Abstract
We prove the existence of three distinct solutions for a biharmonic nonlocal elliptic system with singular terms under the Navier boundary conditions, by using variational methods and the theory of the variable exponent Sobolev space.
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1 Introduction
In this paper, we consider the following biharmonic system
for \(i=1,\ldots ,n\), where \(\Omega\) is a bounded domain in \({{\mathbb {R}}^{N}}(N\ge 2)\) with smooth boundary and \(\Delta ^2_{p_i(x)}u:=\Delta (\vert \Delta u\vert ^{p_i(x)-2}\Delta u)\) for each \(p_i\in C({\overline{\Omega }})\), \(i=1,\ldots ,n\). For \(i=1,\ldots ,n\), we assume that \(1<s_i<\frac{N}{2}\) and nonnegative function \(b_i\) belongs to \(L^\infty (\Omega )\). \(\lambda\) is a positive parameter and the function
is a measurable function with respect to \(x\in \Omega\) for each \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) and is \(C^1\) with respect to \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) for a.e. \(x\in \Omega\); \(F_{u_i}\) denotes the partial derivative of F with respect to \(u_i\).
Problems involving biharmonic operator arise in the study of traveling waves in suspension bridge and the study of static deflection of plate.
Singular boundary value problems arise in the context of chemical heterogeneous catalysts and chemical catalyst kinetics, in the theory of heat conduction in electrical conducting materials, as well as in the study of non-Newtonian fluids and boundary layer phenomena for viscous fluids.
In the last years, study of the biharmonic problems in various spaces is one of the interesting objects; for instance, authors of [10] studied the following fourth-order elliptic problem with Navier boundary conditions
where \(\Delta ^{2}_{p}u= \Delta |\Delta u|^{p-2} \Delta u\) denotes the p-biharmonic operator, \(\Omega\) is a bounded domain in \({\mathbb {R}}^N\) \((N\ge 5)\) containing the origin and with smooth boundary, \(1<p<\frac{N}{2}\), \(\lambda >0\) is a parameter and \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function such that
for some non-negative constants \(a_1\), \(a_2\) and \(q \in ]p,p^*[\), where
The existence of the solutions to the following weighted (p(x), q(x))-Laplacian problem consisting of a singular term
have been proved [11], where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with smooth boundary, \(a,b\in L^\infty (\Omega )\) are positive functions with \(a(x)\ge 1\) a.e. on \(\Omega\); \(\lambda >0\) is a real parameter, \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function satisfying the following growth condition
for all \((x,t)\in \Omega \times {\mathbb {R}}\) (see also [1, 17] and the references therein). Recently the existence of at least one positive radial solution of the weighted p-biharmonic problem
with Navier boundary conditions on a Korányi ball has been proved [19] via a variational principle, where \(w\in A_s\) is a Muckenhoupt weight function and \(\Delta ^2_{{\mathbb {H}}^n, p}\) is the Heisenberg p-biharmonic operator.
Motivated by their works, we study the existence of multiplicity of weak solutions for the problem (1.1), which is consisting of biharmonic type operator and singular terms. Our main tool in this study is the following theorem [2].
Theorem 1.1
Let X be a reflexive real Banach space, \(\Phi :X \rightarrow {\mathbb {R}}\) be a coercive, continuously G\({\hat{a}}\)teaux differentiable and sequentially weakly lower semi-continuous functional whose G\({\hat{a}}\)teaux derivative admits a continuous inverse on \(X^{*}\). Let \(\Psi :X\rightarrow {\mathbb {R}}\) be a continuously G\({\hat{a}}\)teaux differentiable functional whose G\({\hat{a}}\)teaux derivative is compact such that
Assume that there exist \(r>0\) and \({\bar{x}}\in X\), with \(r<\Phi ({\bar{x}})\), such that
-
(i)
\(\dfrac{sup_{\Phi (x)<r}\Psi (x)}{r}<\dfrac{\Psi ({\bar{x}})}{\Phi ({\bar{x}})}\);
-
(ii)
For each
$$\begin{aligned} \lambda \in \Lambda _{r}:=\left]\dfrac{\Phi ({\bar{x}})}{\Psi ({\bar{x}})}, \dfrac{r}{sup_{\Phi (x)<r}\Psi (x)} \right[, \end{aligned}$$the functional \(I_{\lambda }=\Phi -\lambda \Psi\) is coercive.
Then, for each \(\lambda \in \Lambda _{r}\), the functional \(I_\lambda =\Phi -\lambda \Psi\) has at least three distinct critical points in X.
The structure of this paper is the following: In Sect. 2, we present preliminaries and some basic facts. We also introduce a suitable function space for the solution and we prove some remarks which we need for the last section. In Sect. 3, the existence of multiple weak solutions for Problem (1.1) is prove by variational methods and three critical points result mentioned above.
2 Preliminaries
During the note, \(\Omega\) is a bounded domain in \({{\mathbb {R}}^{N}}(N\ge 2)\) with smooth boundary. We suppose that \(1<s_i<\frac{N}{2}\) and \(p_i \in C({\overline{\Omega }})\) for \(i=1,\ldots ,n\), satisfy the following inequalities:
Define the variable exponent Lebesgue space \(L^{p_i(x)}(\Omega )\), \(i=1,\ldots ,n\), by
We difine a norm, the so-called Luxemburg norm, on this space by the formula
Let us point out that if \(q(\cdot )\equiv q, \; q\in \{s_i:i=1,\ldots ,n\}\cup \{1\}\), this norm is equal to the standard norm on \(L^q(\Omega )\) that we denote it by \(|\cdot |_q\); that is
For any \(u\in L^{p_i(x)}(\Omega )\) and \(v \in L^{p_i' (x)}(\Omega )\), where \(L^{p_i' (x)}(\Omega )\) is the conjugate space of \(L^{p_i (x)}(\Omega )\), the Hölder type inequality
holds true. The following theorem is in [9, Theorem 2.8].
Theorem 2.1
Assume that \(\Omega\) is a bounded and smooth in \({\mathbb {R}}^N\) and \(p, q \in C_+({\bar{\Omega }})=\{g\in C({\overline{\Omega }}): g^->1\}\). Then
if and only if \(q(x) \le p(x)\) a.e. \(x \in \Omega\); moreover, there exists constant \(M_q\) such that
Following the authors of [18], for any \(\kappa >0\), we put
and
for \(r\in \{p_i: i=1,\ldots ,n\}\). Then the well-known proposition 2.7 of [7] will be rewritten as follows.
Proposition 2.1
For each \(u\in L^{p(x)}(\Omega )\), we have
We denote the first order variable exponent Sobolev space by
endowed with the norm
where
is the gradient of u at \(x=(x_1, \ldots ,x_n)\) and as usual \(|\nabla u|= \big ( \sum _{i=1}^N |\frac{\partial u}{\partial x_i}|^2 \big )^\frac{1}{2}\). And, the second order variable exponent Sobolev space is defined as follows
with the norm
where \(\Delta u=\sum _{i=1}^N \frac{\partial ^2 u}{\partial x_i^2}\) is Laplace operator. It is well known that the spaces \(L^{p(x)}(\Omega )\) and \(W^{m,p(x)}(\Omega ), m=1,2,\) are separable, reflexive and uniform convex Banach spaces [4]. Let \(W_0^{1,p(x)}(\Omega )\) be the closure of \(C^{\infty }_{0}(\Omega )\) in \(W^{1,p(x)}(\Omega )\). We set
for \(p\in \{p_i: i=1,\ldots ,n\}\), which is a reflexive Banach space with respect to the norm
By using the Poincaré inequality and [21], the norms \(\Vert \cdot \Vert _{Z}\) and
are equivalent on Z.
Remark 2.1
As a consequence of Theorem 2.1, we have
if \(p(x) \le q(x)\) a.e. \(x \in \Omega\). In a special case, for \(p_i\), \(i=1,\ldots ,n\), with the condition (2.1),
is embedded continuously and since \(p_i^->\frac{N}{2}\), one has the following compact embedding
Then
So, in particular, there exist positive constants \(k_i>0, \; i=1,\ldots ,n\), such that
for each \(u \in W^{2,p_i(x)}(\Omega )\cap W_{0}^{1,p_i(x)}(\Omega )\) where \(| u |_{\infty }:=\sup _{x \in \Omega }|u(x)|\).
Remark 2.2
From Proposition 2.1, for each \(u\in Z\), we have
Here, we recall the classical Hardy–Rellich inequality mentioned in [3].
Lemma 2.1
Let \(1<s<\frac{N}{2}\). Then for \(u \in W^{1,s}_{0}(\Omega )\cap W^{2,s}(\Omega )\), one has
where \({\mathcal {H}}_s:= (\frac{N(s-1)(N-2s)}{s^2})^s\).
Remark 2.3
Suppose \(1<s_i<\frac{N}{2}\) and \(p_i \in C(\Omega )\) be as in (2.1), for \(i=1,\ldots ,n\). Then there exists \(\alpha\) such that
for \(u \in W^{1,p_i(x)}_{0}(\Omega )\cap W^{2,p_i(x)}(\Omega )\), where
Proof
Since for each \(i=1,\ldots , n\), \(s_i<p_i(x)\) a.e. in \(\Omega\), so by (2.5), we have
moreover, there exist constants \(\alpha _{s_i}\) such that
By using classical Hardy–Rellich inequality, we have
for \(u \in W^{1,s_i}_{0}(\Omega )\cap W^{2,s_i}(\Omega )\). Then we gain
Put \(\alpha =\max \{\alpha ^{s_i}_{s_i}, i=1,\ldots ,n\}\) and \({\mathcal {H}}_s=\min \{{\mathcal {H}}_{s_i}=(\frac{N(s_i-1)(N-2s_i)}{s_i^2})^{s_i}, i=1,\ldots ,n\}\), this completes the proof. \(\square\)
In what follows, we set
endowed with the norm
for \(u=(u_1,\ldots ,u_n)\in X\). From Remark 2.1, we conclude that the embedding
is compact and if we put
where \(k_i, 1\le i\le n\) are as in relation (2.6), it is clear that \(K>0\) and one has
We mean by a weak solution to the problem (1.1) is as follows:
Definition 2.1
We say that \(u=(u_1,\ldots ,u_n)\in X\) is a weak solution of problem (1.1) if
for every \(v=(v_1,\ldots ,v_n)\in X\).
We introduce the functional \(\Phi :X\longrightarrow {\mathbb {R}}\) as follows
Remark 2.4
There exists positive constant \({\hat{C}}\) such that
for each \(1\le i\le n\) and \(u=(u_1,\ldots ,u_n)\in X\).
Proof
From (2.2) and Remark 2.3, for every \(1\le i\le n\), one has the following estimate
where \(p^{-}:=\min \{p_i^-:i=1,\ldots ,n\}\). Now, it is enough to set
\(\square\)
Remark 2.5
Due to the Remark 2.4\(\Phi\) is coercive.
Proof
Let \(u=(u_1,\ldots ,u_n)\in X\) and \(\Vert u\Vert \rightarrow \infty\). By definition of \(\Vert \cdot \Vert\), there exists \(1\le i_0\le n\) such that \(|\Delta u_{i_0}|_{p_{i_0}(x)}\rightarrow \infty\). Then Remark 2.4 result in \(\Phi (u)\rightarrow \infty\). \(\square\)
Furthermore, \(\Phi\) is sequentially weakly lower semicontinuous and it is known that \(\Phi\) is continuously Gâteaux differentiable functional whose derivative given by
for each \((v_1,\ldots ,v_n)\in X\).
Now imagine that the function
is a measurable function with respect to \(x\in \Omega\) for each \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) and is \(C^1\) with respect to \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) for a.e. \(x\in \Omega\); \(F_{u_i}\) denotes the partial derivative of F with respect to \(u_i\). We define \(\Psi :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) with
The functional \(\Psi\) is well defined, continuously Gâteaux differentiable with compact derivative, whose Gâteaux derivative at point \(u=(u_1,\ldots ,u_n)\in X\) is as follows
for every \((v_1,\ldots ,v_n)\in X\). Define
for each \(u=(u_1,\ldots ,u_n)\); if \(I_{\lambda }^{\prime }(u)=0\) we have
then the critical points of \(I_{\lambda }\) are exactly the weak solutions of the problem (1.1). We set
and we define
Obviously, there exists \(x^0=(x^0_1,\ldots ,x^0_N)\in \Omega\) such that
Now, we are ready to sketch the main result of the paper.
3 Main approach
We formulate our main result as follows.
Theorem 3.1
Assume that \(F:{\overline{\Omega }}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) satisfies the following conditions
- (F1):
-
\(F(x,0,\ldots ,0)=0\), for a.e. \(x\in \Omega\);
- (F2):
-
There exist \(\eta \in L^1(\Omega )\) and n positive continuous functions \(\gamma _i\), \(1\le i\le n\), with \(\gamma _i(x)<p_i(x)\) a.e in \(\Omega\) such that
$$\begin{aligned} 0\le F(x,u_1,\ldots ,u_n)\le \eta (x) \left( 1+\sum _{i=1}^n \vert u_i \vert ^{\gamma _i(x)}\right) ; \end{aligned}$$ - (F3):
-
There exist \(r>0\), \(\delta >0\) and \(1\le i_*\le n\) such that
$$\begin{aligned} \frac{1}{p_{i_*}^+}\left( \dfrac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\check{p_{i_*}}}m\left( R^N-\left( \frac{R}{2}\right) ^N\right) >r, \end{aligned}$$where \(m:=\frac{\pi ^{\frac{N}{2}}}{\frac{N}{2}\Gamma (\frac{N}{2})}\) is the measure of unit ball of \({\mathbb {R}}^{N}\) and \(\Gamma\) is the Gamma function.
Such that
where
and
Then for each
the problem (1.1) admits at least three distinct weak solutions in X.
Proof
Our aim is to apply Theorem 1.1. According to previous section, the space
and the functionals \(\Phi ,\Psi :X\rightarrow {\mathbb {R}}\) defined as above satisfy the regularity assumptions of Theorem 1.1. By condition (F1) and definition of \(\Phi , \Psi\), it is clear that
Now Fix \(\delta >0\) and R defined as in (2.8). By w, we consider the function of the space \(W^{2,p_i(x)}(\Omega )\cap W_0^{1,p_i(x)}(\Omega ), 1\le i\le n\), defined by
where \(x=(x_1,\ldots ,x_N)\in \Omega\). Then
So, by Remark 2.4, for \(1\le i_*\le n\), one has
then by assumption (F3), we gain \(\Phi (w,\ldots ,w)>r\). On the other hand, we have
where m is the measure of unit ball of \({\mathbb {R}}^{N}\) and so,
Now, let \(u=(u_1,\ldots ,u_n)\in \Phi ^{-1}(-\infty ,r)\), from Remark 2.4, for each \(i=1,\ldots ,n\) we have
Then for every \(u=(u_1,\ldots , u_n)\in \Phi ^{-1}(-\infty ,r)\), from condition (F2), Hölder inequality and (2.6), we gain
So,
From assumption (3.1), relations (3.3) and (3.5), one has
Therefore, the assumption (i) of Theorem 1.1 is satisfied.
Now, we prove that the functional \(I_{\lambda }\) for all \(\lambda >0\) is coercive.
With the same arguments as used before, we have
The last inequality and Remark 2.4 lead to
for each \(i=1,\ldots ,n\). Now, suppose that \(u\in X\) and \(\Vert u\Vert \rightarrow \infty\). So, there exists \(1\le i_0\le n\) such that \(|\Delta u_{i_0}|_{p_{i_0}(x)}\rightarrow \infty\). Since \(\gamma _{i_0}(x)<p_{i_0}(x)\) a.e. in \(\Omega\), so \(I_{\lambda }\) is coercive.
Then condition (ii) holds. So for each
Theorem 1.1 ensures that for each \(\lambda \in \Lambda _{r,\delta }\), the functional \(I_{\lambda }\) admits at least three critical points in X that are weak solutions of the problem (1.1). \(\square\)
Remark 3.1
An interesting problem is to probe the existence and multiplicity of solutions of this system under the Steklov boundary conditions [8] or in the Heisenberg Sobolev spaces and Orlicz Sobolev spaces. Interested reader can see details of these spaces in [5, 6, 12,13,14,15,16, 20] and references therein.
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Khaleghi, A., Razani, A. Multiple Solutions for a Class of Biharmonic Nonlocal Elliptic Systems. J Nonlinear Math Phys 31, 31 (2024). https://doi.org/10.1007/s44198-024-00199-9
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DOI: https://doi.org/10.1007/s44198-024-00199-9