A fractional-order model of COVID-19 with a strong Allee effect considering the fear effect spread by social networks to the community and the existence of the silent spreaders during the pandemic stage

1.   Introduction

In this article, we study the following anisotropic singular $\overset{\rightarrow }{p}(\cdot)$-Laplace equation

where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 3)$ with smooth boundary $\partial\Omega$; $f \in L^{1}(\Omega)$ is a positive function; $g \in L^{\infty}(\Omega)$ is a nonnegative function; $\beta \in C(\overline{\Omega})$ such that $1 < \beta(x) < \infty$ for any $x \in \overline{\Omega}$; $q \in C(\overline{\Omega})$ such that $0 < q(x) < 1$ for any $x \in \overline{\Omega}$; $p_{i} \in C(\overline{\Omega })$ such that $2\leq p_{i}(x) < N$ for any $x \in \overline{\Omega}$, $i\in \left\{1, ..., N\right\}$.

The differential operator

that appears in problem (1.1) is an anisotropic variable exponent $\overset{\rightarrow }{p}\left(\cdot \right)$-Laplace operator, which represents an extension of the $p(\cdot)$-Laplace operator

obtained in the case for each $i\in \left\{ 1, ..., N\right\}$, $p_{i}(\cdot) = p(\cdot)$.

In the variable exponent case, $p(\cdot)$, the integrability condition changes with each point in the domain. This makes variable exponent Sobolev spaces very useful in modeling materials with spatially varying properties and in studying partial differential equations with non-standard growth conditions ^{[1,2,3,4,5,6,7,8]}.

Anisotropy, on the other hand, adds another layer of complexity, providing a robust mathematical framework for modeling and solving problems that involve complex materials and phenomena exhibiting non-uniform and direction-dependent properties. This is represented mathematically by having different exponents for different partial derivatives. We refer to the papers ^{[9,10,11,12,13,14,15,16,17,18,19,20,21]} and references for further reading.

The progress in researching anisotropic singular problems with $\overset{\rightarrow }{p}(\cdot)$-growth, however, has been relatively slow. There are only a limited number of studies available on this topic in academic literature. We could only refer to the papers ^[22,23,24] that were published recently. In ^[22], the author studied an anisotropic singular problems with constant case $p(\cdot) = p$ but with a variable singularity, where existence and regularity of positive solutions was obtained via the approximation methods. In ^[23], the author obtained the existence and regularity results of positive solutions by using the regularity theory and approximation methods. In ^[24], the authors showed the existence of positive solutions using the regularity theory and maximum principle. However, none of these papers studied combined effects of variable singular and sublinear nonlinearities.

We would also like to mention that the singular problems of the type

have been intensively studied because of their wide applications to physical models in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogenous catalysts, glacial advance, etc. (see, e.g., ^{[25,26,27,28,29,30]}).

These studies, however, have mainly focused on the case $0 < \beta < 1$, i.e., the weak singularity (see, e.g. ^{[31,32,33,34,35,36]}), and in this case, the corresponding energy functional is continuous.

When $\beta > 1$ (the strong singularity), on the other hand, the situation changes dramatically, and numerous challenges emerge in the analysis of differential equations of the type (1.2), where the primary challenge encountered is due to the lack of integrability of $u^{-\beta}$ for $u \in H_{0}^{1}(\Omega)$ ^{[37,38,39,40,41]}.

To overcome these challenges, as an alternative approach, the so-called "compatibility relation" between $f(x)$ and $\beta$ has been introduced in the recent studies ^[37,40,42]. This method, used along with a constrained minimization and the Ekeland's variational principle ^[43], suggests a practical approach to obtain solutions to the problems of the type (1.2). In the present paper, we generalize these results to nonstandard $p(\cdot)$-growth.

The paper is organized as follows. In Section 2, we provide some fundamental information for the theory of variable Sobolev spaces since it is our work space. In Section 3, first we obtain the auxiliary results. Then, we present our main result and obtain a positive solution to problem (1.1). In Section 4, we provide an example to illustrate our results in a concrete way.

2.   Preliminaries

We start with some basic concepts of variable Lebesgue-Sobolev spaces. For more details, and the proof of the following propositions, we refer the reader to ^[1,2,44,45].

For $p\in C_{+}(\overline{\Omega})$ denote

For any $p\in C_{+}\left(\overline{\Omega }\right)$, we define the variable exponent Lebesgue space by

then, $L^{p(\cdot)}(\Omega)$ endowed with the norm

becomes a Banach space.

Proposition 2.1. For any $u\in L^{p(\cdot) }(\Omega)$ and $v\in L^{p^{\prime}(\cdot)}(\Omega)$, we have

where $L^{p^{\prime }(x) }(\Omega)$ is the conjugate space of $L^{p(\cdot) }(\Omega)$ such that $\frac{1}{p(x) }+\frac{1}{p^{\prime}(x)} = 1$.

The convex functional $\Lambda :L^{p(\cdot) }(\Omega) \rightarrow \mathbb{R}$ defined by

is called modular on $L^{p(\cdot) }(\Omega)$.

Proposition 2.2. If $u, u_{n}\in L^{p(\cdot) }(\Omega)$ ($n = 1, 2, ...$), we have

$(i)$ $|u|_{p(\cdot) } < 1 ( = 1; > 1) \Leftrightarrow \Lambda(u) < 1 ( = 1; > 1)$;

$(ii)$ $|u|_{p(\cdot) } > 1 \implies |u|_{p(\cdot)}^{p^{-}}\leq \Lambda(u) \leq |u|_{p(\cdot) }^{p^{+}}$;

$(iii)$ $|u|_{p(\cdot) }\leq1 \implies |u|_{p(\cdot) }^{p^{+}}\leq \Lambda(u) \leq |u|_{p(\cdot) }^{p^{-}}$;

$(iv)$ $\lim\limits_{n\rightarrow \infty }|u_{n}|_{p(\cdot)} = 0\Leftrightarrow \lim\limits_{n\rightarrow \infty }\Lambda (u_{n}) = 0;\lim\limits_{n\rightarrow \infty }|u_{n}|_{p(\cdot)} = \infty \Leftrightarrow \lim\limits_{n\rightarrow \infty }\Lambda (u_{n}) = \infty.$

Proposition 2.3. If $u, u_{n}\in L^{p(\cdot)}(\Omega)$ ($n = 1, 2, ...$), then the following statements are equivalent:

$(i)$ $\lim\limits_{n\rightarrow \infty}|u_{n}-u|_{p(\cdot)} = 0$;

$(ii)$ $\lim\limits_{n\rightarrow \infty }\Lambda (u_{n}-u) = 0$;

$(iii)$ $u_{n}\rightarrow u\mathit{\ }$in measure in$\mathit{\ }\Omega \mathit{\ }$and $\mathit{\ }\lim\limits_{n\rightarrow \infty}\Lambda (u_{n}) = \Lambda(u)$.

The variable exponent Sobolev space $W^{1, p(\cdot)}(\Omega)$ is defined by

with the norm

or equivalently

for all $u\in W^{1, p(\cdot)}(\Omega)$.

As shown in ^[46], the smooth functions are in general not dense in $W^{1, p(\cdot)}(\Omega)$, but if the variable exponent $p\in C_{+}(\overline{\Omega})$ is logarithmic Hölder continuous, that is

then the smooth functions are dense in $W^{1, p(\cdot)}(\Omega)$ and so the Sobolev space with zero boundary values, denoted by $W_{0}^{1, p(\cdot)}(\Omega)$, as the closure of $C_{0}^{\infty}(\Omega)$ does make sense. Therefore, the space $W_{0}^{1, p(\cdot)}(\Omega)$ can be defined as $\overline{C_{0}^{\infty }(\Omega)}^{\|\cdot\|_{1, p(\cdot)}} = W_{0}^{1, p(\cdot)}(\Omega)$, and hence, $u\in W_{0}^{1, p(\cdot)}(\Omega)$ iff there exists a sequence $(u_{n})$ of $C_{0}^{\infty }(\Omega)$ such that $\|u_{n}-u\|_{1, p(\cdot)}\rightarrow0$.

As a consequence of Poincaré inequality, $\|u\|_{1, p(\cdot)}$ and $|\nabla u|_{p(\cdot)}$ are equivalent norms on $W_{0}^{1, p(\cdot)}(\Omega)$ when $p\in C_{+}(\overline{\Omega})$ is logarithmic Hölder continuous. Therefore, for any $u\in W_{0}^{1, p(\cdot)}(\Omega)$, we can define an equivalent norm $\|u\|$ such that

Proposition 2.4. If $1 < p^{-}\leq p^{+} < \infty$, then the spaces $L^{p(\cdot)}(\Omega)$ and $W^{1, p(\cdot)}(\Omega)$ are separable and reflexive Banach spaces.

Proposition 2.5. Let $q\in C(\overline{\Omega })$. If $1\leq q(x) < p^{\ast }(x)$ for all $x\in \overline{\Omega }$, then the embedding $W^{1, p(\cdot)}(\Omega) \hookrightarrow L^{q(\cdot)}(\Omega)$ is compact and continuous, where

Finally, we introduce the anisotropic variable exponent Sobolev spaces.

Let us denote by $\overset{\rightarrow }{p}:\overline{\Omega }\rightarrow \mathbb{R}^{N}$ the vectorial function $\overset{\rightarrow }{p}(\cdot) = (p_{1}(\cdot), ..., p_{N}(\cdot))$ with $p_{i}\in C_{+}(\overline{\Omega})$, $i\in \left\{1, ..., N\right\}$. We will use the following notations.

Define $\overrightarrow{P}_{+}, \overrightarrow{P}_{-}\in \mathbb{R}^{N}$ as

and $P_{+}^{+}, P_{-}^{+}, P_{-}^{-}\in \mathbb{R}^{+}$ as

Below, we use the definitions of the anisotropic variable exponent Sobolev spaces as given in ^[12] and assume that the domain $\Omega \subset \mathbb{R}^N$ satisfies all the necessary assumptions given in there.

The anisotropic variable exponent Sobolev space is defined by

which is associated with the norm

$W^{1, \overset{\rightarrow }{p}(\cdot) }(\Omega)$ is a reflexive Banach space under this norm.

The subspace $W_{0}^{1, \overset{\rightarrow }{p}(\cdot) }(\Omega)\subset W^{1, \overset{\rightarrow }{p}(\cdot) }(\Omega)$ consists of the functions that are vanishing on the boundary, that is,

We can define the following equivalent norm on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot) }(\Omega)$

since the smooth functions are dense in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot) }(\Omega)$, as the variable exponent $p_{i}\in C_{+}(\overline{\Omega})$, $i\in \left\{1, ..., N\right\}$ is logarithmic Hölder continuous.

The space $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is also a reflexive Banach space (for the theory of the anisotropic Sobolev spaces see, e.g., the monographs ^[2,47,48] and the papers ^[12,15]).

Throughout this article, we assume that

and define $P_{-}^{\ast }\in\mathbb{R}^{+}$ and $P_{-, \infty }\in\mathbb{R}^{+}$ by

Proposition 2.6. [^[15], Theorem 1] Suppose that $\Omega \subset\mathbb{R}^{N}$($N\geq 3$) is a bounded domain with smooth boundary and relation (2.2) is fulfilled. For any $q\in C\left(\overline{\Omega } \right)$ verifying

the embedding

is continuous and compact.

3.   The main results

We define the singular energy functional $\mathcal{J}:W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)\rightarrow \mathbb{R}$ corresponding to equation (1.1) by

Definition 3.1. A function $u$ is called a weak solution to problem (1.1) if $u\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ such that $u > 0$ in $\Omega$ and

for all $\varphi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Definition 3.2. Due to the singularity of $\mathcal{J}$ on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$, we apply a constrained minimization for problem (1.1). As such, we introduce the following constrains:

and

Remark 1. $\mathcal{N}_{2}$ can be considered as a Nehari manifold, even though in general it may not be a manifold. Therefore, if we set

then one might expect that $c_{0}$ is attained at some $u \in \mathcal{N}_{2}$ (i.e., $\mathcal{N}_{2} \neq \varnothing$) and that $u$ is a critical point of $\mathcal{J}$.

Throughout the paper, we assume that the following conditions hold:

$(A_{1})$ $\beta:\overline{\Omega}\rightarrow (1, \infty)$ is a continuous function such that $1 < \beta^{-} \leq \beta(x) \leq \beta^{+} < \infty$.

$(A_{2})$ $q:\overline{\Omega}\rightarrow (0, 1)$ is a continuous function such that $0 < q^{-} \leq q(x) \leq q^{+} < 1$ and $q^{+}+1 \leq\beta^{-}$.

$(A_{3})$ $2\leq P_{-}^{-}\leq P_{+}^{+} < P_{-}^{*}$ for almost all $x\in \overline{\Omega}.$

$(A_{4})$ $f \in L^{1}(\Omega)$ is a positive function, that is, $f(x) > 0$ a.e. in $\Omega$.

$(A_{5})$ $g \in L^{\infty}(\Omega)$ is a nonnegative function.

Lemma 3.3. For any $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ satisfying $\int_{\Omega}f(x)|u|^{1-\beta(x)}dx < \infty$, the functional $\mathcal{J}$ is well-defined and coercive on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Proof. Denote by $\mathcal{I}_{1}, \mathcal{I}_{2}$ the indices sets $\mathcal{I}_{1} = \{i \in \{1, 2, ..., N\}: |\partial_{x_i}u|_{p_i(\cdot)}\leq1 \}$ and $\mathcal{I}_{2} = \{i \in \{1, 2, ..., N\}: |\partial_{x_i}u|_{p_i(\cdot)} > 1 \}$. Using Proposition 2.2, it follows

which shows that $\mathcal{J}$ is well-defined on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Applying similar steps and using the generalized mean inequality for $\sum_{i = 1}^{N}|\partial_{x_i} u|^{P^-_-}_{p_i(\cdot)}$ gives

That is, $\mathcal{J}$ is coercive (i.e., $\mathcal{J}(u) \to \infty \text{ as } \|u\|_{\overset{\to }{p}(\cdot)}\to \infty$), and bounded below on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$. □

Next, we provide a-priori estimate.

Lemma 3.4. Assume that $(u_{n}) \subset \mathcal{N}_{1}$ is a nonnegative minimizing sequence for the minimization problem $\lim_{n \to \infty}\mathcal{J}(u_{n}) = \inf_{\mathcal{N}_{1}}\mathcal{J}$. Then, there are positive real numbers $\delta_{1}, \delta_{2}$ such that

Proof. We assume by contradiction that there exists a subsequence $(u_{n})$ (not relabelled) such that $u_{n}\rightarrow 0$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$. Thus, we can assume that $\|u_{n}\|_{\overset{\to }{p}(\cdot)} < 1$ for $n$ large enough, and therefore, $|\partial_{x_i} u_n|_{{L^{p_i(\cdot)}}} < 1$. Then, using Proposition 2.2, we have

We recall the following elementary inequality: for all $r, s > 0$ and $m > 0$ it holds

where $K: = \max \{1, 2^{1-m}\}$. If we let $r = |\partial_{x_1} u_n|^{P^-_-}_{{L^{p_1(\cdot)}}}$, $s = |\partial_{x_2} u_n|^{P^-_-}_{{L^{p_2(\cdot)}}}$ and $m = P^-_-$ in (3.5), it reads

where $K = \max \{1, 2^{1-P^-_-}\} = 1$. Applying this argument to the following terms in the sum $\sum_{i = 1}^{N}|\partial_{x_i} u_n|_{p_i(\cdot)}^{P^-_-}$ consecutively leads to

Now, using (3.7) and the reversed Hölder's inequality, we have

By the assumption, $(u_{n}) \subset \mathcal{N}_{1}$. Thus, using (3.8) and Proposition 2.2 leads to

Considering the assumption $(A_{2})$, this can only happen if $\int_{\Omega}|u_{n}|dx\rightarrow \infty$, which is not possible. Therefore, there exists a positive real number $\delta_1$ such that $\|u_{n}\|_{\overset{\to }{p}(\cdot)}\geq \delta_1$.

Now, let's assume, on the contrary, that $\|u_{n}\|_{\overset{\to }{p}(\cdot)} > 1$ for any $n$. We know, by the coerciveness of $\mathcal{J}$, that the infimum of $\mathcal{J}$ is attained, that is, $\infty < m: = \inf_{u\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)}\mathcal{J}(u)$. Moreover, due to the assumption $\lim_{n\rightarrow \infty}\mathcal{J}(u_{n}) = \inf_{\mathcal{N}_{1}}\mathcal{J}$, $(\mathcal{J}(u_{n}))$ is bounded. Then, applying the same steps as in (3.3), it follows

for some constant $C > 0$. If we drop the nonnegative terms, we obtain

Dividing the both sides of the above inequality by $\|u_{n}\|_{\overset{\to }{p}(\cdot)}^{q^{+}+1}$ and passing to the limit as $n \to \infty$ leads to a contradiction since we have $q^{-}+1 < P_{-}^{-}$. Therefore, there exists a positive real number $\delta_2$ such that $\|u_{n}\|_{\overset{\to }{p}(\cdot)}\leq \delta_2$.

Lemma 3.5. $\mathcal{N}_{1}$ is closed in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Proof. Assume that $(u_{n}) \subset \mathcal{N}_{1}$ such that $u_{n} \to \hat{u}\, (strongly)$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$. Thus, $u_{n}(x) \to \hat{u}(x)$ a.e. in $\Omega$, and $\partial_{x_i} u_{n} \to \partial_{x_i} \hat {u}$ in $L^{p_i(\cdot)}(\Omega)$ for $i = 1, 2, ..., N$. Then, using Fatou's lemma, it reads

and hence,

which means $\hat{u} \in \mathcal{N}_{1}$. $\mathcal{N}_{1}$ is closed in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Lemma 3.6. For any $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ satisfying $\int_{\Omega}f(x)|u|^{1-\beta(x)}dx < \infty$, there exists a unique continuous scaling function $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)\rightarrow (0, \infty):u\longmapsto t(u)$ such that $t(u)u\in \mathcal{N}_{2}$, and $t(u) u$ is the minimizer of the functional $\mathcal{J}$ along the ray $\{tu:t > 0\}$, that is, $\inf_{t > 0}\mathcal{J}(tu) = \mathcal{J}(t(u)u)$.

Proof. Fix $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ such that $\int_{\Omega}f(x)|u|^{1-\beta(x)}dx < \infty$. For any $t > 0$, the scaled functional, $\mathcal{J}(tu)$, determines a curve that can be characterized by

Then, for a $t \in [0, \infty)$, $tu\in \mathcal{N}_{2}$ if and only if

First, we show that $\Phi(t)$ attains its minimum on $[0, \infty)$ at some point $t = t(u)$.

Considering the fact $0 < \int_{\Omega}f(x)|u|^{1-\beta(x)}dx < \infty$, we will examine two cases for $t$.

For $0 < t < 1$:

Then, $\Psi_{0}:(0, 1)\to\mathbb{R}$ is continuous. Taking the derivative of $\Psi_{0}$ gives

It is easy to see from (3.12) that $\Psi_{0}^{\prime}(t) < 0$ when $t > 0$ is small enough. Therefore, $\Psi_{0}(t)$ is decreasing when $t > 0$ is small enough. In the same way,

Then, $\Psi_{1}:(0, 1)\to\mathbb{R}$ is continuous. Taking the derivative of $\Psi_{1}$ gives

But (3.13) also suggests that $\Psi_{1}^{\prime}(t) < 0$ when $t > 0$ is small enough. Thus, $\Psi_{1}(t)$ is decreasing when $t > 0$ is small enough. Therefore, since $\Psi_{0}(t)\leq \Phi(t)\leq \Psi_{1}(t)$ for $0 < t < 1$, $\Phi(t)$ is decreasing when $t > 0$ is small enough.

For $t > 1$: Following the same arguments shows that $\Psi_{0}^{\prime}(t) > 0$ and $\Psi_{1}^{\prime}(t) > 0$ when $t > 1$ is large enough, and therefore, both $\Psi_{0}(t)$ and $\Psi_{1}(t)$ are increasing. Thus, $\Phi(t)$ is increasing when $t > 1$ is large enough. In conclusion, since $\Phi(0) = 0$, $\Phi(t)$ attains its minimum on $[0, \infty)$ at some point, say $t = t(u)$. That is, $\frac{d}{dt}\Phi(t)|_{t = t(u)} = 0$. Then, $t(u)u\in \mathcal{N}_{2}$ and $\inf_{t > 0}\mathcal{J}(tu) = \mathcal{J}(t(u)u)$.

Next, we show that scaling function $t(u)$ is continuous on $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Let $u_{n}\rightarrow u$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)\backslash \{0\}$, and $t_{n} = t(u_{n})$. Then, by the definition, $t_{n}u_{n}\in \mathcal{N}_{2}$. Defined in this way, the sequence $t_{n}$ is bounded. Assume on the contrary that $t_{n}\rightarrow \infty$ (up to a subsequence). Then, using the fact $t_{n}u_{n}\in \mathcal{N}_{2}$ it follows

which suggests a contradiction when $t_{n}\rightarrow \infty$. Hence, sequence $t_{n}$ is bounded. Therefore, there exists a subsequence $t_{n}$ (not relabelled) such that $t_{n}\rightarrow t_{0}$, $t_{0}\geq 0$. On the other hand, from Lemma 3.4, $\|t_{n}u_{n}\|_{\overset{\to }{p}(\cdot)}\geq \delta_1 > 0$. Thus, $t_{0} > 0$ and $t_{0}u\in\mathcal{N}_{2}$. By the uniqueness of the map $t(u)$, $t_{0} = t(u)$, which concludes the continuity of $t(u)$. In conclusion, for any $\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ satisfying $\int_{\Omega}f(x)|u|^{1-\beta(x)}dx < \infty$, the function $t(u)$ scales $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ continuously to a point such that $t(u)u\in \mathcal{N}_{2}$.

Lemma 3.7. Assume that $(u_{n}) \subset \mathcal{N}_{1}$ is the nonnegative minimizing sequence for the minimization problem $\lim_{n \to \infty}\mathcal{J}(u_{n}) = \inf_{\mathcal{N}_{1}}\mathcal{J}$. Then, there exists a subsequence $(u_{n})$ (not relabelled) such that $u_{n}\rightarrow u^{*}$ (strongly) in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

Proof. Since $(u_{n})$ is bounded in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ and $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is reflexive, there exists a subsequence $(u_{n})$, not relabelled, and $u^{*}\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ such that

● $u_{n}\rightharpoonup u^{*}$ (weakly) in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$,

● $u_{n}\rightarrow u^{*}$ in $L^{s(\cdot)}(\Omega)$, $1 < s(x) < P_{-, \infty}$, for all $x \in \overline{\Omega}$,

● $u_{n}(x) \rightarrow u^{*}(x)$ a.e. in $\Omega$.

Since the norm $\|\cdot\|_{\overset{\to }{p}(\cdot)}$ is a continuous convex functional, it is weakly lower semicontinuous. Using this fact along with the Fatou's lemma, and Lemma 3.4, it reads

The above result implies, up to subsequences, that

Thus, (3.15) along with $u_{n}\rightharpoonup u^{*}$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ show that $u_{n}\to u^{*}$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$.

The following is the main result of the present paper.

Theorem 3.8. Assume that the conditions $(A_{1})-(A_{5})$ hold. Then, problem (1.1) has at least one positive $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$-solution if and only if there exists $\overline{u} \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ satisfying $\int_{\Omega}f(x)|\overline{u}|^{1-\beta(x)}dx < \infty$.

Proof. $(\Rightarrow):$ Assume that the function $u \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is a weak solution to problem (1.1). Then, letting $u = \varphi$ in Definition (3.1) gives

where $P_{M}: = \max\{P^{-}_{-}, P^{+}_{+}\}$ and $q_{M}: = \max\{q^-, q^+\}$, changing according to the base.

$(\Leftarrow):$ Assume that there exists $\overline{u} \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ such that $\int_{\Omega}f(x)|\overline{u}|^{1-\beta(x)}dx < \infty$. Then, by Lemma 3.6, there exists a unique number $t(\overline{u}) > 0$ such that $t(\overline{u})\overline{u}\in \mathcal{N}_{2}$.

The information we have had about $\mathcal{J}$ so far and the closeness of $\mathcal{N}_{1}$ allow us to apply Ekeland's variational principle to the problem $\inf_{\mathcal{N}_{1}}\mathcal{J}$. That is, it suggests the existence of a corresponding minimizing sequence $(u_{n}) \subset \mathcal{N}_{1}$ satisfying the following:

$(E_{1})$ $\mathcal{J}(u_{n})-\inf_{\mathcal{N}_{1}}\mathcal{J}\leq \frac{1}{n}$,

$(E_{2})$ $\mathcal{J}(u_{n})-\mathcal{J}(\nu)\leq \frac{1}{n}\|u_{n}-\nu\|_{\overset{\to }{p}(\cdot)}, \, \, \, \forall \nu \in \mathcal{N}_{1}$.

Due to the fact $\mathcal{J}(|u_{n}|) = \mathcal{J}(u_{n})$, it is not wrong to assume that $u_{n}\geq0$ a.e. in $\Omega$. Additionally, considering that $(u_{n}) \subset \mathcal{N}_{1}$ and following the same approach as it is done in the $(\Rightarrow)$ part, we can obtain that $\int_{\Omega}f(x)|u_{n}|^{1-\beta(x)}dx < \infty$. If all this information and the assumptions $(A_{1})$, $(A_{2})$ are taken into consideration, it follows that $u_{n}(x) > 0$ a.e. in $\Omega$.

The rest of the proof is split into two cases.

Case Ⅰ: $(u_{n}) \subset \mathcal{N}_{1}\setminus \mathcal{N}_{2}$ for $n$ large.

For a function $\varphi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ with $\varphi\geq 0$, and $t > 0$, we have

Therefore, using $(A_{1})$, $(A_{2})$ gives

Then, when $t > 0$ is small enough in (3.16), we obtain

which means that $\nu: = u_{n}+t \varphi \in \mathcal{N}_{1}$. Now, using $(E_{2})$, it reads

Dividing the above inequality by $t$ and passing to the infimum limit as $t \to 0$ gives

Calculation of $I_{1}, I_{2}$ gives

and

For $I_{3}$: Since for $t > 0$ it holds

we can apply Fatou's lemma, that is,

Now, substituting $I_{1}, I_{2}, I_{3}$ gives

From Lemma 3.7, we know that $u_{n}\to u^{*}$ in $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$. Thus, also considering Fatou's lemma, we obtain

for any $\varphi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ with $\varphi\geq 0$. Letting $\varphi = u^{*}$ in (3.21) shows clearly that $u^{*} \in \mathcal{N}_{1}$.

Lastly, from Lemma 3.7, we can conclude that

which means

Case Ⅱ: There exists a subsequence of $(u_{n})$ (not relabelled) contained in $\mathcal{N}_{2}$.

For a function $\varphi\in W_{0}^{1, p(x)}(\Omega)$ with $\varphi\geq 0$, $t > 0$, and $u_{n} \in \mathcal{N}_{2}$, we have

and hence, there exists a unique continuous scaling function, denoted by $\theta_{n}(t): = t(u_{n}+t \varphi) > 0$, corresponding to $(u_{n}+t \varphi)$ so that $\theta_{n}(t)(u_{n}+t \varphi)\in \mathcal{N}_{2}$ for $n = 1, 2, ...$. Obviously, $\theta_{n}(0) = 1$. Since $\theta_{n}(t)(u_{n}+t \varphi)\in \mathcal{N}_{2}$, we have

and

where $\beta_{m}: = \min\{\beta^-, \beta^+\}$. Then, using (3.24) and (3.25) together gives

for some constants $\tau_1, \tau_2 \in (0, 1)$. To proceed, we assume that $\theta^{\prime}_{n}(0) = \frac{d}{dt}\theta_{n}(t)|_{t = 0} \in [-\infty, \infty]$. In case this limit does not exist, we can consider a subsequence $t_{k} > 0$ of $t$ such that $t_{k}\to 0$ as $k \to \infty$.

Next, we show that $\theta^{\prime}_{n}(0)\neq \infty$.

Dividing the both sides of (3.26) by $t$ and passing to the limit as $t \to 0$ leads to

or

which, along with Lemma 3.4, concludes that $-\infty \leq \theta^{\prime}_{n}(0) < \infty$, and hence, $\theta^{\prime}_{n}(0) \leq \overline{c}$, uniformly in all large $n$.

Next, we show that $\theta^{\prime}_{n}(0)\neq -\infty$.

First, we apply Ekeland's variational principle to the minimizing sequence $(u_{n}) \subset \mathcal{N}_{2}(\subset \mathcal{N}_{1})$. Thus, letting $\nu: = \theta_{n}(t)(u_{n}+t \varphi)$ in $(E_{2})$ gives

If we use Lemma 3.4 to manipulate the norm $\|u+t\varphi\|_{\overset{\to }{p}(\cdot)}$, the integral in the last line of (3.29) can be written as follows

Then,

Dividing by $t$ and passing to the limit as $t \to 0$ gives

which concludes that $\theta^{\prime}_{n}(0)\neq-\infty$. Thus, $\theta^{\prime}_{n}(0) \geq \underline{c}$ uniformly in large $n$.

In conclusion, there exists a constant, $C_{0} > 0$ such that $|\theta^{\prime}_{n}(0)|\leq C_{0}$ when $n \geq N_0, \, \, N_0 \in \mathbb{N}$.

Next, we show that $u^{*} \in \mathcal{N}_{2}$.

Using $(E_{2})$ again, we have

Dividing by $t$ and passing to the limit as $t \to 0$ gives

If we consider that $|\theta^{\prime}_{n}(0)|\leq C_{0}$ uniformly in $n$, we obtain that $\int_{\Omega}f(x)u_{n}^{-\beta(x)}dx < \infty$. Therefore, for $n \to \infty$ it reads

for all $\varphi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$, $\varphi \geq 0$. Letting $\varphi = u^{*}$ in (3.35) shows clearly that $u^{*} \in \mathcal{N}_{1}$.

This means, as with the Case Ⅰ, that we have

By taking into consideration the results (3.21), (3.22), (3.35), and (3.36), we infer that $u^{*} \in \mathcal{N}_{2}$ and (3.35) holds, in the weak sense, for both cases. Additionally, since $u^{*}\geq 0$ and $u^{*}\neq0$, by the strong maximum principle for weak solutions, we must have $u^{*}(x) > 0\, \, \, \text {almost everywhere in}\, \, \Omega.$

Next, we show that $u^{*} \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is a weak solution to problem (1.1).

For a random function $\phi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$, and $\varepsilon > 0$, let $\varphi = (u^{*}+\varepsilon \phi)^{+} = \max\{0, u^{*}+\varepsilon \phi\}$. We split $\Omega$ into two sets as follows:

and

If we replace $\varphi$ with $(u^{*}+\varepsilon \phi)$ in (3.35), it follows

Since $u^{*} \in \mathcal{N}_{2}$, we have

Dividing by $\varepsilon$ and passing to the limit as $\varepsilon \to 0$, and considering that $|\Omega_{ < }| \to 0$ as $\varepsilon \to 0$ gives

However, since the function $\phi\in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is chosen randomly, it follows that

which concludes that $u^{*} \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(\Omega)$ is a weak solution to problem (1.1).

4.   Example

Suppose that

Then equation (1.1) becomes

Theorem 4.1. Assume that the conditions $(A_{1})-(A_{3})$ hold. If $1 < \beta^{+} < 1+\frac{k+1}{\alpha}$ and $\alpha > 1/2$, then, problem (4.1) has at least one positive $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(B_1(0))$-solution.

Proof. Function $f(x) = \frac{(1-|x|)^{k}}{\beta(x)}\leq \frac{(1-|x|)^{k}}{\beta^{-}}$ is clearly non-negative and bounded above within the unit ball $B_1(0)$ since $|x| < 1$. Hence, $f(x) \in L^{1}(B_1(0))$.

Now, let's choose $\overline{u} = (1-|x|)^{\alpha}$. Since $\overline{u}$ is also non-negative and bounded within $B(0, 1)$, it is in $\overline{u} \in L^{P_{+}^{+}}(B(0, 1))$. Indeed,

Next, we show that $\partial_{x_{i}}\overline{u} \in L^{p_i(\cdot)}(B_1(0))$ for $i\in \left\{1, ..., N\right\}$. Fix $i\in \left\{1, ..., N\right\}$. Then

Considering that $x \in B_1(0)$, we obtain

Therefore,

if $\alpha > \frac{P^{-}_{-}-1}{P^{-}_{-}}$. Thus, $\partial_{x_{i}}\overline{u} \in L^{p_i(\cdot)}(B_1(0))$ for $i\in \left\{1, ..., N\right\}$, and as a result, $\overline{u} \in W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(B_1(0))$.

Finally, we show that $\int_{B(0, 1)}\frac{(1-|x|)^{k}(1-|x|)^{\alpha(1-\beta(x))}}{\beta(x)}dx < \infty$. Then,

Thus, by Theorem 3.8, problem (4.1) has at least one positive $W_{0}^{1, \overset{\rightarrow }{p}(\cdot)}(B_1(0))$-solution.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Athabasca University Research Incentive Account [140111 RIA].

Conflict of interest

The author declares there is no conflict of interest.