Abstract
In this paper, we are concerned with a class of discrete problems involving the p(k)-Laplacian-like operators. Under some suitable conditions on the nonlinearity f, we study the existence and multiplicity of solutions by using some results of the critical point theory and variational methods.
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Introduction
We consider the discrete anisotropic problem governed by the p(k)-Laplacian-like operators with Dirichlet-type boundary condition given as follows:
where \(N \ge 2\) is a positive integer, \([1,N]_{\mathbb {N}}:=\lbrace 1,2, \ldots ,N\rbrace\) is the discrete interval, \(\Delta\) denotes the forward difference operator defined by \(\Delta u (t)=u (t+1)-u (t)\). For every fixed \(t \in [0,N]_{\mathbb {N}}\), the function \(f(t,.): \mathbb {R} \rightarrow \mathbb {R}\) is continuous and \(p:[0,N+1]_\mathbb {N}\rightarrow [2,+\infty )\) is a given bounded function. \(\phi _c\) is the so-called mean curvature operator [19] defined by
For any bounded function \(h: [0,N+1]_{\mathbb {N}} \rightarrow \mathbb {R}\), we consider the symbols
Furthermore, we will consider the following assumptions:
- \((H_1)\)::
-
There exist \(C>0\) and \(2\le \alpha (t)< p^-\) for all \(t \in [0,N+1]_{\mathbb {N}}\) such that
$$\vert f(t,x)\vert \le C(1+\vert x \vert ^{\alpha (t)-1}) ~~ \text {for all} ~(t,x) \in [0,N]_{\mathbb {N}} \times \mathbb {R}.$$ - \((H_2)\)::
-
\(f(r, 0) \ne 0\) for any \(r \in [1, N]_\mathbb {N}\).
- \((H_3)\)::
-
There exists \(\kappa\) such that \(\kappa > \frac{2^{p^+ +1}}{p^-} \left( N+1\right) ^{\frac{p^+}{2}}\) and
$$\liminf _{\vert x\vert \rightarrow \infty } \frac{F(r, x)}{\vert x\vert ^{p^+}} \ge \kappa , \text { for all } r\in [1, N]_{\mathbb {N}},$$where \(F(r,x)= \int _0^x f(r,s)ds.\)
In recent years, a great deal of work has been devoted to the study of difference equations, because it is an interesting topic and plays an important role in various research domains, such as the dynamic problems of combustible gas [28], electrorheological fluids [27] and image restoration [9].
Problem \((P_\lambda )\) makes a part of discrete nonlinear problems involving the p-Laplacian-like operator and the specified mean curvature operator, which has its origin from the study of capillary phenomena. These phenomena can be briefly explained by considering the effects of two opposing forces: cohesion, i.e., the attractive force between the molecules of the liquid; and adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container. The study of these phenomena has gained more attention recently, motivated by its importance in applied fields such as industrial, biomedical, economic systems [11, 26], the analysis of capillary surfaces [4, 13, 15] and the capillarity problem in hydrodynamics [21]. We refer the reader to [1, 10, 11, 16, 18] and references therein.
For the continuous counterpart of problem \((P_\lambda )\), we refer the reader to recent works (see [12, 24, 25]). For instance, in [24], Rodrigues has studied the existence and multiplicity of solutions for the following problem involving the p(x)-Laplacian-like operators arising from capillary phenomena:
where \(\lambda >0\), \(\Omega \subset \mathbb {R}^N\) is a bounded domain with smooth boundary \(\partial \Omega\), \(p \in C(\overline{\Omega })\) and \(p(x)>2\), \(\forall x \in \Omega\). Based on the mountain pass theorem, the author showed that the problem has at least one nontrivial solution. The author also proved the existence of a sequence of solutions using the Fountain theorem. The problem \((P_\lambda )\) may be regarded as the discrete analog of the above problem.
Despite the interest mentioned above, the number of works dealing with problems involving p-Laplacian-like in the discrete case remains relatively small. For some related papers in the area of discrete problems, we can mention, for instance, [5, 17, 29]. In [29], the authors have treated the following discrete problem involving the mean curvature operator with Dirichlet boundary conditions:
where \(\lambda >0\), \(N \in \mathbb {Z}^{+}, ~ \mathbb {Z}(1, N)=\{1,2, \cdots , N\}\) and for each \(k \in \mathbb {Z}(1, N)\), \(f(k, \cdot ) \in C(\mathbb {R}, \mathbb {R})\). By using the critical point theory, they proved the existence of infinitely many positive solutions under certain sufficient conditions.
In this work, we are inspired by the above-mentioned works and the ideas introduced in [17], where the author studied the problem
where \(\phi _{p_i(k)} (t)= \vert t \vert ^{p_i(k) -2}t\) for \(i=1, 2\) and \(t\in \mathbb {R}\). The author proved the existence and multiplicity results by using some critical point theorems and based on the hypotheses \((H_1)\), \((H_2)\) and the so-called Ambrosetti-Rabinowitz condition ((AR)-condition for short [3]) given as:
- (AR)::
-
There exist two positive constants \(\theta >p^{+}\)and \(t^*>0\) such that for every \(t \in [1, N]_\mathbb {N}\) and \(\vert s \vert \ge t^*\)
$$0<\theta F(t, s) \le s f(t, s).$$
It is well known that the (AR)-condition has a crucial role in the application of variational methods; it is extensively used to ensure the boundedness of the Palais-Smale sequences of the energy functional. Despite this, it is still restrictive and eliminates many nonlinearities. For example, the function
with a primitive
does not satisfy the (AR)-condition. However, it satisfies the hypothesis \((H_3)\), which is weaker and covers a wider functional space. For this reason, in our work, we have taken hypothesis \((H_3)\) instead of the (AR)-condition.
The rest of this article is structured as follows. In the “Preliminaries” section, we present some basic preliminaries and provide several inequalities useful in our approach. In the “Variational framework” section, we give the variational framework corresponding to the problem (P). Our main results will be established in the “Main results and proofs” section, where some examples are also provided.
Preliminaries
Consider E the N-dimensional Hilbert space [2]:
endowed with the inner product
The associated norm is defined by
Also, for \(r\ge 2\), we define the norm
Since E is a finite-dimensional space, all norms are equivalent.
For our purpose, it is useful to use the following inequalities.
Proposition 1
-
(a)
Let \(u \in E\) and \(\Vert u\Vert >1\). Then
$$\begin{aligned} \sum _{t=1}^{N+1} \frac{\vert \Delta u(t-1)\vert ^{p(t-1)}}{p(t-1)} \ge N^{\frac{2-p^- }{2}}\Vert u\Vert ^{p^{-}}-N. \end{aligned}$$ -
(b)
Let \(u \in E\) and \(\Vert u\Vert <1\). Then
$$\begin{aligned} \sum _{t=1}^{N+1} \frac{\vert \Delta u(t-1)\vert ^{p(t-1)}}{p(t-1)} \ge N^{\frac{p^+ -2}{2}}\Vert u\Vert ^{p^{+}}. \end{aligned}$$ -
(c)
For all \(u\in E\), we have
$$\begin{aligned} \Vert u\Vert _{\infty }:=\max _{t\in [1,N]_\mathbb {N}}\vert u(t)\vert \le \sqrt{N+1} \Vert u \Vert . \end{aligned}$$ -
(d)
For all \(u\in E\), we have
$$\sum _{t=1}^{N+1} \vert \Delta u(t-1)\vert ^{p^+} \le (N+1)\Vert u \Vert ^{p^+}.$$ -
(e)
For all \(u\in E\), \(m\ge 2\) we have
$$\sum _{t=1}^{N} \vert u(t)\vert ^{m} \ge 2^{-m}\left( N+1\right) ^{\frac{2-m}{2}}\Vert u \Vert ^{m}.$$
In this article, we are based on three local minimum theorems due to Bonanno and D’Agui [6, 7], which are a simple extension of the Ricceri’s variational principle [23].
Let E be a finite-dimensional Banach space and let \(F, G: E\rightarrow \mathbb {R}\) two functions of class \(C^1\) on E with F be coercive.
Theorem 2.1
(see [6]) Suppose that there exist \(r \in \mathbb {R}\) and \(\omega \in E\), with \(0<F(\omega )<r\) such that
Then, for each
the function \(I_\lambda =F-\lambda G\) admits at least one local minimum \(u_0 \in E\backslash \{ 0\}\) such that \(F(u_0)<r, I_\lambda (u_0) \le I_\lambda (u)\) for all \(u \in F^{-1}([0, r])\) and \(I_\lambda ^{\prime }(u_0)=0\).
Theorem 2.2
(see [6]) Let \(r>0\). Suppose that the function \(I_\lambda =F-\lambda G\) satisfies the (PS)-condition and is unbounded from below for all
Then, for any \(\lambda \in \Lambda _2\), the function \(I_\lambda\) has at least two distinct critical points.
We will also use this result, which guarantees the existence of at least three critical points.
Theorem 2.3
[7, 23] Let W be a reflexive real Banach space, \(F: W \rightarrow \mathbb {R}\) be a continuously Gâteaux differentiable, coercive and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on \(W^*\), \(G: W \rightarrow \mathbb {R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, moreover
Assume that there exist \(r \in \mathbb {R}\) and \(\omega \in W\), with \(0<r<F(\omega )\), such that
-
1.
$$\frac{\sup _{ F^{-1}(]-\infty , r ] )} G}{r}<\frac{G(\omega )}{F(\omega )}.$$
-
2.
The functional \(I_\lambda = F-\lambda G\) is coercive for any
$$\lambda \in \Lambda :=\bigg ] \frac{F(\omega )}{G(\omega )}, \frac{r}{\sup _{u \in \Phi ^{-1}(]-\infty , r ])} G(u)}\bigg [.$$
Then, for any \(\lambda \in \Lambda\), the functional \(I_\lambda\) admits at least three distinct critical points in W.
Variational framework
Let \(T_\lambda : E \rightarrow \mathbb {R}\) the functional associated with problem \((P_\lambda )\) defined in the following way
with
The functional \(T_\lambda\) is well defined, of class \(C^1\) and its Gâteaux derivative is given by:
for all \(u,v \in E\).
By a standard argument, using summation by parts, it can be shown that the critical points of the functional \(T_\lambda\) are exactly the solutions of problem \((P_\lambda )\).
In the remainder of the article, we put
for any \(\xi \in \mathbb {R}\).
Main results and proofs
Before giving our main results, we give and demonstrate some lemmas.
Lemma 4.1
The functional A is coercive on E.
Proof
We recall that for all \(s\ge 0\), we have \(\max {(0, s-1)} \le \sqrt{1+s^2}-1 \le s\).
Let \(u\in E\) with \(\Vert u \Vert\) be large enough such that \(\vert \Delta u(t-1)\vert \ge 1\) for all \(k\in [1,N]_\mathbb {N}\), in view of (a), we have
So, A is coercive. \(\square\)
Lemma 4.2
Suppose that condition \((H_3)\) is verified, then the functional \(T_\lambda\) is unbounded from below and satisfies the (PS)-condition, i.e., any sequence \((u_n)\subset E\) such that \((T_\lambda (u_n))\) is bounded and \((T_\lambda ^{\prime }(u_n)) \longrightarrow 0\) as \(n \rightarrow +\infty\), has a subsequence which converges in E.
Proof
According to the hypothesis \((H_3)\), for any \(\varepsilon >0\) there exists \(R_0>0\) such that
so,
By continuity of the function \(s\rightarrow F(t, s)\), there exists \(K>0\) such that
Then,
and by relation (e), we get
On the other hand, for \(u \in E\) with \(\Vert u \Vert > 1\) such that \(\vert \Delta u(r-1) \vert > 1\) for all \(r\in [1, N]_{\mathbb {N}}\), by (d), we have
it follows that,
Consequently, for \(\varepsilon <\kappa - \frac{2^{p^+ +1}}{p^-} \left( N+1\right) ^{\frac{p^+}{2}}\), we have
Hence \(T_\lambda\) is anti-coercive on E.
Therefore, the functional \(T_\lambda\) is unbounded from below. Furthermore, any (PS) sequence \(\left( u_n\right)\) associated with \(T_\lambda\) will be bounded in E, which is a finite-dimensional space. Thus, the functional \(T_\lambda\) satisfies the (PS)-condition. \(\square\)
Now, we state our main result as follows.
Theorem 4.1
Assume that there exist two real constants \(\xi\) and \(\delta \ge 1\), with
and
such that
Therefore, for every
problem \((P_\lambda )\) has at least one nontrivial solution \(u_0 \in E\).
Proof
In this proof, we will apply Theorem 2.1. By Lemma 4.1, the functional A is coercive, it remains to verify that there exist \(r \in \mathbb {R}\) and \(w \in E\), with \(0<A(w)<r\), such that
For this reason, we put \(r:= R (\xi )\) and choosing \(w\in E\) such that
Obviously with \(\delta \ge 1\), we obtain
So, from (5) it follows that \(0<A(w)<r\).
Now, let \(u \in E\) such that \(u \in A^{-1}([0, r])\) and \(\Vert u\Vert <1\), by using inequality (b), we get
Then as well in (3), from (a) we have
for all \(u \in E\) with \(\Vert u\Vert >1\).
Then
By virtue of (4), we obtain
this together with (c), gives
for all \(t \in [1, N]_\mathbb {N}\). Therefore, we have
In view of (9) and (11), taking in consideration \((F_1)\) and \((F_2)\), we have
Therefore, according to Theorem 2.1, the functional \(T_\lambda\) has at least one critical point \(u_0\) with \(0< A(u_0) < r\), and thus, \(u_0\) is a nontrivial solution of problem \((P_\lambda )\). \(\square\)
Theorem 4.2
We suppose that hypotheses \((H_2)\) and \((H_3)\) hold. Moreover, suppose that inequalities (4) and (5) of Theorem 4.1 are satisfied. Then, for any
the problem \((P_\lambda )\) admits at least two nontrivial solutions.
Proof
Fix \(\lambda \in \Lambda _2\). The functional A and B defined in (1) verify all regularity assumptions requested in Theorem 2.2. Choosing \(r=R(\xi )\) and w(t) as in (8), we have
Moreover, by Lemma 4.2, the function \(T_\lambda\) satisfies the (PS) condition and is unbounded from below. Therefore, in view of Theorem 2.2, for any \(\lambda \in \Lambda _2\), the problem \((P_\lambda )\) admits at least two distinct solutions. \(\square\)
Theorem 4.3
Assume that there exist two real numbers \(\xi\) and \(\delta \ge 1\) satisfy the first inequality in (4) and
such that the assertions \((F_1)\) and \((F_2)\) in Theorem 4.1 hold. Moreover, if \((H_1)\) is verified, then, for any \(\lambda \in \Lambda _1\), where \(\Lambda _1\) is defined in (6), the problem \((P_\lambda )\) has at last three solutions.
Proof
Similar to the proof of Theorem 4.1, we set w(k) as in (8) and \(r=R(\xi )\). Taking into account (13), we have \(A(w)>r>0\). Then, (12) is satisfied.
Now, in order to apply Theorem 2.3, it remains to establish that the functional \(T_\lambda\) is coercive. Let \(u \in E\) such that \(\Vert u \Vert >1\). We point out that
On the other hand, the relation (c) implies that for \(m\ge 2\)
and thus
this combined with (14), gives
Now, from \((H_1)\) there exists \(C_1>0\) such that
which implies that
In addition, the combination of (3) and (15) leads us to
Since \(p ^{-}>\alpha ^{+}\), the above inequality means that \(T_{\lambda }\) is coercive.
Therefore, for any \(\lambda \in \Lambda _1\), the functional \(T_{\lambda }\) possesses at least three critical points which are exactly solutions for the problem \((P_\lambda )\). \(\square\)
Now, we provide two examples illustrating Theorems 4.1 and 4.2.
Example 1
Let \(N=3\) and \(p(r)= 3+\frac{r}{4}\) for \(r\in [0, 4]_{\mathbb {N}}\). Then, \(p^- = 3 \text { and } p^+ = 4\).
We consider the function
with
In addition, if we choose \(\delta =1\) and \(\xi = (10\sqrt{3})^{\frac{1}{3}}\), we obtain \(R(\xi )=\frac{8}{3}\), and inequalities (4)–(5) are verified. Moreover, \(F(r,1)\ge 0\) for \(r=1, 2, 3\) and
Then, in view of Theorem 4.1, for all \(\lambda \in \left]\frac{4}{9}, \frac{16}{21} \right[\), the problem \((P_\lambda )\) has at least one nontrivial solution.
Example 2
We keep as in Example 1, \(N=3\), \(p(r)= \frac{r}{4} +3\) for \(r\in [0, 4]_{\mathbb {N}}\), \(\delta =1\) and \(\xi = (10\sqrt{3})^{\frac{1}{3}}\). We get \(p^- = 3\), \(p^+ = 4\) and \(R(\xi )=\frac{8}{3}\).
Let f be the function defined as:
with
We have
where \(\kappa\) is a constant such that \(\kappa > \frac{2^{p^+ +1}}{p^-} (N+1)^{\frac{p^+}{2}} = \frac{2^{9}}{3}.\)
Then, conditions \((H_2)\) and \((H_3)\) hold. So, by virtue of Theorem 4.2, for any parameter \(\lambda \in \biggl ]0, \frac{8/3}{\sum _{t=1}^3 \max _{\vert s\vert \le (10\sqrt{3})^{\frac{1}{3}}} F(t, s)}\biggr [\), the problem \((P_\lambda )\) has at least two nontrivial solutions.
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Barghouthe, M., Ayoujil, A. & Berrajaa, M. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF DISCRETE PROBLEMS WITH THE p(k)-LAPLACIAN-LIKE OPERATORS. J Math Sci (2024). https://doi.org/10.1007/s10958-024-07188-9
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DOI: https://doi.org/10.1007/s10958-024-07188-9