in $L^{h\left(\zeta \right)}(\Lambda )$. So, using the continuous embedding

$$E\hookrightarrow L^{h\left(z\right)}(\Lambda ),$$

we obtain that $|{\varphi }_n{|}^{1-m\left(z\right)}$ is bounded in $L^{{\displaystyle \frac{h\left(z\right)}{1-m\left(z\right)}}}(\Lambda )$, which is crucial when estimating the singular term. Now, hypothesis $\left(H_1\right)$ ensures that

$$\mu \in L^{{\displaystyle \frac{h\left(z\right)}{h\left(z\right)+m\left(z\right)-1}}}(\Lambda ).$$

By applying Hölder’s inequality, together with Lemmas 2.2 and 2.3, we now explicitly show that

$${\int }_{\Lambda }\mu \left(z\right)|{\varphi }_n{|}^{1-m\left(z\right)}dz\le C{\parallel \mu \parallel }_{L^{{\displaystyle \frac{h\left(z\right)}{h\left(z\right)+m\left(z\right)-1}}}}{∥|{\varphi }_n{|}^{1-m\left(z\right)}∥}_{L^{{\displaystyle \frac{h\left(z\right)}{1-m\left(z\right)}}}}.$$

Since the second factor is uniformly bounded due to the embedding property noted above, the sequence is uniformly integrable on measurable subsets of arbitrarily small measure. Finally, from Vitali’s theorem (see [35], p. 113), the combination of almost-everywhere convergence and uniform integrability of the set $\left\{\mu \left(z\right)\right|{\varphi }_n{|}^{1-m\left(z\right)}\}$ implies that we have

$$\underset{n\to \infty }{lim}{\int }_{\Lambda }\mu \left(z\right)|{\phi }_n{|}^{1-m\left(z\right)}dz={\int }_{\Lambda }\mu \left(z\right){\left|{\psi }_1\right|}^{1-m\left(z\right)}dz.$$

(12)

Now, from $\left(H_3\right)$, we have

$$|F(\zeta ,{\phi }_n\left(\zeta \right))|\le {\displaystyle \frac{c_{ϵ}}{v^{-}}}\left|\sigma \left(\zeta \right)\right||{\phi }_n{|}^{v\left(\zeta \right)},\mathrm{where}{c}_{ϵ}>0.$$

Since ${\phi }_n\rightharpoonup {\psi }_1$ weakly in E, and using the fact that $S^{\prime }\left(\zeta \right)v\left(\zeta \right)<p^{*}\left(\zeta \right)$, then from the compact embedding, we deduce the strong convergence in $L^{S^{\prime }\left(\zeta \right)v\left(\zeta \right)}(\Lambda )$. So, up to a sub-sequence, we have ${\phi }_n\to {\psi }_1$ a.e in $\Lambda$, and there exists $\mu \in L^{v\left(\zeta \right)S^{\prime }\left(\zeta \right)}$ such that

$$|{\phi }_n\left(\zeta \right)|\le \mu \left(\zeta \right).$$

Therefore, we get

$$|F(\zeta ,{\phi }_n\left(\zeta \right))|\le {\displaystyle \frac{c_{ϵ}}{v^{-}}}{\left|\sigma \left(\zeta \right)\right|\left|\mu \left(\zeta \right)\right|}^{v\left(\zeta \right)}.$$

On the other hand, from Lemma 1, we obtain

$${\int }_{\Lambda }|F(\zeta ,{\phi }_n\left(\zeta \right))|d\zeta \le {\displaystyle \frac{c_{ϵ}}{v^{-}}}{\left|\sigma \right|}_{S\left(\zeta \right)}{\left|\mu \right|}_{S^{\prime }\left(\zeta \right)}^{v\left(\zeta \right)}.$$

Finally, from the Lebesgue dominated convergence theorem and Lemma 3, we conclude that

$$\underset{n\to +\infty }{lim}{\int }_{\Lambda }F(\zeta ,{\phi }_n\left(\zeta \right))d\zeta ={\int }_{\Lambda }F(\zeta ,{\psi }_1\left(\zeta \right))d\zeta .$$

(13)

Let

$$\begin{array}{c}\hfill I\left(u\right)=\widehat{G}\left(u\right)+{\int }_{\lambda }{\displaystyle \frac{{\left|u\left(\zeta \right)\right|}^{r\left(z\right)}}{r\left(z\right)}}d\zeta .\end{array}$$

Since $\widehat{G}$ is continuous and $u_n\to {\psi }_1$ a.e in $\Lambda$, then from Fatou’s lemma we have

$$I\left(\psi \right)\le \underset{n\to \infty }{lim}infI\left(u_n\right).$$

(14)

Now, by combining Equation (14) with Equations (12) and (13), we find that ${\mathcal{I}}_{\lambda }$ is weakly lower semi-continuous. Hence, we obtain

$${\mathcal{I}}_{\lambda }\left({\psi }_1\right)\le \underset{n\to +\infty }{lim}inf{\mathcal{I}}_{\lambda }\left({\phi }_n\right)=l.$$

and we have from the definition of l

$${\mathcal{I}}_{\lambda }\left({\psi }_1\right)\ge l.$$

So, by Lemma 7, we deduce that

$${\mathcal{I}}_{\lambda }\left({\psi }_1\right)=l<0.$$

□

Now we show that ${\psi }_1$ obtained in Lemma 8 is a nontrivial solution of our Equation (1).

Proof of Theorem 2: 

From Lemma 8, we see that the function ${\psi }_1$ is a global minimizer for ${\mathcal{I}}_{\lambda }$ that satisfies ${\mathcal{I}}_{\lambda }\left({\psi }_1\right)<0$. In particular, we have that ${\psi }_1$ is nontrivial.

Now, let $\varphi$ be an arbitrary function in E and $t>0$, then we have

$$0\le \underset{t\to 0^{+}}{lim}{\displaystyle \frac{{\mathcal{I}}_{\lambda }({\psi }_1+t\varphi )-{\mathcal{I}}_{\lambda }\left({\psi }_1\right)}{t}}.$$

So, by replacing ${\mathcal{I}}_{\lambda }$ by its expression, we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left({\psi }_1\right)\right){\int }_{\Lambda \times \Lambda }{A}_i({\psi }_1\left(z\right)-{\psi }_1\left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(z,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|{\psi }_1\left(\zeta \right)\right|}^{r\left(\zeta \right)-2}{\psi }_1\left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|{\psi }_1\right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,{\psi }_1\left(\zeta \right))\varphi \left(\zeta \right)d\zeta \ge 0.\end{array}$$

Since the function $\varphi$ is arbitrary in E, then we can replace $\varphi$ by $-\varphi$ in the last inequality, which yields to

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left({\psi }_1\right)\right){\int }_{\Lambda \times \Lambda }{A}_i({\psi }_1\left(z\right)-{\psi }_1\left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(z,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|{\psi }_1\left(\zeta \right)\right|}^{r\left(\zeta \right)-2}{\psi }_1\left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|{\psi }_1\right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,{\psi }_1\left(\zeta \right))\varphi \left(\zeta \right)d\zeta \le 0.\end{array}$$

Hence, we deduce that

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left({\psi }_1\right)\right){\int }_{\Lambda \times \Lambda }{A}_i({\psi }_1\left(z\right)-{\psi }_1\left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(z,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|{\psi }_1\left(\zeta \right)\right|}^{r\left(\zeta \right)-2}{\psi }_1\left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|{\psi }_1\right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,{\psi }_1\left(\zeta \right))\varphi \left(\zeta \right)d\zeta =0.\end{array}$$

That is, ${\psi }_1$ is a nontrivial solution of Equation (1). □

4. Existence Result for $\lambda <0$

In this section, we present and prove the first main result of this work that concerns negative values of the parameter $\lambda$. For this, we assume the following hypothesis:

(H₄)

There exists $L>0$ such that

$${\int }_{\Lambda }F(\zeta ,t)d\zeta \ge 0,\mathrm{for}\mathrm{any}t>L.$$

Theorem 3. 

Under the same hypotheses of Theorem 2, if in addition $\left(H_4\right)$ is satisfied. Then, for each $\lambda >0$, Equation (1) admits a nontrivial solution.

Since $\lambda <0$, the ${\mathcal{I}}_{\lambda }$ can be written as

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\phi \right)& =& \widehat{G}\left(\phi \right)+{\int }_{\Lambda }{\displaystyle \frac{{\left|\phi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}}d\zeta -{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\left|\phi \right|}^{1-m\left(\zeta \right)}d\zeta +\left|\lambda \right|{\int }_{\Lambda }F(\zeta ,\phi \left(\zeta \right))d\zeta .\hfill \end{array}$$

Before proving Theorem 3, we need to prove three lemmas.

Lemma 9. 

Let hypotheses $\left(A_1\right)$–$\left(A_2\right)$ and $\left(H_0\right)$–$\left(H_4\right)$ are satisfied. Then, for all $\lambda <0$, the functional ${\mathcal{I}}_{\lambda }$ is coercive in E.

Proof. 

Let $\varphi \in E$ with $\left|\right|\varphi \left|\right|>max(1,L)$, where L is defined in hypothesis $\left(H_4\right)$. Then, by Lemma 5, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\phi \right)& =& \widehat{G}\left(\phi \right)+{\int }_{\Lambda }{\displaystyle \frac{{\left|\phi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}}d\zeta -{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\left|\phi \right|}^{1-m\left(\zeta \right)}d\zeta +\left|\lambda \right|{\int }_{\Lambda }F(\zeta ,\phi \left(\zeta \right))d\zeta \hfill \\ & \ge & \widehat{G}\left(\phi \right)-{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\left|\phi \right|}^{1-m\left(\zeta \right)}d\zeta ,\hfill \\ & \ge & {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}-{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\left|\phi \right|}^{1-m\left(\zeta \right)}d\zeta ,\hfill \end{array}$$

(15)

On the other hand, by Equation (8), there exists $c>0$ such that,

$$\begin{array}{ccc}\hfill {\int }_{\Lambda }{\displaystyle \frac{\mu \left(\zeta \right)}{1-m\left(\zeta \right)}}{\varphi }^{1-m\left(\zeta \right)}d\zeta & \le & {\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}\left({\left|\right|\varphi \left|\right|}^{1-m^{+}}+{\left|\right|\varphi \left|\right|}^{1-m^{-}}\right).\hfill \end{array}$$

Finally, by combining Equation (8) with Equation (15), we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\phi \right)& \ge & {\displaystyle \frac{1}{{\beta }_1{\left(p^{+}\right)}^{{\beta }_1}}}min({\displaystyle \frac{{\left(a_{11}{c}_{A_1}\right)}^{{\beta }_1}}{{\theta }_1}},{\displaystyle \frac{{\left(a_{12}{c}_{A_2}\right)}^{{\beta }_2}}{{\theta }_2}}){\left|\right|\phi \left|\right|}^{{\beta }_2{p}^{-}}\hfill \\ & & -{\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\zeta \right)}{t\left(\zeta \right)+m\left(\zeta \right)-1}}}\left({\left|\right|\varphi \left|\right|}^{1-m^{+}}+{\left|\right|\varphi \left|\right|}^{1-m^{-}}\right).\hfill \end{array}$$

Since $1-m^{-}<{\beta }_2{p}^{-},$ then we obtain ${\mathcal{I}}_{\lambda }\left(\varphi \right)\to \infty$ as $\left|\right|\varphi \left|\right|\to \infty$. This means that ${\mathcal{I}}_{\lambda }$ is coercive on E. □

Lemma 10. 

Assume that hypotheses $\left(A_1\right)$–$\left(A_2\right)$ and $\left(H_0\right)$–$\left(H_4\right)$ hold. Then, for each $\lambda <0$, there exists a positive function $\varphi \in E$ such that ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0$ for $t>0$ small enough.

Proof. 

We assume that $t\in (0,1)$, and we let $\varphi \in C_0^{\infty }(\Lambda )$ be such that $supp\left(\varphi \right)\subset {\Lambda }_1\subset \subset \Lambda$ and $\varphi =1$ in a subset ${\Lambda }^{\prime }\subset supp\left(\varphi \right)$, $\varphi \in (0,1)$ in $\Lambda$. Then, from Equation (9) and hypotheses $\left(A_1\right)$–$\left(A_2\right)$, we deduce the existence of $c>0$ such that

$${\int }_{\Lambda }F(\zeta ,t\varphi \left(\zeta \right))d\zeta \le {c\left|\sigma \right|}_{S\left(\zeta \right)}{\left(\right|\left|t\varphi \right||}^{v^{+}}+{\left|\right|t\varphi \left|\right|}^{v^{-}}).$$

(16)

We have to discuss two cases:

• If $\left|\right|\varphi \left|\right|\le 1$, then from Equations (10) and (11), we get

  $$\begin{array}{ccc}& & \widehat{G_1}\left(L_{p_1}\left(t\varphi \right)\right)+\widehat{G_2}\left(L_{p_2}\left(t\varphi \right)\right)+{\int }_{{\Lambda }_1}{\displaystyle \frac{t^{r\left(\zeta \right)}{\left|\varphi \left(\zeta \right)\right|}^{r\left(\zeta \right)}}{r\left(\zeta \right)}}d\zeta \hfill \\ & \le & t^{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-})}\left({\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta \right):=Q.\hfill \end{array}$$

  Since, $min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-},v^{-})>1-m^{-},$ then by Equation (16), there exists $C>0$ such that

  $$\begin{array}{ccc}& & {\mathcal{I}}_{\lambda }\left(t\varphi \right)\le Q-t^{1-m^{-}}{\int }_{{\Lambda }_1}{\displaystyle \frac{\mu \left(\zeta \right)\varphi {\left(\zeta \right)}^{1-m\left(\zeta \right)}}{1-m\left(\zeta \right)}}d\zeta ,\hfill \\ & \le & t^{1-m^{-}}\left(t^{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-},v^{-})-1+m^{-}}{A}_3-B\right),\hfill \end{array}$$

  where

  $$A_3={\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{-}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta +{C\left|\lambda \right|\left|\right|\varphi \left|\right|}^{v^{-}},$$

  and

  $$B={\int }_{{\Lambda }_1}{\displaystyle \frac{\mu \left(\zeta \right)\varphi {\left(\zeta \right)}^{1-m\left(\zeta \right)}}{1-m\left(\zeta \right)}}d\zeta .$$

  So, it is clear that if

  $$0<t<min\left(1,{\left({\displaystyle \frac{B}{A_3}}\right)}^{{\displaystyle \frac{1}{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-},v^{-})}}}\right),$$

  then we obtain ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0.$
• If $\left|\right|\varphi \left|\right|>1$, then if we proceed as in the first step, we can prove that ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0,$ provided that

  $$0<t<min\left(1,{\left({\displaystyle \frac{B}{A_4}}\right)}^{{\displaystyle \frac{1}{min({\beta }_1{p}_1^{-},{\beta }_2{p}_2^{-},r^{-},v^{-})}}}\right),$$

  where

  $$A_4={\displaystyle \frac{{\theta }_1}{{\beta }_1}}{\displaystyle \frac{a_{21}{C}_{A_1}}{p_1^{+}}}{\left|\right|\varphi \left|\right|}^{{\beta }_1{p}_1^{+}}+{\displaystyle \frac{{\theta }_2}{{\beta }_2}}{\displaystyle \frac{a_{22}{C}_{A_2}}{p_2^{-}}}{\left|\right|\varphi \left|\right|}^{{\beta }_2{p}_2^{+}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{{\Lambda }_1}{\left|\varphi \right|}^{r\left(\zeta \right)}d\zeta +{C\left|\lambda \right|\left|\right|\varphi \left|\right|}^{v^{+}}.$$

□

Put

$${\displaystyle l_1=\underset{\varphi \in E}{inf}{\mathcal{I}}_{\lambda }\left(\varphi \right).}$$

The next lemma shows that this minimum is achieved. Since the proof is similar to the one in the Lemma 8. Then, we omit it here.

Lemma 11. 

Assume that hypotheses $\left(H_1\right)$–$\left(H_4\right)$ hold. Then, for any $\lambda <0,$ there exist ${\psi }_2\in E$ such that, ${\mathcal{I}}_{\lambda }\left({\psi }_2\right)=l_1<0$.

Proof of Theorem 3: 

From Lemma 11, we see that ${\psi }_2$ is a global minimizer for ${\mathcal{I}}_{\lambda }$ that satisfies ${\mathcal{I}}_{\lambda }\left({\psi }_2\right)<0$. In particular, ${\psi }_2$ is nontrivial. Moreover, for each arbitrary $\varphi \in E$, and for each $t>0$, we have

$$0\le \underset{t\to 0^{+}}{lim}{\displaystyle \frac{{\mathcal{I}}_{\lambda }({\psi }_2+t\varphi )-{\mathcal{I}}_{\lambda }\left({\psi }_2\right)}{t}}.$$

If we replace ${\mathcal{I}}_{\lambda }$ by its expression, we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left({\psi }_2\right)\right){\int }_{\Lambda \times \Lambda }{A}_i({\psi }_2\left(z\right)-{\psi }_2\left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(z,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|{\psi }_2\left(\zeta \right)\right|}^{r\left(\zeta \right)-2}{\psi }_2\left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|{\psi }_2\right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,{\psi }_2\left(\zeta \right))\varphi \left(\zeta \right)d\zeta \ge 0.\end{array}$$

If we proceed as in the proof of Theorem 2, and using the fact that $\varphi$ is arbitrary, we deduce that

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^2{G}_i\left(L_{p_i}\left({\psi }_2\right)\right){\int }_{\Lambda \times \Lambda }{A}_i({\psi }_2\left(z\right)-{\psi }_2\left(\zeta \right))(\varphi \left(z\right)-\varphi \left(\zeta \right)){\kappa }_i(z,\zeta )dzd\zeta +{\int }_{\Lambda }{\left|{\psi }_2\left(\zeta \right)\right|}^{r\left(\zeta \right)-2}{\psi }_2\left(\zeta \right)\varphi \left(\zeta \right)d\zeta }\\ \hfill -{\int }_{\Lambda }\mu \left(\zeta \right){\left|{\psi }_2\right|}^{-m\left(\zeta \right)}\varphi \left(\zeta \right)d\zeta -\lambda {\int }_{\Lambda }f(\zeta ,{\psi }_2\left(\zeta \right))\varphi \left(\zeta \right)d\zeta =0.\end{array}$$

So, ${\psi }_2$ is a nontrivial solution for Equation (1). □

5. Example

In this section, we provide a concrete application of the main results by considering specific cases. For this, let $1<{\beta }_2\ge {\beta }_1$, $\psi$ be an integrable function, $m\in C(\Lambda ,(0,1\left)\right)$, and $(r,S,v,h)\in {\left(C\left(\overline{\Lambda }\right)\right)}^4$, such that

$$p^{*}\left(\zeta \right)>r\left(\zeta \right)\ge {\beta }_{1}p(\zeta ,\zeta )\ge {\beta }_{2}p(\zeta ,\zeta ),\mathrm{and}sp(z,\zeta )<N,$$

(17)

and

$$1<v\left(z\right)<p_i(z,z)<{\displaystyle \frac{N}{s}}<S\left(z\right).$$

(18)

Consider the problem

$$\left(P\right)\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^2{\left(T_i\left(\chi \right)\right)}^{{\beta }_i-1}{(-\Delta )}_{p_i(z,.)}^s\chi +{\left|\chi \right|}^{r\left(z\right)-2}\chi =\mu \left(z\right){\chi }^{-m\left(z\right)}+\lambda \sigma \left(z\right){\left|z\right|}^{v\left(z\right)-2}z,\mathrm{in} \Lambda ,}\hfill \\ \\ \chi =0,\mathrm{on} \partial \Lambda ,\hfill \end{array}\right.$$

where $\mu ={\psi }^{{\displaystyle \frac{m\left(z\right)+h\left(z\right)-1}{h\left(z\right)}}}$, $\sigma \in L^{S\left(z\right)}(\Lambda )$, and

$$T_i\left(u\right)={\int }_{\Omega \times \Omega }{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p_i(x,y)}}{p_i(x,y){|x-y|}^{N+s{p}_i(x,y)}}}dxdy.$$

(19)

As mentioned above, the operator ${(-\Delta )}_{p_i(z,.)}^s$ correspond to

$$A_i\left(z\right)={\left|z\right|}^{p_i\left(z\right)-2}z,\mathrm{and}\kappa (z,\zeta )={\displaystyle \frac{1}{{|z-\zeta |}^{N+sp\left(z\right)}}}.$$

So, hypothesis $\left(A_1\right)$ is satisfied for any $c_{A_i}<1<C_{A_i}$, and hypothesis $\left(A_1\right)$ is satisfied for any $a_{1i}<1<a_{2i}>1$.

Now, from Equation (17) and the fact that $1<{\beta }_2\le {\beta }_1$, we obtain that the function $G_i\left(t\right)=t^{{\beta }_i-1}$, so hypothesis $\left(H_0\right)$ holds for any ${\theta }_i\ge 1$.

We have ${\mu }^{{\displaystyle \frac{h\left(z\right)}{m\left(z\right)+h\left(z\right)-1}}}=\psi$ is integrable, which implies that the hypothesis $\left(H_1\right)$ holds.

Since $\sigma \in L^{S\left(z\right)}(\Lambda )$, and the function $f(z,\zeta )={\sigma \left(z\right)\left|\zeta \right|}^{v\left(\zeta \right)-2}\zeta$ satisfies the inequality in hypothesis $\left(H_2\right)$ for any $c\le 1$, then from Equation (18), we see that hypothesis $\left(H_2\right)$ is also satisfied.

Finally, a simple calculation shows that $F(z,\zeta )=\sigma \left(z\right){\displaystyle \frac{{\left|\zeta \right|}^{v\left(\zeta \right)}}{v\left(\zeta \right)}}$, which satisfies hypothesis $\left(H_4\right)$. Moreover, if we assume in addition that $\sigma$ is nonnegative, then hypothesis $\left(H_3\right)$ holds. Finally, Theorems 2 and 3 can be applied. So, we conclude that Equation (4) admits a nontrivial solution for every $\lambda \in \mathbb{R}$.

6. Conclusions

In this work, we discussed the existence of solutions for two types of singular fractional problems of Kirchhoff type. Since the functional is not of class $C^1$, to prove the existence of solutions, we used the min–max method. These types of problems have garnered significant attention, especially in recent years, with a growing focus on methods that incorporate fractional operators. Due to the importance of these problems, we will continue studying similar problems by considering the following:

• Perturbation of this equation by a singular critical logarithmic term.
• Other types of boundary conditions, like Neumann and Styklov boundary conditions.