Dynamic study of the pathogen-immune system interaction with natural delaying effects and protein therapy

1.   Introduction

The study of solutions to superlinear problems driven by the double phase operator is a new and important topic, since it sheds light on a range of applications in the field of mathematical physics such as elasticity theory, strongly anisotropic materials, Lavrentiev's phenomenon, etc. (see ^[1,2,3]).

In the present paper, we study the existence and multiplicity of solutions for the following double–phase problems with mixed boundary conditions:

where $\Omega \subset \mathbb{R}^N (N \geq 2)$ is a bounded domain with Lipschitz boundary $\partial \Omega$, $\Lambda _1, \Lambda _2$ are disjoint open subsets of $\partial \Omega$ such that $\partial \Omega = \overline{\Lambda _1} \cup \overline{\Lambda _2}$ and $\Lambda _1 \neq \emptyset$, $1 < p(x) < q(x) < N$ for all $x \in \overline{\Omega}$, $b: \overline{\Omega} \mapsto[0, +\infty)$ is Lipschitz continuous, $\nu$ denotes the outer unit normal of $\Omega$ at the point $x \in \Lambda _2$, $f: \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ and $g: \Lambda _2 \times \mathbb{R} \rightarrow \mathbb{R}$ are Carathéodory functions, and $D$ is the double phase variable exponents operator given by

Note that the differential operator defined above in (1.2) is called the double phase operator with variable exponents, which is a natural generalization of the classical double phase operator when $p$ and $q$ are constant functions

From the physical point of view, while studying the behavior of strongly anisotropic materials, V.V. Zhikov ^[2] in 1986 discovered that their hardening properties changed radically point by point, what is known as the Lavrentiev phenomenon ^[3]. To describe this phenomenon, he initially introduced the functional

where the integrand changes its ellipticity and growth properties according to the point in the domain. In the framework of mathematics, the functional (1.3) has been investigated by many authors with respect to regularity and nonstandard growth. For instance, we refer to the papers of P. Baroni et al. ^[4,5], P. Baroni et al. ^[6], G. Cupini ^[7], and the references therein.

Multiple authors have recently concentrated on the study of double phase problems in the case when the exponents $p$ and $q$ are constants, and a plethora of results have been obtained; see, for example, W. Liu and G. Dai ^[8], M. El Ahmadi et al. ^[9], L. Gasiński and P. Winkert ^[10], N. Cui and H.R. Sun ^[11], Y. Yang et al. ^[12], and the references therein. For example, N. Cui and H.R. Sun ^[11] considered the following problem in the particular case: $p(x) = p$, $q(x) = q$, and $\lambda = 1$

where $D(u): = \mathop{\rm{div}} \left(\vert \nabla u \vert ^{p-2} \nabla u + b(x) \vert \nabla u \vert ^{q-2} \nabla u \right).$ The authors have proved the existence and multiplicity of nontrivial weak solutions for the above problem with superlinear nonlinearity. Their approach was based on critical point theory with Cerami condition.

Very recently, Y. Yang et al. ^[12] considered the problem (1.1) in the particular case of $p(x) = p$ and $q(x) = q$. Based on the maximum principle and homological local linking, they proved the existence of at least two bounded nontrivial weak solutions.

The main novelty of the current paper is the combination of the double phase variable exponents operator with mixed boundary conditions, that is, the Dirichlet condition on $\Lambda _1$ and the Steklov condition on $\Lambda _2$, which is different from ^[13]. To the best of our knowledge, there are only a few results related to the study of such problems.

To state our results, we make the subsequent hypotheses on $f$ and $g$:

$(H_0)$ There exist $C_1, C_2 > 0$, $s_1 \in C_+(\Omega)$, and $s_2 \in C_+(\Lambda _2)$ such that

(ⅰ) $\vert f(x, t) \vert \leq C_1 (1+\vert t \vert ^{s_1(x)-1}) \ \ \text{ for all } (x, t) \in \Omega \times \mathbb{R},$

(ⅱ) $\vert g(x, t) \vert \leq C_2 (1+\vert t \vert ^{s_2(x)-1}) \ \ \text{ for all } (x, t) \in \Lambda _2 \times \mathbb{R}$.

$(H_1)$ (ⅰ) $1 < p^+ \leq q^+ < s_1^- \leq s_1^+ < p^*(x) \quad \text{ for all } \ x \in \Omega$, $

(ⅱ) $1 < p^+ \leq q^+ < s_2^- \leq s_2^+ < p_*(x) \quad \text{ for all } \ x \in \Lambda _2$,

where

$(H_2)$ (ⅰ) $\underset{\vert t \vert \rightarrow \infty}{\liminf} \frac{F(x, t)}{\vert t \vert ^{q^+}} = +\infty$ uniformly a.e. $x \in \Omega$,

(ⅱ) $\underset{\vert t \vert \rightarrow \infty}{\liminf} \frac{G(x, t)}{\vert t \vert ^{q^+}} = +\infty$ uniformly a.e. $x \in \Lambda _2$,

where $F(x, t) = {\int}_0^t f(x, s)ds$ and $G(x, t) = {\int}_0^t g(x, s)ds$.

$(H_3)$ (ⅰ) There exist $c_1, r_1 \geq 0$ and $l_1 \in L^{\infty} (\Omega)$ with $l_1(x) > \frac{N}{p^{-}}$ such as

for all $(x, t) \in \Omega \times \mathbb{R}$, $\vert t \vert \geq r_1$ and $\mathcal{F}(x, t) : = \frac{1}{q^+} f(x, t)t-F(x, t) \geq 0.$

(ⅱ) There exist $c_2, r_2 \geq 0$ and $l_2 \in L^{\infty} (\Lambda _2)$ with $l_2(x) > \frac{N-1}{p^{-}-1}$ such as

for all $(x, t) \in \Lambda _2 \times \mathbb{R}$, $\vert t \vert \geq r_2$ and $\mathcal{G}(x, t) : = \frac{1}{q^+} g(x, t)t-G(x, t) \geq 0.$

$(H_4)$ (ⅰ) $f(x, t) = \circ (\vert t \vert ^{ p^+-1})$ as $t \rightarrow 0$ uniformly for a.e. $x \in \Omega$,

(ⅱ) $g(x, t) = \circ (\vert t \vert ^{ p^+-1})$ as $t \rightarrow 0$ uniformly for a.e. $x \in \Lambda _2$.

$(H_5)$ (ⅰ) $f(x, -t) = -f(x, t)$ for all $(x, t) \in \Omega \times \mathbb{R}$,

(ⅱ) $g(x, -t) = -g(x, t)$ for all $(x, t) \in \Lambda _2 \times \mathbb{R}$.

Let us consider $\phi : X_0 \rightarrow \mathbb{R}$ the Euler functional corresponding to problem (1.1), which is defined as follows:

where

and

with $X_0$ will be defined in preliminaries and $d \sigma$ is the measure on the boundary.

Then, it follows from the hypothesis $(H_0)$ that the functional $\phi \in C^1(X_0, \mathbb{R})$, and its Fréchet derivative is

for any $u, v \in X_0$. It is clear that any critical point of $\phi$ is a weak solution to the problem (1.1).

Now, we present the main results of this paper.

Theorem 1. Suppose that $(H_0), (H_1), (H_2), (H_3)$ and $(H_4)$ hold. Then problem (1.1) has at least one nontrivial weak solution.

Theorem 2. Suppose that $(H_0), (H_1), (H_2), (H_3)$ and $(H_5)$ hold. Then problem (1.1) possesses a sequence of weak solutions $(u_n)$ such that $\phi (u_n) \rightarrow + \infty$ as $n \rightarrow + \infty$.

2.   Preliminaries

To study double phase problems, we need some definitions and basic properties of $W^{1, \mathcal{D}} (\Omega)$, which are called Musielak–Orlicz–Sobolev spaces. For more details, see ^{[14,15,16,17,18,19]} and references therein.

First, we recall the definition of variable exponent Lebesgue space. For $p \in C_+(\overline{\Omega}) : = \lbrace p \in C (\overline{\Omega}) : p^- : = \underset{x \in \overline{\Omega}}{\inf} p(x) > 1 \rbrace$, we designate the variable exponent Lebesgue space by

equipped with the Luxemburg norm

Proposition 1. ^[20]

1. The Sobolev space $(L^{p(x)} (\Omega), \vert. \vert _{p(x)})$ is defined as the dual space $L^{q(x)} (\Omega)$, where $q(x)$ is conjugate to $p(x)$, i.e., $\frac{1}{p(x)}+ \frac{1}{q(x)} = 1.$ For any $u \in L^{p(x)} (\Omega)$ and $v \in L^{q(x)} (\Omega)$, we have

2. If $p_1, p_2 \in C_+(\overline{\Omega})$, $p_1(x) \leq p_2(x)$, for all $x \in \overline{\Omega}$, then $L^{p _2 (x)}(\Omega) \hookrightarrow L^{p _1 (x)}(\Omega)$ and the embedding is continuous.

Let $p\in C_+(\partial \Omega) : = \lbrace p \in C (\partial \Omega) : p^- : = \underset{x \in \partial \Omega}{\inf} p(x) > 1 \rbrace$ and denote by $d \sigma$ the Lebesgue measure on the boundary. We define

with the norm

Now, we give the main properties of the Musielak–Orlicz–Sobolev functional space that we will use in the rest of this paper. Denote by $N(\Omega)$ the set of all generalized $N$-functions. Let us denote by

the function defined as

where the weight function $b(.)$ and the variable exponents $p(.), q(.) \in C_+ (\overline{\Omega})$ satisfies the following hypothesis:

Note that the role of assuming the inequality $\frac{Nq(x)}{N+q(x)-1} < p(x)$ is to ensure that $q(x) < p_*(x)$ and $q(x) < p^*(x)$ for all $x \in \overline{\Omega}$, where $p^*(x) = \frac{Np(x)}{N-p(x)}$ and $p_*(x) = \frac{(N-1)p(x)}{N-p(x)}$.

It is clear that $\mathcal{D}$ is a generalized $N$-function, locally integrable, and

which is called condition $(\Delta _2)$.

We designate the Musielak–Orlicz space by

equipped with the so-called Luxemburg norm

The Musielak–Orlicz–Sobolev space $W^{1, \mathcal{D}}(\Omega)$ is defined as

endowed with the norm

With such norms, $L^{\mathcal{D}} (\Omega)$ and $W^{1, \mathcal{D}}(\Omega)$ are separable, uniformly convex, and reflexive Banach spaces.

On $L^{\mathcal{D}} (\Omega)$, we consider the function $\rho : L^{\mathcal{D}} (\Omega) \rightarrow \mathbb{R}$ defined by

The relationship between $\rho$ and $\vert. \vert _{\mathcal{D}}$ is established by the next result.

Proposition 2. (see ^[16]) For $u \in L^{\mathcal{D}} (\Omega)$, $(u_n) \subset L^{\mathcal{D}} (\Omega)$, and $\mu > 0$, we have

1. For $u \neq 0$, $\vert u \vert _{\mathcal{D}} = \mu \Longleftrightarrow \rho (\frac{u}{\mu}) = 1;$

2. $\vert u \vert _{\mathcal{D}} < 1 \left( = 1, > 1 \right) \Longleftrightarrow \rho (u) < 1 \left( = 1, > 1 \right);$

3. $\vert u \vert _{\mathcal{D}} > 1 \Longrightarrow \vert u \vert _{\mathcal{D}} ^{p^-} \leq \rho (u) \leq \vert u \vert _{\mathcal{D}} ^{q^+};$

4. $\vert u \vert _{\mathcal{D}} < 1 \Longrightarrow \vert u \vert _{\mathcal{D}} ^{q^+} \leq \rho (u) \leq \vert u \vert _{\mathcal{D}} ^{p^-};$

5. $\underset{n \rightarrow + \infty}{\lim} \vert u_n \vert _{\mathcal{D}} = 0 \Leftrightarrow \underset{n \rightarrow + \infty}{\lim} \rho (u_n) = 0$ and $\underset{n \rightarrow + \infty}{\lim} \vert u_n \vert _{\mathcal{D}} = + \infty \Leftrightarrow \underset{n \rightarrow + \infty}{\lim} \rho (u_n) = + \infty$.

On $W^{1, \mathcal{D}}(\Omega)$, we introduce the equivalent norm by

Similar to Proposition (2), we have

Proposition 3. (see ^[16]) Let

For $u \in W^{1, \mathcal{D}}(\Omega)$, $(u_n) \subset W^{1, \mathcal{D}}(\Omega)$, and $\mu > 0$, we have

1. For $u \neq 0$, $\Vert u \Vert = \mu \Longleftrightarrow \hat{\rho} (\frac{u}{\mu}) = 1;$

2. $\Vert u \Vert < 1 \left( = 1, > 1 \right) \Longleftrightarrow \hat{\rho} (u) < 1 \left( = 1, > 1 \right);$

3. $\Vert u \Vert > 1 \Longrightarrow \Vert u \Vert ^{p^-} \leq \hat{\rho} (u) \leq \Vert u \Vert ^{q^+};$

4. $\Vert u \Vert < 1 \Longrightarrow \Vert u \Vert ^{q^+} \leq \hat{\rho} (u) \leq \Vert u \Vert ^{p^-};$

5. $\underset{n \rightarrow + \infty}{\lim} \Vert u_n \Vert = 0 \Leftrightarrow \underset{n \rightarrow + \infty}{\lim} \hat{\rho} (u_n) = 0$ and $\underset{n \rightarrow + \infty}{\lim} \Vert u_n \Vert = + \infty \Leftrightarrow \underset{n \rightarrow + \infty}{\lim} \hat{\rho} (u_n) = + \infty$.

We recall that problem (1.1) has a mixed boundary condition. For this, our Banach space workspace is given by

endowed with the equivalent norm (2.2). Obviously, since $X_0$ is a closed subspace of $W^{1, \mathcal{D}} (\Omega)$, then $(X_0, \Vert. \Vert)$ is a reflexive Banach space.

Proposition 4. (see ^[16]) Let hypothesis (2.1) be satisfied. Then the following embeddings hold:

1. There is a continuous embedding $L^{\mathcal{D}} (\Omega) \hookrightarrow L^{r(x)}(\Omega)$ for $r \in C(\overline{\Omega})$ with $1\leq r(x) \leq p(x)$ for all $x \in \overline{\Omega}$.

2. There is a compact embedding $W^{1, \mathcal{D}}(\Omega) \hookrightarrow L^{r(x)}(\Omega)$ for $r \in C(\overline{\Omega})$ with $1 \leq r(x) < p^{*}(x)$ for all $x \in \overline{\Omega}$.

3. If $p \in C_{+}(\overline{\Omega}) \cap W^{1, \gamma}(\Omega) \mathit{\text{for some}} \gamma \geq N.$ Then, there is a continuous embedding $W^{1, \mathcal{D}}(\Omega) \hookrightarrow L^{r(x)}(\partial \Omega)$ for $r \in C(\partial \Omega)$ with $1 \leq r(x) \leq p_{*}(x)$ for all $x \in \partial \Omega$.

4. There is a compact embedding $W^{1, \mathcal{D}}(\Omega) \hookrightarrow L^{r(x)}(\partial \Omega)$ for $r \in C(\partial\Omega)$ with $1 \leq r(x) < p_{*}(x)$ for all $x \in \partial \Omega$.

It is important to note that when we replace $W^{1, \mathcal{D}}(\Omega)$ by $X_0$ in Proposition 4, the embeddings 2 and 4 remain valid.

Let $A : W^{1, \mathcal{D}}(\Omega) \rightarrow \left(W^{1, \mathcal{D}}(\Omega) \right) ^*$ be defined by

for all $u, v \in W^{1, \mathcal{D}}(\Omega)$, where $\left(W^{1, \mathcal{D}}(\Omega) \right) ^*$ denotes the dual space of $W^{1, \mathcal{D}}(\Omega)$ and $\langle., .\rangle$ stands for the duality pairing between $W^{1, \mathcal{D}}(\Omega)$ and $\left(W^{1, \mathcal{D}}(\Omega) \right) ^*$.

Proposition 5. (see [16,Proposition 3.4]) Let hypothesis (2.1) be satisfied.

1. $A : W^{1, \mathcal{D}}(\Omega) \rightarrow \left(W^{1, \mathcal{D}}(\Omega) \right) ^*$ is a continuous, bounded, and strictly monotone operator.

2. $A : W^{1, \mathcal{D}}(\Omega) \rightarrow \left(W^{1, \mathcal{D}}(\Omega) \right) ^*$ satisfies the $(S_+)$-property, i.e., if $u_n \rightharpoonup u$ in $W^{1, \mathcal{D}}(\Omega)$ and $\overline{\underset{n \rightarrow + \infty }{\lim}} \langle A(u_n)-A(u), u_n-u \rangle \leq 0$, then $u_n \rightarrow u$ in $W^{1, \mathcal{D}}(\Omega)$.

Definition 1. Let $u \in X_0$. We say that $u$ is a weak solution to the problem (1.1) if

for all $v \in X_0$.

Now, we give the definition of the Cerami condition that was first introduced by G. Cerami in ^[21].

Definition 2. Let $(X, \Vert. \Vert)$ be a real Banach space and $\phi \in C^1(X, \mathbb{R})$. We say that $\phi$ satisfies the Cerami condition (we denote $(C)-$condition) in $X$, if any sequence $(u_n) \subset X$ such that $(\phi (u_n))$ is bounded and $\Vert \phi '(u_n) \Vert (1+ \Vert u_n \Vert) \rightarrow 0$ as $n \rightarrow + \infty$ has a strong convergent subsequence in $X$.

Remark 1. 1. It is clear from the above definition that if $\phi$ satisfies the $(PS)$-condition, then it satisfies the $(C)$-condition. However, there are functionals that satisfy the $(C)$-condition but do not satisfy the $(PS)$-condition (see ^[21]). Consequently, the $(PS)$-condition implies the $(C)$-condition.

2. The $(C)$-condition and the $(PS)$-condition are equivalent if $\phi$ is bounded below (see ^[22]).

Next, we present the following theorems, which will play a fundamental role in the proof of the main theorems.

Theorem 3. (see ^[23]) Let $(X, \Vert. \Vert)$ be a real Banach space; $\phi \in C^1 (X, \mathbb{R})$ satisfies the $(C)$-condition; $\phi(0) = 0$, and the following conditions hold:

1. There exist positive constants $\rho$ and $\alpha$ such that $\phi(u) \geq \alpha$ for any $u \in X$ with $\Vert u \Vert = \rho$.

2. There exists a function $e \in X$ such that $\Vert e \Vert > \rho$ and $\phi(e) \leq 0.$

Then, the functional $\phi$ has a critical value $c \geq \alpha$, that is, there exists $u \in X$ such that $\phi(u) = c$ and $\phi'(u) = 0$ in $X^*$.

Let $X$ be a real, reflexive, and separable Banach space. Then there exist $\lbrace e_j \rbrace _{j \in \mathbb{N}} \subset X$ and $\lbrace e_j ^* \rbrace _{j \in \mathbb{N}} \subset X^*$ such that

and $\langle e_i^*, e_j \rangle = 1$ if $i = j$, $\langle e_i^*, e_j \rangle = 0$ if $i \neq j$.

We denote $X_j = span \lbrace e_j \rbrace$, $Y_k = \bigoplus _{j = 1}^k X_j$, and $Z_k = \overline{\bigoplus _{j = k}^{+ \infty } X_j}$.

Theorem 4. (see ^[24]) Assume that $X$ is a Banach space, and let $\phi : X \rightarrow \mathbb{R}$ be an even functional of class $C^1 (X, \mathbb{R})$ that satisfies the $(C)-$condition. For every $k \in \mathbb{N}$, there exists $\gamma _k > \eta _k > 0$ such that

$(A_1)$ $b_k : = \inf \lbrace \phi(u): u \in Z_k, \Vert u \Vert = \eta _k \rbrace \rightarrow + \infty$ as $k \rightarrow +\infty$;

$(A_2)$ $c_k : = \max \lbrace \phi (u): u \in Y_k, \Vert u \Vert = \gamma _k \rbrace \leq 0$.

Then, $\phi$ has a sequence of critical values tending to $+ \infty.$

3.   Proofs of Theorems 1-2

First of all, we are going to show that the functional $\phi$ fulfills the $(C)$-condition.

Lemma 3.1. If assumptions $(H_0), (H_1), (H_2)$ and $(H_3)$ hold, then the functional $\phi$ satisfies the $(C)$-condition.

Proof. Let $(u_n) \subset X_0$ be a Cerami sequence for $\phi$, namely,

which implies that

where $\underset{n \rightarrow + \infty }{\lim} \circ _n (1) = 0$ and $M > 0$.

We need to prove the boundedness of the sequence $(u_n)$ in $X_0$. To this end, assume to the contrary, that the sequence $(u_n)$ is unbounded in $X_0$. Without loss of generality, we can assume that $\Vert u_n \Vert > 1$. By virtue of $(H_3)$, for $n$ large enough, we have

where $\mathcal{F}(x, u_n) : = \frac{1}{ q^+} f(x, u_n)u_n-F(x, u_n)\geq 0$ and $\mathcal{G} (x, u_n) = \frac{1}{ q^+} g(x, u_n)u_n-G(x, u_n) \geq 0$.

Then, we obtain

which implies

and

On the other hand, by Proposition 3, we have

Because $\Vert u_n \Vert > 1$, we can obtain

Since $\Vert u_n \Vert \rightarrow + \infty$ as $n \rightarrow + \infty$, we deduce that

Furthermore, using Proposition 3, we have

Then, we obtain

In view of condition $(H_2)$, there exist $T_1, T_2 > 0$ such that

Since $F(x, .)$ and $G(x, .)$ are continuous functions on $\left[ - T_1, T_1 \right]$ and $\left[- T_2, T_2 \right]$, respectively, there exist $C_0, C_0^* > 0$ such that

Then, there exist two real numbers $K$ and $K'$, such that

Hence,

for all $(x, n)\in \overline{\Omega} \times \mathbb{N}$.

Put $\beta _n = \frac{u_n}{\Vert u_n \Vert }$, so $\Vert \beta _n \Vert = 1$. Up to subsequences, for some $\beta \in X_0$, we have

for $s(x) < p^*(x)$ and $r(x) < p_*(x)$.

Define the sets $\Omega _0 = \lbrace x \in \Omega: \beta (x) \neq 0 \rbrace$ and $\Gamma = \lbrace x \in \Lambda_2 : \beta (x) \neq 0 \rbrace$.

Obviously, since $\Vert u_n \Vert \rightarrow + \infty$ as $n \rightarrow + \infty$, we have

for any $x \in \Omega _0 \cup \Gamma$.

Therefore, due to $(H_2)$, for all $x \in \Omega _0 \cup \Gamma$, we deduce

Thus, $\vert \Omega _0 \vert = 0$ and $\vert \Gamma \vert = 0$. In fact, suppose by contradiction that $\vert \Omega _0 \vert \neq 0$ or $\vert \Gamma \vert \neq 0$. Using (3.6), (3.7), (3.10), and Fatou's lemma, we get

which is a contradiction. Therefore, $\beta (x) = 0$ for a.e. $x \in \Omega$ and for a.e. $x \in \Lambda _2$.

From (3.5) and (3.9), respectively, we can deduce that

for $s(x) < p^*(x)$, $r(x) < p_*(x)$, and

Using $(H_0)$ and $(H_1)$, we obtain

where $C_3, C_4 > 0$, $s$ is either $s_1^+$ or $s_1^-$ and $r_1$ comes from $(H_3)$.

Put $l_1'(x) = \frac{l(x)}{l(x)-1}$. Since $l_1 \in L^{\infty} (\Omega)$ with $l_1(x) > \frac{N}{p^-}$, it follows that $l_1'(x) p^- < p^* (x).$

On the other hand, by virtue of hypothesis $(H_3)(i)$, (3.3), and (3.11), we deduce

Combining this with (3.13), we obtain

Similarly, let $l_2'(x) = \frac{l_2(x)}{l_2(x)-1}$. Since $l_2 \in L^{\infty} (\Omega)$ with $l_2(x) > \frac{N-1}{p^{-}-1}$, it follows that $l_2'(x) p^- < p_* (x).$ Then, by $(H_3)(ii)$, (3.4), and (3.11), we can prove in a similar way that

Consequently, combining (3.14) with (3.15), we obtain

which is a contradiction to (3.12). Thus, $(u_n)$ is bounded in $X_0$.

Finally, we need to prove that any $(C)$-sequence has a convergent subsequence. Let $(u_n) \subset X_0$ be a $(C)$-sequence. Then, $(u_n)$ is bounded in $X_0$. Passing to the limit, if necessary, to a subsequence, from Proposition 4, we have

for $1 \leq s_1(x) < p^*(x)$ and $1 \leq s_2(x) < p_*(x)$. It is easy to check from $(H_0)$, (3.16) and Hölder's inequality that

and

where $\frac{1}{s_1(x)}+\frac{1}{s_1'(x)} = 1$ and $\frac{1}{s_2(x)}+\frac{1}{s_2'(x)} = 1$.

Next, since $u _n \rightharpoonup u$, from (3.1), we have

Then

where $A$ is given in Proposition 5.

Finally, the combination of (3.17), (3.18), and (3.19) implies

Since the operator $A$ satisfies the $(S_+)$ property in view of Proposition 5, we can obtain that $u_n \rightarrow u$ in $X_0$. The proof is complete. □

Proof of Theorem 1

Let us check that the functional $\phi$ satisfies the geometric conditions of the mountain pass in Theorem 3. By Lemma 3.1, $\phi$ satisfies the $(C)-$condition. According to the definition of $\phi$, we have $\phi (0) = 0$. Then, to apply Theorem 3, it remains to prove that

(ⅰ) There exist positive constants $\rho$ and $\alpha$ such that $\phi(u) \geq \alpha$ for any $u \in X_0$ with $\Vert u \Vert = \rho$.

(ⅱ) There exists a function $e \in X_0$ such that $\Vert e \Vert > \rho$ and $\phi(e) \leq 0.$

For $(i)$, let $\Vert u \Vert < 1$. Then, by Proposition 3, we have

Using $(H_0)$ and $(H_4)$, for $\varepsilon > 0$ be small enough, there exist $C_1(\varepsilon), C_2(\varepsilon) > 0$ such that

Since $p^+ < s_1(x) < p^*(x)$ and $p^+ < s_2(x) < p_*(x)$ for all $x \in \overline{\Omega}$ and for all $x \in \overline{\Lambda _2}$ in view of condition $(H_1)$, we have from Proposition 4 that

So, there exist $c_i > 0(i = 3, ...6)$ such that

Therefore, by (3.20) and (3.21), for $\Vert u \Vert < 1$ sufficiently small, we obtain

Since $s_1^- > p^+$ in view of condition $(H_1)$ and $\Vert u \Vert < 1$, then $\Vert u \Vert ^{s_1^-} < \Vert u \Vert ^{ p^+}$. Thus, we obtain

Since $s_2^- > q^+ \geq p^+$, then by the standard argument, there exist positive constants $\rho$ and $\alpha$ such that $\phi(u) \geq \alpha$ for any $u \in X_0$ with $\Vert u \Vert = \rho$.

Next, we affirm that there exists $e \in X_0$ with $\Vert u \Vert > \rho$ such that

In fact, from $(H_2)$, it follows that for every $k > 0$, there exist constants $T_k$ and $T_k^*$ such that

Since $F(x, .)$ and $G(x, .)$ are continuous functions on $\left[ - T_k, T_k \right]$ and $\left[- T_k^*, T_k^* \right]$, respectively, there exist constants $C_0, C_0^* > 0$ such that

Thus,

Let $w \in X_0 \setminus \lbrace 0 \rbrace$ such that $\Vert w \Vert = 1$ and $l > 1$ be large enough. Using the above inequality, we obtain

As

for $k$ large enough, we deduce

Thus, there exist $t_0 > 1$ and $e = t_0 w \in X_0 \setminus \overline{B_{\rho} (0)}$ such that $\phi(e) < 0$.

Proof of Theorem 2

To prove Theorem 2, we need the following auxiliary lemmas:

Lemma 3.2. (see ^[25,26]) For $s \in C_+(\overline{\Omega})$ and $r \in C_+(\overline{\Lambda _2})$ such that $s(x) < p^*(x)$ for all $x \in \overline{\Omega}$ and $r(x) < p_*(x)$ for all $x \in \overline{\Lambda _2}$. Let

Then, $\underset{k \rightarrow + \infty}{\lim} \delta _k = \underset{k \rightarrow + \infty}{\lim} \delta_k' = 0$.

Lemma 3.3. (see ^[27]) For all $s \in C_+(\overline{\Omega})$ ($r \in C_+(\overline{\Lambda _2})$) and $u \in L^{s(x)}(\Omega)$ ($v \in L^{r(x)}(\Lambda _2)$), there exists $y \in \Omega$ ($z \in \Lambda _2$) such that

Now, we return to the proof of Theorem 2. To this end, based on Fountain Theorem 4, we will show that the problem (1.1) possesses infinitely many weak solutions with unbounded energy. Evidently, according to $(H_5)$, $\phi$ is an even functional. By Lemma 3.1, we know that $\phi$ satisfies the $(C)$-condition. Then, to prove Theorem 2, it only remains to verify the following assertions:

$(A_1)$ $b_k : = \inf \lbrace \phi (u): u \in Z_k, \Vert u \Vert = \eta _k \rbrace \rightarrow + \infty$ as $k \rightarrow +\infty$,

$(A_2)$ $c_k : = \max \lbrace \phi (u): u \in Y_k, \Vert u \Vert = \gamma _k \rbrace \leq 0$.

$(A_1)$ For any $u \in Z_k$ such that $\Vert u \Vert = \eta _k > 1$. It follows from $(H_0)$, Proposition 3, and Lemma 3.3 that

where $c_3 = \max \lbrace C_1c_1, C_2c_2 \rbrace$.

Then, it follows that

Let us consider the following equations:

and

Let $a_k$ and $d_k$ be the two non-zero solutions of (3.26) and (3.27), respectively. Then, we obtain

We fix $\eta _k$ as follows

Then, by Lemma 3.2, (3.25) and $s_1^+, s_2^+ > q^+ > p^-$, we obtain

where $C_6 > 0$. Hence, $(A_1)$ holds.

$(A_2)$ In view of Proposition 3 and (3.23), for $u\in Y_k$ with $\Vert u \vert > 1$, we have

Since $dim Y_k < \infty$, then all norms are equivalent in $Y_k$. Therefore, as $\frac{1}{p^-} < 1$, for $k$ large enough, we obtain

Finally, the assertion $(A_2)$ is also valid.

This completes the proof.

Author contributions

Mahmoud El Ahmadi: Writing-original draft, Writing-review & editing; Mohammed Barghouthe: Formal Analysis, Methodology; Anass Lamaizi: Formal Analysis; Mohammed Berrajaa: Supervision, Validation.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which have improved the quality of this paper.

Conflict of interest

The authors declare there is no conflict of interest.