Existence of multiple solutions for a singular $p(\cdot)$-biharmonic problem with variable exponents

1.   Introduction

The biharmonic problem with variable exponents usually arises in the context of partial differential equations where the biharmonic operator is applied to a function, but the coefficients or exponents of the problem are not constant. This type of problem has important applications in several fields, such as mathematical physics, engineering, and differential geometry, especially in modeling phenomena in which the underlying material or medium has non-uniform properties. In particular, the biharmonic equation describes the deflection of thin plates subjected to forces ^[1]. Su et al. ^[2] established the solution of the thin film epitaxy equation, and the biharmonic equation can include variable exponents to account for this heterogeneity. This is particularly useful for modeling composite materials where the material properties change gradually with position. For example, a composite plate may have different stiffnesses in different regions, and the biharmonic problem with variable exponents can model the deformation of the plate under external loads. Also, the biharmonic equation appears in the study of flow around objects ^[3,4,5]. When the viscosity of the fluid varies in space, the biharmonic operator with variable exponents can be used to model the diffusion of momentum in such media. This would allow a more realistic description of flow around objects with variable surface characteristics. The biharmonic equation was used for smoothing, denoising, and segmentation tasks ^[1]. When applied to images with nonuniform texture or intensity, a biharmonic operator with variable exponents can better model local image properties. For example, areas of an image with low contrast may have different smoothing behavior than areas with high contrast, leading to a variable exponent formulation in partial differential equations. A biharmonic operator with variable exponents can also be applied in the analysis of electromagnetic wave propagation in media with varying permeability, which is particularly useful in the design of optical and microwave devices with spatially graded materials.

In this paper, we fix a bounded domain $\Omega\subset \mathbb{R}^{N}$ with $N\geq 3$ and consider the following system

where $\delta(\tau)$ is the distance between $\tau$ and the boundary of $\Omega$. The operator $\Delta^{2}_{p(.)}$ is defined by:

The functions $a$, $b$, and $\theta$ are non-negative and satisfy suitable hypotheses that are fixed later, and the functions $p$ and $r$ are continuous on $\overline{\Omega}$, such that for any $\tau \in \overline{\Omega}$, we have

For a given function $Z$, and for any $\tau$ in $\Omega$, we denote

Hereafter, we assume that

and

where

We recall that for each $\varphi\in W_{0}^{2, p(\tau)}(\Omega)$, we have

The last inequality was introduced in the reference ^[6] for the case $N = 1$, and in the reference ^[7] for the case $N \geq 2$.

Recently, problems related to biharmonic operators have been extensively studied by several authors, and by using several methods, we cite for instance the papers of Su and Chen ^[8], who used a combination of the bipolar Rellich inequality with Gigliardo-Nirenberg inequality and Ekeland variational principle; Su and Shi ^[9], who used a combination of the non-vanishing and the structure of a Palais-Smale sequence to prove ground state solutions to prove the existence of nontrivial solutions; Su and Feng ^[10], who used the generalized versions of Lions-type theorem to prove ground state solutions; Alsedi ^[11], who used a fountain theorem to prove the existence of infinitely many solutions; Dhifli and Alsedi ^[12], who used the Nehari manifold to prove a multiplicity result; Ji and Wang ^[13], who used the Nehari manifold to prove the existence of two nontrivial solutions.

Very recently, singular biharmonic equations have attracted the attention of several authors, we cite for example the works of Ghanmi and Sahbani ^[14], who used the mountain pass and the symmetric mountain pass theorems to prove their results; Alsaedi et al. ^[15], who used the Nehari manifold to prove some multiplicity results; Drissi et al.^[16], who used the Nehari manifold to study biharmonic problems with Hardy nonlinearities, and Rǎdulescu and Repovš ^[17], who used variational and topological arguments to study singular biharmonic problem including the capillarity equation and the mean curvature problem. In particular, Ghanmi and Sahbani ^[14] considered the following singular problem:

where $a$, $b$, and $\theta$ are three non-negative functions, $\theta \in (0, 1)$. Using the condition $1 < r(\tau) < p(\tau)$, for all $\tau \in \overline{\Omega}$, the authors proved that the functional associated with the problem (1.2), is coercive and bounded below in its domain, and based on the min-max theorem, they proved that, for all $\lambda\in (0, C_{H})$, the problem (1.2) admits a nontrivial weak solution. For interested readers, we cite ^[18,19,20] for other interesting related works.

Motivated by the above-mentioned results, our aim in this paper is to investigate a more general problem. More precisely, problem $P_{\lambda, \mu}$ contains two types of singularities (i.e., a power singularity and a singularity of Hardy type), and this makes our study more difficult. Moreover, the associated functional energy is not of class $C^1$, so the direct variational method cannot be applied. In particular, the energy functional does not satisfy the mountain pass geometry, which implies also that the Ekland's variational principle cannot be applied. On the other hand, the exponent $r$ is assumed to satisfy $r > p$, and this implies that the functional energy is not coercive in its domain. To guarantee the coercivity of the associated functional energy, we will work in some subsets called Nehari manifold sets. By these sets, we prove the multiplicity of solutions in the space $X = W_{0}^{2, p(\tau)}(\Omega)$. For this aim, we assume the following hypotheses:

$(H_{1})$ $\bullet$ $a:\Omega \to \mathbb{R},$ is such that

with

$\bullet$ There exist $b_{1}$, $b_{2} > 0$, such that

$(H_{2})$

Remark 1.1. If $a$ is a continuous function on $\overline{\Omega}$, then, the function $a$ satisfies the hypothesis $(H_{1})$ for any

with $a_1 > a_2 > a_3 > 1.$ On the other hand, there are several functions satisfying hypothesis $(H_{2})$. So, we can construct a class of functions satisfying hypotheses $(H_{1})$ and $(H_{2})$.

Definition 1.1. A function $\ \varphi$ in the space $X$ is a weak solution of the problem $(P_{\lambda, \mu})$ if, for each function $v$ in $X$, we have

Our main result of this paper is the following.

Theorem 1.1. Assume assumptions $(H_{1})$ and $(H_{2})$. Then, for each $\lambda\in (0, C_{H})$, the problem $(P_{\lambda, \mu})$ admits two non-trivial solutions, provided that $\mu \in (0, \mu^{*})$ for some positive constant $\mu^{*}$.

In Section 2 we introduce some results on functional spaces. In Section 3, we present and prove the main result of this work.

2.   Preliminaries

In this section, we recall some preliminaries on the Lebesgue and Sobolev spaces. For interested readers, we refer to the works ^[21,22,23].

The sets $C_{+}(\Omega)$ will denote the set of all functions $\mu$ that are continuous on $\Omega$, and satisfy

For each $\mu \in C_{+}(\Omega)$, we define the space $L^{\mu(\tau)}(\Omega)$ by:

We equip the space $L^{\mu(\tau)}(\Omega)$ with the following norm,

Equipped with the last norm, the space $L^{\mu(\tau)}(\Omega)$ becomes a Banach space. Moreover, it is separable and reflexive if and only if $\mu$ satisfies

Proposition 2.1. ^[24,25] For any $\varphi\in L^{\mu(\tau)}(\Omega)$ and $v\in L^{\mu^{'}(\tau)}(\Omega)$, where $\frac{1}{\mu(\tau)}+ \frac{1}{\mu^{'}(\tau)} = 1,$ we have,

Let

the modular on the space $L^{\mu(\tau)}(\Omega).$

Proposition 2.2. ^[22,24] For all $\varphi\in L^{\mu(\tau)}(\Omega)$, we have,

(1) $|\varphi|_{\mu(\tau)} < 1 (resp = 1, > 1)\Leftrightarrow \rho_{\mu(\tau)}(\varphi) < 1 (resp = 1, > 1)$;

(2) $|\varphi|_{\mu(\tau)} > 1\Rightarrow |\varphi|_{\mu(\tau)}^{\mu^{-}}\leq \rho_{\mu(\tau)}(\varphi), \leq |\varphi|_{\mu(\tau)}^{\mu^{+}}$;

(3) $|\varphi|_{\mu(\tau)} < 1\Rightarrow |\varphi|_{\mu(\tau)}^{\mu^{+}}\leq \rho_{\mu(\tau)}(\varphi)\leq |\varphi|_{\mu(\tau)}^{\mu^{-}}$.

Also, we get the following proposition.

Proposition 2.3. ^[22] Let $\mu$ and q be measurable functions such that $q\in L^{\infty}(\Omega)$ and $1\leq \mu(\tau).q(\tau)\leq\infty$ for all $\tau \in \Omega$.

Let $\varphi\in L^{\mu(\tau)}(\Omega), \varphi\neq 0$. Then,

(1) $|\varphi|_{\mu(\tau)q(\tau)}\leq1\Rightarrow |\varphi|_{\mu(\tau)q(\tau)}^{\mu^{+}}\leq ||\varphi|^{\mu(\tau)}|_{q(\tau)}\leq |\varphi|_{\mu(\tau)q(\tau)}^{\mu^{-}},$

(2)$|\varphi|_{\mu(\tau)q(\tau)}\geq 1\Rightarrow |\varphi|_{\mu(\tau)q(\tau)}^{\mu^{-}}\leq ||\varphi|^{\mu(\tau)}|_{q(\tau)}\leq |\varphi|_{\mu(\tau)q(\tau)}^{\mu^{+}}$.

Let us define the space

equipped with the norm

$W^{2, \mu(\tau)}(\Omega)$ is a separable and reflexive Banach space (see ^[21,23]).

Let $W_{0}^{2, \mu(\tau)}(\Omega)$ be the closure of $C_{0}^{\infty}(\Omega)$ in $W^{2, \mu(\tau)}(\Omega)$. Then, $W_{0}^{2, \mu(\tau)}(\Omega)$ is a Banach and reflexive space with the norm

Theorem 2.1. ^[26] If $q\in C_{+}(\overline{\Omega})$ with $q(\tau) < \mu^{*}(\tau)$, for any $\tau\in \overline{\Omega}$, then the embedding from $W^{2, \mu(\tau)}(\Omega)$ into $L^{q(\tau)}(\Omega)$ is compact and continuous.

Let

Then, we have the following result.

Proposition 2.4. ^[21,23]

(1) If $\Theta(\varphi)\geq 1$, then $||\varphi||^{\mu^{-}}\leq \Theta(\varphi)\leq ||\varphi||^{\mu^{+}},$

(2) If $\Theta(\varphi)\leq 1$, then $||\varphi||^{\mu^{+}}\leq \Theta(\varphi)\leq ||\varphi||^{\mu^{-}},$

(3) $\Theta(\varphi)\geq 1( = 1, \leq 1)\Leftrightarrow ||\varphi||\geq 1( = 1, \leq 1)$.

3.   Main results

In this section, we prove the main result of this paper. We begin by defining the functional $\Phi_{\lambda, \mu}: X \rightarrow \mathbb{R},$ associated with the problem $(P_{\lambda, \mu})$, as follows:

We note that, due to the singular term, the functional $\ \Phi_{\lambda, \mu}$ is not of class $C^{1}$. Moreover, from the fact that $0 < 1-\theta(\tau) < p(\tau) < r(\tau)$, we see that $\ \Phi_{\lambda, \mu}$ is not bounded below in $X$. So, we cannot use the direct variational method to prove the existence of solutions. Throughout this paper, we consider for $\varphi\in\; X$, the function $J_{\varphi}: [0, \infty[ \rightarrow \mathbb{R},$ defined by

We note that, by the fact that $0 < 1-\theta(\tau) < p(\tau) < r(\tau)$, the functional $\ \Phi_{\lambda, \mu}$ is not bounded below in $X$, so we will prove that $\Phi_{\lambda, \mu}$ is bounded on the Nehari manifold $\aleph$ which is defined as:

Now, we split $\aleph$ into the following parts:

It is clear that:

and

Moreover, we have

where

and

Next, we will prove the following lemma.

Lemma 3.1. Assume assumption $(H_{1})$ holds. Let $\varphi\in X$, for $\lambda\in(0, C_{H})$. Then, we have the following:

(1) There exists a constant $C^{'}_{H} = \dfrac{p^{-}}{p^{+}}(1-\frac{\lambda}{C_{H}}) > 0$ such that,

(2) There exists a constant $C_{1} > 0$ such that,

(3) There exists a constant $C_{2} > (C^{'}_{H})^{2}$ such that,

Proof. (1) Let $\varphi\in X$ and $0 < \lambda < C_{H}$. By (1.3), we have

This implies that

Moreover, we have

From (3.2) and (3.3), we get

So,

Then, by Proposition 2.4, we deduce the assertion (1).

(2) By $(H_{1})$ and Proposition 2.1, we get

Using, $1 < t(\tau) < p^{*}(\tau)$, Proposition 2.2, and Theorem 2.1, we have

(3) By $(H_{1})$, Proposition 2.2, and Theorem 2.1, we get

where, $C_{2} = 2C+(C^{'}_{H})^{2}.$

□

Lemma 3.2. Assume that assumption $(H_{1})$ holds. For $\lambda\in(0, C_{H})$, $\Phi_{\lambda, \mu}$ is coercive and bounded below on $\aleph$.

Proof. Let $\varphi\in N$, for $||\varphi|| > 1$, from (3.1) and Lemma 3.1, we have

Since $0 < 1-\theta^{+} < 1-\theta^{-} < p^{-} < p^{+} < r^{-} < r^{+}$, we get,

Hence, the lemma is proved. □

Lemma 3.3. Under assumption $(H_{1})$, there exists $\mu_{0} > 0$, given by

such that for any $\lambda \in (0, C_{H})$, and $\mu \in (0, \mu_{0})$, we have $\aleph^{0} = \emptyset$. The positive constants $C_{H}, \; C_{1}, C_{2}$, and $C^{'}_{H}$ are given in (1.3) and in Lemma 3.1.

Proof. We suppose that there exists $0 < \mu < \mu_{0}$ such that $\aleph^{0}\neq\emptyset$. Then, there exists $\varphi\in \aleph^{0}$. So, if $||\varphi|| < 1$, then by (3.1), we have

Then

By Lemma 3.1, we have

Then,

Again from (3.1), we have

Then, using Lemma 3.1, we obtain

Then,

Combining (3.4) and (3.5), we get

Now, if $||\varphi||\geq 1$, we have

and

Using $C_{2} > (C^{'}_{H})^{2}$ and $0 < 1-\theta^{+} < 1-\theta^{-} < 1 < q^{-} < q^{+} < p^{-} < p^{+} < r^{-} < r^{+}$, we have,

Combining, (3.6), (3.8), (3.9), and (3.10), we get

which is a contradiction. □

Remark 3.1. For any $0 < \lambda < C_{H}$, $0 < \mu < \mu_{0}$ and from Lemmas 3.2 and 3.3, we obtain that

(1) $N = \aleph^{+}\cup \aleph^{-}$.

(2) $\Phi_{\lambda, \mu}$ is coercive and bounded below on $\aleph^{+}$ and $\aleph^{-}$.

Then, we note,

Lemma 3.4. If $(H_{1})$ and $(H_{2})$ hold, then for any $0 < \lambda < C_{H}$ and $0 < \mu < \mu_{0}$, we have

Proof. If $\varphi\in \aleph^{+}$, from (3.1) we have,

Then,

On the other hand, we also have from (3.1) that:

By to (3.11), we get,

Then,

From $(H_{2})$, we have,

So, $\Phi_{\lambda, \mu}(\varphi) < 0$, $\forall\; \; \varphi\in \aleph^{+}$. Thus $m\leq m^{+} < 0$. □

Lemma 3.5. If $(H_{1})$ and $(H_{2})$ hold, then for $0 < \lambda < C_{H}$, $0 < \mu < \frac{1-\theta^{+}}{p^{+}}\mu_{0}$, there exists $k_{0} > 0$, such that

Proof. Let $\varphi\in \aleph^{-}$. Then we know that $\varphi^{''}_{\varphi}(1) < 0$. Moreover, we have two cases:

Case 1. $||\varphi|| < 1$. From Lemma 3.2 and (3.5), we have

So, if

we conclude that $\Phi_{\lambda, \mu}(\varphi)\geq d_{1} > 0$.

Case 2. $||\varphi|| > 1$. From Lemma 3.2 and (3.7), we have

So, if we have

then we obtain $J(\varphi)\geq d_{2} > 0$.

Now, a simple calculation shows that

Hence, if we put $k_{0} = \min(d_{1}, d_{2})$, then from the above study, we obtain $\Phi_{\lambda, \mu}(\varphi)\geq k_{0} > 0$, which implies that

□

Lemma 3.6. If $(H_{1})$ and $(H_{2})$ hold, then for $\varphi\in X \backslash \{0\}$, there exists $\mu^{'} > 0$, such that, for $0 < \mu < \mu^{'}$ and $0 < \lambda < C_{H}$, there exists $t^{*} > 0$ and $t^{+} < t^{-}$ such that, $t^{-}\varphi\in \aleph^{-}, \; \; t^{+}\varphi\in \aleph^{+}$,

Proof. Let $\varphi\in X \backslash \{0\}$. For all $t > 1$, we have

and for all $0 < t\leq1$, we get

Now, we introduce the following function

where, $\beta > \alpha > \theta > 0, A(\varphi), B(\varphi), C(\varphi)\geq 0,$ we have

Define $\tau(t) = A(\varphi)t^{\alpha+\theta}-C(\varphi)t^{\beta+\theta}$. So, $\tau$ possesses a unique maximum point at

$B(\varphi) = \mu \int_{\Omega} a(\tau)|\varphi|^{1-\theta(\tau)}\tau > 0$ and if $\mu > 0$, is small enough such that $B(\varphi) < \tau(t_{\max})$, then there exist $0 < t_{1} < t_{\max} < t_{2} < \infty$ such that $\tau(t^{+}) = \tau(t^{-}) = B(\varphi), \; \tau'(t^{+}) > 0,$ and $\tau'(t^{-}) < 0$, that is, $t^{+}$ and $t^{-}$ are two solutions of the equation $h(t) = 0$, for all $t > 0$.

The graph $J'_{\varphi}(t)$ is between two graphs

and

Now, using the discussion as $\tau$, there exists $\mu^{'} > 0$ and $0 < t^{+} < t^{-} < \infty$ such that $J'_{\varphi}(t^{-}) = J'_{\varphi}(t^{+}) = 0,$ and $t^{+}\varphi\in \aleph^{+}\; \; t^{-}\varphi\in \aleph^{-}$ $\lambda$ for all $\mu \in (0, \mu^{'})$. □

Nextly, we pose $\mu^{*} = \min\{\mu^{'}, \frac{1-\theta^{+}}{p^{+}} \mu_{0}\}$, where $\mu^{'}$ and $\mu_{0}$, are given in Lemmas 3.6 and 3.3, respectively.

Proposition 3.1. If $(H_{1})$ and $(H_{2})$ hold, then for $\lambda\in(0, C_{H})$ and $\mu\in(0, \mu^{*})$ the functional $\Phi_{\lambda, \mu}$ has a minimizer $\varphi_{1}\in \aleph^{+}$, such that,

Proof. The functional $\Phi_{\lambda, \mu}$ is bounded below in $\aleph^{+}$. So, there exists $\{v_{n}\}$, such that, $\Phi_{\lambda, \mu}(v_{n})\rightarrow m^{+}$. By Lemma 3.2, we conclude that $\{v_{n}\}$ is bounded on reflexive space $X$, so, up to a sub-sequence, there exist $\{v_n\}$ and $\varphi_{1}$ in $X$ such that

Next, by Proposition 2.3, we get

We prove that $\varphi_{1}\neq 0$ and $B(\varphi_{1}) > 0$. If $B(\varphi_{1}) = 0$, since $(v_{n})\in \aleph^{+}$ and by (3.1) and (3.12), we get

Then,

From, Lemma 3.4, $\lim_{n\rightarrow \infty}\Phi_{\lambda, \mu}(v_{n}) = m^{+} < 0$. This is a contradiction. Then, $B(\varphi_{1}) > 0$ and $\varphi_{1}\in X \backslash \{0\}$.

Now, we will be showing that $v_{n}\rightarrow \varphi_{1}$ strongly in $X$.

Supposing the contrary, then $v_{n}\nrightarrow \varphi_{1}$ strongly in $X$. By the Brezis-Lieb Lemma (see ^[27]), we get

Using (3.12)–(3.14), we get

From Lemma 3.6, for $\varphi_{1}\in X\backslash \{0\}$, there exists $t^{+} > 0$, such that $t^{+}\varphi_{1}\in \aleph^{+}$. Since, $v_{n}\nrightarrow \varphi_{1}$ in $X$, we concluded that,

Then, by Proposition 2.3, we get

By (3.16) and (3.17), we obtain

Then for $n$ large enough, we obtain

Now, since $v_{n}\in \aleph^{+}$ for all $n\in \mathbb{N}$, we have, $J^{'}_{v_{n}}(1) = 0$. Thus $t^{+}\neq 1$. Clearly, $t^{+}$ is a minimizer of $g(t) = \Phi_{\lambda, \mu}(tz_{1})$, for $t > 0$. Then, by (3.15), we get

This is a contracted $t^{+}\varphi_{1}\in \aleph^{+}$. So, $v_{n}\rightarrow \varphi_{1}$ strongly in $\aleph^{+}\cup \aleph^{-}$ as $n\rightarrow +\infty$, and by Lemma 3.2, $\aleph^{0} = {\emptyset}$. Then, $\varphi_{1}\in \aleph^{+}$, and by Lemma 3.4,

□

Proposition 3.2. If $(H_{1})$ and $(H_{2})$ hold, then for $\lambda\in(0, C_{H})$ and $\mu\in(0, \mu^{*})$ the functional $\Phi_{\lambda, \mu}$ has a minimizer $\varphi_{2}\in \aleph^{-}$, such that

Proof. Let $\{v_{n}\}$ in $\aleph^{-}$, such that, $\Phi_{\lambda, \mu}(v_{n})\rightarrow m^{-}$. Lemma 3.2 implies that $\{v_{n}\}$ is bounded in reflexive space $X$. So, up to a sub-sequence, there exist $\{v_n\}$ and $\varphi_{2}$ in $X$ such that,

and

Next, we have $\varphi_{2}\neq 0$. Indeed, if $\varphi_{2} = 0$, from (3.19), we obtain,

Using the fact that $\{v_{n}\}\in \aleph^{-}$, Eq (3.1), and Lemma 3.5, we have

Then, by (3.20) and the fact that $1-\theta^{-} < p^{-}$, we obtain

which is a contradiction. So, $\varphi_{2}\in X \backslash \{0\}$. On the other hand, by Lemma 3.6 there exists a positive real $t^{-}$ such that $t^{-}\varphi_{2}\in \aleph^{-}$.

Next, we will prove that $v_{n}$ converges strongly to $\varphi_{2}$ in $X$. Assume that this is not true. Then, by the Brezis-Lieb Lemma (see ^[27]), we have

On the other hand, from Eq (3.21) and Proposition 2.3, we obtain

Thus for $n$ large enough, we conclude that

Since $v_{n}\in \aleph^{-}$, then we have $J^{'}_{v_{n}}(1) = 0$ and using (3.23), we get $t^{-}\neq 1$. Observe that the function $G(t) = \Phi_{\lambda, \mu}(tv_{n})$, for $t > 0$, attains its maximum at $t = 1$ and using (3.22), we get

This contradicts the fact that $t^{-}\varphi_{2}\in \aleph^{-}$. Then, $v_{n}\rightarrow \varphi_{2}$ strongly in $X$ as $n\rightarrow +\infty$, and thus $\varphi_{2}\in \aleph$. By Lemma 3.2 we have, $\aleph^{0} = {\emptyset}$ and by Lemma 3.4, we get,

So, we conclude that $\varphi_{2}\in \aleph^{-}$. □

Lemma 3.7. If $(H_{1})$ and $(H_{2})$ hold and $\varphi\in \aleph^{\pm}$, there exist $\epsilon > 0$, and a continuous function $\alpha: B_{\epsilon}(0)\rightarrow (0, +\infty)$, such that

where

Proof. Let $\varphi\in \aleph^{+}$ and let the function $f:X\times \mathbb{R} \rightarrow \mathbb{R}$ defined by:

Since $\varphi\in \aleph^{+}$, ve have $f(0, 1) = J^{'}_{\varphi}(1) = 0$ and $f^{'}(0, 1) = J^{''}_{\varphi}(1) > 0$.

Then, from the implicit function theorem, there exist $\epsilon > 0$ and a continuous function $\alpha: B_{\epsilon}(0)\rightarrow (0, +\infty)$, such that

Using (3.24), we get

Taking $\epsilon > 0$ even smaller if necessary, we can also have

The proof for the case $\varphi\in \aleph^{-}$ is very similar, so we omit it. □

Proof of Theorem 1.1. By Lemma 3.7, we can find $\vartheta(t) > 0, \; t\in[0, t_{0}]$ such that,

Then, by Proposition 3.1, we have, $\forall\; \; t\in[0, t_{0}]$,

So, $\forall\; \; t\in[0, t_{1}]$ with $0 < t_{1}\leq t_{0}$, we get

Then,

So, for $t > 0$, we get

which yields to

Since the function $h$ is arbitrary, then we can change $h$ by $-h$ in the last inequality, and we conclude that $\varphi_{1}\in \aleph^{+}$ is a nontrivial weak solution to the problem $(P_{\lambda, \mu})$.

Now, by Lemma 3.7 and Proposition 3.2, the proof is the same for $\varphi_{2}\in \aleph^{-}$. By this stage, the proof of Theorem 1.2 is now completed.

4.   Conclusions

In this paper, we studied a $p(x)$-biharmonic problem involving two types of nonlinearities: Singular and Hardy type. More precisely, we combine a variational method with the Nehari manifold method to prove that such a problem admits two nontrivial solutions. We will generalize this study to problems involving the $p(\cdot, \cdot)$-Laplace operator, and we will extend this study to double-phase and multi-phase problems.

Use of Generative-AI tools declaration

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

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (GPIP: 1898-130-2024). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Conflict of interest

The author declares that there is no conflict of interest.