Multiple solutions for nonlocal elliptic problems driven by $p(x)$-biharmonic operator

1.   Introduction

In this paper we study the existence of at least three distinct weak solutions for the following problem

where

$\Omega\subset\mathbb{R}^N (N\geq2)$ is an open bounded domain with smooth boundary, $\Delta^2_{p(x)}u$ is the operator defined as $\Delta(|\Delta u|^{p(x)-2}\Delta u)$ and is called the $p(x)$-biharmonic which is a generalization of the $p$-biharmonic, $p(x)\in C(\overline{\Omega})$, $\rho(x)\in L^\infty(\Omega)$, $M:[0, +\infty)\to\mathbb{R}$ is a continuous function such that there are two positive constants $m_{0}$ and $m_{1}$ with $m_{0}\leq M(t)\leq m_{1}$ for all $t\geq 0$, $\dfrac{N}{2} < p^- : = \text{ ess}\inf_{x\in\Omega}p(x)\leq p^+ : = \text{ ess}\sup_{x\in\Omega}p(x) < \infty$, $\lambda > 0$ and $f:{\Omega}\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function.

The Kirchhoff equation refers back to Kirchhoff ^[18] in 1883 in the study on the oscillations of stretched strings and plates, suggested as an extended version of the classical D'Alembert's wave equation by taking into account the effects of the changes in the length of the string during the vibrations. Kirchhoff's equation like problem $\left( {P_\lambda ^f} \right)$ model several physical and biological systems where $u$ describes a process which depend on the average of itself. Lions in ^[23] has proposed an abstract framework for the Kirchhoff-type equations. After the work by Lions, various problems of Kirchhoff-type have been widely investigated, we refer the reader to the papers ^[7,24,27] and the references therein.

The main interest in studying problem $\left( {P_\lambda ^f} \right)$ is given by the presence of the variable exponent $p(\cdot)$. Problems involving such kind of growth conditions benefited by a special attention in the last decade since they can model with sufficient accuracy phenomena arising in different branches of science. Two important models where operators involving variable exponents were considered come from the study of electrorheological fluids ^[8,28] and elastic mechanics ^[34].

Fourth-order equations have various applications in areas of applied mathematics and physics such as micro-electro-mechanical systems, phase field models of multi-phase systems, thin film theory, thin plate theory, surface diffusion on solids, interface dynamics, flow in Hele-Shaw cells (see ^[4,6,26]). The fourth-order equation can also describe the static form change of beam or the sport of rigid body. In ^[22], Lazer and Mckenna have pointed out that this type of nonlinearity furnishes a model to study travelling waves in suspension bridges. Numerous authors investigated the existence and multiplicity of solutions for the problems involving $p(x)$-biharmonic operators. We refer to ^{[10,12,16,19,21,30,31]}. In the last decade, Kirchhoff type equations involving the $p(x)$-Laplacian have been investigated, for instance see ^{[3,9,13,14,15,17,25]}.

In this paper, we are interested to discuss the existence of at least three distinct weak solutions for problem $\left( {P_\lambda ^f} \right)$. No asymptotic condition at infinity is required on the nonlinear term. In Theorem 3.1 we establish the existence of at least three distinct weak solutions for problem $\left( {P_\lambda ^f} \right)$. Theorem 3.3 is a consequence of Theorem 3.1. As a consequence of Theorem 3.3, we obtain Theorem 3.4 for the autonomous case. We present example 3.5 to illustrate Theorem 3.4.

2.   Preliminaries

Let $X$ be a nonempty set and $\Phi, \Psi:X\to \mathbb{R}$ be two functions. For all $r, \ r_1, \ r_2 > \inf_{X}\Phi, \ r_2 > r_1, \ r_3 > 0,$ we define

We shall discuss the existence of at least three distinct solutions to the problem $\left( {P_\lambda ^f} \right)$. Our main tool is based on [1,Theorem 3.3] that we now recall as follows:

Theorem 2.1. Let $X$ be a reflexive real Banach space, $\Phi:X\to \mathbb{R}$ be a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on $X^*, \Psi:X\to \mathbb{R}$ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that $\inf_X\Phi = \Phi(0) = \Psi(0) = 0$ and for every $u_1, u_2\in X$ such that $\Psi(u_1)\geq 0$ and $\Psi(u_2)\geq 0$, one has

Assume that there are three positive constants $r_1, r_2, r_3$ with $r_1 < r_2,$ such that

$(c_1)$ $\varphi(r_1) < \beta(r_1, r_2);$

$(c_2)$ $\varphi(r_2) < \beta(r_1, r_2);$

$(c_3)$ $\gamma(r_2, r_3) < \beta(r_1, r_2).$

Then for each $\lambda\in \left(\dfrac{1}{\beta(r_1, r_2)}, \dfrac{1}{\alpha(r_1, r_2, r_3)}\right)$ the functional $\Phi-\lambda\Psi$ admits three distinct critical points $u_1, u_2, u_3$ such that $u_1\in \Phi^{-1}(-\infty, r_1), \ u_2\in\Phi^{-1}[r_1, r_2)$ and $u_3\in \Phi^{-1}(-\infty, r_2+r_3).$

We refer the interested reader to the papers ^[2,11,20] in which Theorem 2.1 has been successfully used to ensure the existence of at least three solutions for boundary value problems.

Let $\Omega$ be a bounded domain of $\mathbb{R}^N$, denote:

We can introduce the norm on $L^{p(x)}(\Omega)$ by:

Let $X$ be the generalized Lebesgue-Sobolev space $W^{m, p(x)}(\Omega)$ defined by putting $W^{m, p(x)}(\Omega)$ as

which is equipped with the norm:

$\gamma$ is the multi-index and $|\gamma|$ is the order.

The closure of $C^\infty_0(\Omega)$ in $W^{m, p(x)}(\Omega)$ is the $W^{m, p(x)}_0(\Omega)$. It is well known ^[5] that, both $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, with the respective norms, are separable, reflexive and uniformly convex Banach spaces.

Proposition 2.2. ^[5] Suppose $\dfrac{1} {p(x)} +\dfrac{1} {p^0(x)} = 1$, then $L^{p^0(x)}(\Omega)$ and $L^{p(x)}(\Omega)$ are conjugate space, and satisfy the Hölder inequality:

We denote $X : = W^{1, p(x)}_0(\Omega)\cap W^{2, p(x)}(\Omega)$ and has the norm

By ^[32], $\|\cdot\|$, $\|\cdot\|_{2, p(\cdot)}$ and $|\Delta u|_{p(\cdot)}$ are equivalent norms of $X$.

A bounded operator $T :X\to\mathbb{R}$ is said to be compact if $T (B_X)$ has compact closure in $\mathbb{R}$.

Proposition 2.3. ^[31] When $p^- > \dfrac {N}{2}$, $\Omega\subset\mathbb{R}$ is a bounded region, then $X \mapsto C(\overline{\Omega})$ is a compact embedding.

According to 2.3, for each $u\in X,$ there exists a constant $c > 0$ that depends on $p(\cdot), N, \Omega$:

Remark 2.4. We say that $f:\Omega\times \Bbb{R}\rightarrow \Bbb{R}$ is an $L^1$-Carathéodory function if

$(a)$ $t \mapsto f(x, t)$ is measurable for every $t \in \Bbb{R};$

$(b)$ $x \mapsto f(x, t)$ is continuous for a.e. $x \in \Omega;$

$(c)$ for every $\varepsilon > 0$ there exists a function $l_\varepsilon \in L^1(\Omega)$ such that for a.e. $x\in\Omega,$

Corresponding to the functions $f$ and $M$, we introduce the functions $F:\Omega\times \Bbb{R}\to \Bbb{R}$ and $\widetilde{M}:[0, +\infty)\to \Bbb{R}$, respectively, as follows

We say that $u\in X$ is a weak solution of problem $\left( {P_\lambda ^f} \right)$ if for every $v\in X,$

Proposition 2.5. ^[5] Let $J(u) = \int_{\Omega}|u|^{p(x)}dx$ for each $u \in L^{p(x)}(\Omega)$, we have

$(1)$ $|u|_{p(x)} < 1( = 1; > 1)\Leftrightarrow J(u) < 1( = 1; > 1);$

$(2)$ $|u|_{p(x)}\geq1\Longrightarrow|u|_{p(x)}^{p^-}\leq J(u)\leq|u|_{p(x)}^{p^+};$

$(3)$ $|u|_{p(x)}\leq1\Longrightarrow|u|_{p(x)}^{p^+}\leq J(u)\leq|u|_{p(x)}^{p^-};$

$(4)$ $|u|_{p(x)}\longrightarrow0\Leftrightarrow J\longrightarrow0.$

Now for every $u \in X$, we define $I(u): = \Phi(u)-\lambda\Psi(u)$ where

and

For our convenience, set

Proposition 2.6. Let $T = \Phi':X\to X^{*}$ be the operator defined by

for every $u, v\in X$. Then $T$ admits a continuous inverse on $X^{*}$.

Proof. For any $u\in X\setminus\{0\}$,

since $p^- > 1$, it follows that the map $T$ is coercive. Since $T$ is the Fréchet derivative of $\Phi$, it follows that $T$ is continuous and bounded. Using the elementary inequality ^[29]

for all $(x, y) \in \mathbb{R}^{N}\times\mathbb{R}^{N}$, $N\geq1$, we obtain for all $u, v\in X$ such that $u\neq v$,

which means that $T$ is strictly monotone. Thus $T$ is injective. Consequently, thanks to Minty-Browder theorem ^[33], the operator $T$ is a surjection and admits an inverse mapping. Thus it is sufficient to show that $T^{-1}$ is continuous. For this, let $(v_n)^\infty_{n = 1}$ be a sequence in $X^*$ such that $v_n\to v$ in $X^*$. Let $u_n$ and $u$ in $X$ such that

By the coercivity of $T$, we conclude that the sequence $(u_n)$ is bounded in the reflexive space $X$. For a subsequence, we have $u_n\to \tilde{u}$ in $X$, which implies

Therefore, by the continuity of $T$, we have

Moreover, since $T$ is an injection, we conclude that $u = \tilde{u}.$

3.   Main results

Fix $x^0\in \Omega$ and choose $s > 0$ such that $B(x^0, s)\subset\Omega$, where $B(x^0, s)$ denotes the ball with center at $x^0$ and radius of $s$. Put

where $\ell = dist(x, x^{0}) = \sqrt{\sum_{i = 1}^N \left(x_{i}-x_i^0\right) ^2}$ and

$\Gamma$ denotes the Gamma function, and

Theorem 3.1. Assume that there exist positive constants $\theta_1, \theta_2$, $\theta_3$ and $\eta\geq1$ with $\theta_1 < \sqrt[p^-]{L}c\eta$, $\eta < \min\left\{\sqrt[p^+]{\dfrac{p^-M^-}{p^+c^{p^-}M^+L}}\theta_2^{\frac{p^-}{p^+}}, \theta_2\right\}$ and $\theta_2 < \theta_3$ such that

$(A_1)$ $f(x, t)\geq0$ for each $(x, t)\in \overline{\Omega}\setminus B(x^{0}, \frac{s}{2})\times[-\theta_3, \theta_3];$

$(A_2)$

Then for every

problem $\left( {P_\lambda ^f} \right)$ has at least three weak solutions $u_1$, $u_2$ and $u_3$ such that

Proof. Our goal is to apply Theorem 2.1 to the problem $\left( {P_\lambda ^f} \right)$. We consider the auxiliary problem

where $\hat{f}:{\Omega}\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function defined as

If a weak solution of the problem $\left( {P_\lambda ^{\hat f}} \right)$ satisfies the condition $-\theta_3\leq u(x)\leq\theta_3$ for every $x\in \Omega$, then, clearly it turns to be also a weak solution of $\left( {P_\lambda ^f} \right)$. Therefore, it is enough to show that our conclusion holds for $\left( {P_\lambda ^f} \right)$. We define functionals $\Phi$ and $\Psi$ as given in (2.3) and (2.4), respectively. Let us prove that the functionals $\Phi$ and $\Psi$ satisfy the required conditions in Theorem 2.1. It is well known that $\Psi$ is a differentiable functional whose differential at the point $u\in X$ is

for every $v\in X$, as well as it is sequentially weakly upper semicontinuous. Recalling (2.1), we have

for all $u \in X$ with $\|u\| > 1$, which implies $\Phi$ is coercive. Moreover, $\Phi$ is continuously differentiable whose differential at the point $u\in X$ is

for every $v\in X$, while Proposition 2.6 gives that $\Phi'$ admits a continuous inverse on $X^*$. Furthermore, $\Phi$ is sequentially weakly lower semicontinuous. Therefore, we observe that the regularity assumptions on $\Phi$ and $\Psi$, as requested of Theorem 2.1, are verified. Define $w$ by setting

It is easy to see that $w\in X$ and,

and

and so that

It is easy to see that $w\in X$ and, in particular, since

and

In particular, one has

On the other hand, bearing $(A_1)$ in mind, from the definition of $\Psi$, we infer

Choose $r_1 = \dfrac{M^-}{p^+}\left(\dfrac{\theta_1}{c}\right)^{p^-},$ $r_2 = \dfrac{M^-}{p^+}\left(\dfrac{\theta_2}{c}\right)^{p^-}$ and $r_3 = \dfrac{M^-}{p^+}\left(\dfrac{{\theta}^{p^-}_3-\theta^{p^-}_2}{c^{p^-}}\right)$. From the conditions

and $\theta_2 < \theta_3$, we achieve $r_1 < \Phi(w) < r_2$ and $r_3 > 0$. For all $u\in X$ with $\Phi(u) < r_1$, taking (2.1) and (2.2) into account, one has

So, thanks to the embedding $X\hookrightarrow C^0(\overline{\Omega})$, one has $\|u\|_\infty < \theta_1$. From the definition of $r_1$, it follows that

Hence, one has

As above, we can obtain that

and

Therefore, since $0\in \Phi^{-1}(-\infty, r_{1})$ and $\Phi(0) = \Psi(0) = 0$, one has

and

On the other hand, for each $u\in \Phi^{-1}(-\infty, r_1)$ one has

Due to $(A_2)$ we get

Therefore, $(b_1)$ and $(b_2)$ of Theorem 2.1 are verified. Finally, we verify that $\Phi-\lambda\Psi$ satisfies the assumption $2$ of Theorem 2.1. Let $u_1$ and $u_2$ be two local minima for $\Phi-\lambda\Psi$. Then $u_1$ and $u_2$ are critical points for $\Phi-\lambda\Psi$, and so, they are weak solutions of the problem $\left( {P_\lambda ^f} \right)$. Since we assumed $f$ is nonnegative, for fixed $\lambda > 0$, we have $\lambda f (k, su_1 + (1 - s)u_2)\geq0$ for all $s\in[0, 1],$ and consequently, $\Psi(su_1 + (1 - s)u_2)\geq 0$ for every $s\in[0, 1]$. Hence, Theorem 2.1 implies that for every

the functional $\Phi-\lambda\Psi$ has three critical points $u_i, \ i = 1, 2, 3,$ in $X$ such that $\Phi(u_1) < r_1,$ $\Phi(u_2) < r_2$ and $\Phi(u_3) < r_2 + r_3$, that is,

Then, taking into account the fact that the solutions of the problem $\left( {P_\lambda ^f} \right)$ are exactly critical points of the functional $\Phi-\lambda\Psi$ we have the desired conclusion.

Remark 3.2. If $f$ is non-negative, then the weak solution ensured in Theorem 3.1 is non-negative. Indeed, let $u_0$ be the weak solution of the problem $\left( {P_\lambda ^f} \right)$ ensured in Theorem 3.1, then $u_0$ is nonnegative. Arguing by a contradiction, assume that the set $\mathcal{A} = \left\{x \in\Omega : u_0(x) < 0\right\}$ is non-empty and of positive measure. Put $\bar v(x) = \min\left\{0, u_0(x)\right\}$ for all $x \in \Omega$. Clearly, $\bar v \in X$ and one has

for every $\bar v \in X.$ Thus we have

i.e.,

which contradicts with this fact that $u_0$ is a non-trivial weak solution. Hence, the set $\mathcal{A}$ is empty, and $u_0$ is positive.

Theorem 3.3. Assume that there exist positive constants $\theta_1$, $\theta_4$ and $\eta\geq1$ with $\theta_1 < \min\left\{\eta^{\frac{p^+}{p^-}}, \sqrt[p^-]{L}c\eta\right\}$ and $\eta < \min\left\{\sqrt[p^+]{\dfrac{p^-M^-}{2c^{p^-}p^+M^+L}} \theta_4^{\frac{p^-}{p^+}}, \theta_4\right\}$ such that

$(A_3)$ $f(x, t)\geq0$ for each $(x, t)\in\overline{\Omega}\setminus B(x^{0}, \frac{s}{2})\times[-\theta_4, \theta_4];$

$(A_4)$

Then for every

problem $\left( {P_\lambda ^f} \right)$ has at least three weak solutions $u_1$, $u_2$ and $u_3$ such that

Proof. Choose $\theta_2 = \dfrac{1}{\sqrt[p^-]{2}}\theta_4$ and $\theta_3 = \theta_4$. So, from $(A_4)$ one has

and

Moreover, since $\theta_1 < \eta^{\frac{p^+}{p^-}}$, from $(A_4)$ we have

Hence, from $(A_4)$, (3.2) and (3.3), it is easy to observe that the assumption $(A_2)$ of Theorem 3.1 is satisfied, and it follows the conclusion.

The following result is a consequence of Theorem 3.3.

Theorem 3.4. Let $f : {\Omega}\times \mathbb{R}\to\mathbb{R}$ be a continuous function such that $\xi f(x, \xi) > 0$ for all $(x, \xi)\in\Omega\times \mathbb{R}\backslash \{0\}.$ Assume that

$(A_5)$ $\lim_{\xi\to0}\dfrac{f(x, \xi)}{|\xi|^{{p}^{-}-1}} = \lim_{|\xi|\to+\infty}\dfrac{f(x, \xi)} {|\xi|^{{p}^{-}-1}} = 0.$

Then for every $\lambda > \overline{\lambda}$ where

problem $\left( {P_\lambda ^f} \right)$ possesses at least four distinct non-trivial solutions.

Proof. Set

and

and define $F_1(x, \xi): = \int_{0}^{\xi}f_1(x, t)dt$ for every $(x, \xi)\in\Omega\times \mathbb{R}.$ Fix $\lambda > \lambda^*$, and let $\eta\geq1$ such that $\lambda > \dfrac{\left(p^+c^{p^-}M^+L +p^-M^-\right)\eta^{p^+}}{p^-p^+c^{p^-} \int_{B(x^{0}, \frac{s}{2})} F(x, \eta)dx}.$ From

there is $\theta_1 > 0$ such that

and $\theta_4 > 0$ such that

and

Then, $(A_4)$ in Theorem 3.3 is satisfied,

Hence, the problem $(P^{f_1}_{\lambda})$ admits two positive solutions $u_1$, $u_2$, which are positive solutions of the problem $\left( {P_\lambda ^f} \right)$. Next, arguing in the same way, from

we ensure the existence of two positive solutions $u_3$, $u_4$ for the problem $(P^{f_2}_{\lambda})$. Clearly, $-u_3$, $-u_4$ are negative solutions of the problem $\left( {P_\lambda ^f} \right)$ and the conclusion is achieved.

Example 3.5. Let $\Omega = \left\{(x, y)\in\mathbb{R}^2:x^2+y^2\leq9\right\}$. Consider the problem

where $M(t) = \dfrac{3}{2}+\dfrac{\sin(t)}{2}$ for each $t\in[0, \infty)$, $p(x, y) = x^2+y^2+4$ for all $(x, y)\in\Omega$ and

By the expression of $f$, we have

Direct calculations give $M^- = 1,$ $M^+ = 2,$ $p^- = 4$ and $p^+ = 13$. It is clear that

Hence, by applying Theorem 3.4, there is $\lambda^* > 0$ such that for each $\lambda > \lambda^*$, the problem possesses at least four distinct non-trivial solutions.

As a special case, we present a simple consequence of Theorem 3.3 when $f$ dose not depend upon $x$. To be precise, consider the following problem

where $f:\mathbb{R}\to\mathbb{R}$ is a continues function.

Put

Theorem 3.6. Assume that there exist positive constants $\theta_1, \theta_2$, $\theta_3$ and $\eta\geq1$ with $\theta_1 < \sqrt[p^-]{L}c\eta$, $\eta < \min\left\{\sqrt[p^+]{\dfrac{p^-M^-}{p^+c^{p^-}M^+L}} \theta_2^{\frac{p^-}{p^+}}, \theta_2\right\}$ and $\theta_2 < \theta_3$ such that

$(A_7)$ $f(t)\geq0$ for each $t\in[-\theta_3, \theta_3];$

$(A_8)$

Then for every

problem $\left( {P_\lambda ^f} \right)$ has at least three weak solutions $u_1$, $u_2$ and $u_3$ such that

Theorem 3.7. Assume that there exist positive constants $\theta_1$, $\theta_4$ and $\eta\geq1$ with $\theta_1 < \min\left\{\eta^{\frac{p^+}{p^-}}, \sqrt[p^-]{L}c\eta\right\}$ and

such that

$(A_9)$ $f(t)\geq0$ for each $t\in[-\theta_4, \theta_4];$

$(A_{10})$

Then for every

problem $\left( {P_\lambda ^f} \right)$ has at least three weak solutions $u_1$, $u_2$ and $u_3$ such that

Conflict of interest

All authors confirm that there are no competing interests between them.