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Existence and multiplicity of solutions for discrete NeumannSteklov problems with singular ϕLaplacian
Advances in Difference Equations volume 2017, Article number: 312 (2017)
Abstract
This paper establishes the existence and multiplicity of solutions for the discrete NeumannSteklov problem with singular ϕLaplacian by using the method of lower and upper solutions, a priori estimates and Brouwer degree theory.
Introduction
Recently, the various existence results for quasilinear equation problems
and the discrete form
subjected to Dirichlet, periodic or Neumann boundary conditions on \([0,T]\) or \([2,N1]_{\mathbb{N}}\) have been studied by many authors such as Mawhin, Bereanu, Jebelean, Torres, Thompson, Coelho, Corsato, Obersnel, Omari, Rivetti, Ma and so on [1–9]. Here \(\phi:(a,a)\to\mathbb{R}\) is an increasing homeomorphism with \(\phi(0)=0\), the model example is
Δ is a forward difference operator with \(u_{k}=u(t_{k})\), \(\Delta u_{k}=u(t_{k+1})u(t_{k})\), \(t_{N}=T\) and ∇ is a backward difference operator with \(\nabla u_{k}=u(t_{k})u(t_{k1})\), \(t_{1}=0\), \(f:[0,T]\times\mathbb{R}^{2}\to\mathbb {R}\) is continuous. In addition, the nonlinear difference equations play an important role in many fields such as biology, engineering, science and technology where discrete phenomena abound, meanwhile, from the advent and rise of computers, differential equations have been solved by employing their approximative difference equations formulations, e.g., see [7–18] and the references therein.
This paper focuses on the existence and multiplicity of solutions for the discrete NeumannSteklov problem with singular ϕLaplacian operator
where \(h_{1}, h_{N}\in C(\mathbb{R},\mathbb{R})\) and \(f:[2,N1]_{\mathbb{Z}}\times\mathbb{R}^{2}\to\mathbb{R}\) is continuous with respect to the second and third variables.
Obviously, the NeumannSteklov conditions of (1.1) can be written in the equivalent classical form
with \(g_{0}:\mathbb{R}\to(a,a)\), and \(g_{N}:\mathbb{R}\to(a,a)\) given by \(g_{1}=\phi^{1}\circ h_{1}\), \(g_{N}=\phi^{1}\circ h_{N}\). In addition, \(u\in \mathbb{R}^{N}\) satisfies \(u_{2}=u_{1}+g_{1}(u_{1})\), \(u_{N1}=u_{N}g_{N}(u_{N})\). Notice that if \(h_{1},h_{N}\equiv0\), then problem (1.1) is degenerate to the Neumann problem, which was studied in [7–9]. However, as far as we know, there is very little work on the existence of solutions of difference equation with nonlinear boundary value conditions. Motivated by the above works [2, 7–9], we shall discuss the existence and multiplicity of solutions of (1.1).
The rest of this paper is organized as follows. In Section 2, we state some notations and preliminary results. Section 3 contains the proof of the existence of one solution of (1.1) when the nonlinearity f and \(h_{0}\), \(h_{N}\) satisfy some suitable sign conditions. In Section 4, we extend the classical method of upper and lower solutions to the NeumannSteklov problem, and we obtain AmbrosettiProdi type results for the NeumannSteklov problem (1.1) in Section 5.
Preliminaries
For convenience, we list a few notations that will be used throughout this paper. Let \(a,b\in\mathbb{N}\) with \(a< b\), we denote \([a,b]_{\mathbb{Z}}:=\{a,a+1,\ldots, b\}\). In addition, we denote \(\sum_{s=a}^{b}u_{s}=0\) with \(b< a\) and \(\prod_{s=a}^{b}u_{s}=1\) with \(b< a\).
For \(\mathbf{u}=(u_{1},\ldots, u_{p})\in\mathbb{R}^{p}\), set \(\Vert \mathbf{u} \Vert _{\infty}=\max_{1\leq k\leq p} \vert u_{k} \vert \). If \(\boldsymbol{\alpha},\boldsymbol{\beta}\in\mathbb{R}^{p}\), we write \(\boldsymbol{\alpha}\leq\boldsymbol{\beta}\) (respectively \(\boldsymbol{\alpha}<\boldsymbol{\beta}\)) if \(\alpha_{k}\leq\beta_{k}\) (resp. \(\alpha_{k}<\beta_{k}\)) for all \(1\leq k\leq p\). The following assumption upon ϕ (called singular) is made throughout the paper:
 (\(H_{\phi}\)):

\(\phi:(a,a)\to\mathbb{R}\) (\(0< a<\infty\)) is an increasing homeomorphism with \(\phi(0)=0\).
Let \(N\in\mathbb{N}\) with \(N\geq4\) be fixed and \(\mathbf{u}=(u_{1}, u_{2},\ldots, u_{N})\in\mathbb{R}^{N}\). Then we denote
by \(\Delta u_{k}= u_{k+1}u_{k}\) for \(k\in[1,N1]_{\mathbb{Z}}\), \(\Vert\Delta \mathbf{u}\Vert_{\infty}:=\max_{k\in[1,N1]_{\mathbb{Z}}}\vert\Delta u_{k}\vert\) and \(\Vert \mathbf{u} \Vert = \Vert \mathbf {u} \Vert _{\infty}+\Vert\Delta \mathbf{u}\Vert_{\infty}\); if \(\Vert\Delta \mathbf{u}\Vert_{\infty}< a\), then we define
by \(\nabla(\phi(\Delta u_{k}))=\phi(\Delta u_{k})\phi(\Delta u_{k1})\) for \(k\in[2,N1]_{\mathbb{Z}}\).
Let \(f:[2,N1]_{\mathbb{Z}}\times\mathbb{R}^{2}\to\mathbb{R}\) be a continuous function with respect to the second and third variables. Then its Nemytskii operator \(N_{f}(\mathbf{u}):\mathbb{R}^{N}\to\mathbb{R}^{N2}\) is given by
It follows that \(N_{f}\) is continuous and takes bounded sets into bounded sets.
Let P, Q be the projectors defined by
We also define the linear mapping
Moreover,
which means Q̃ is also a projector. Let the linear operator \(H_{i}:\mathbb{R}^{N}\to\mathbb{R}^{N1}\) as follows:
The vector space \(\mathbb{R}^{N}\) will be a Banach space endowed with the norm \(\Vert \mathbf{u} \Vert \). For \(\mathbf{u}\in\mathbb {R}^{N}\), we also give some notations as follows:
We set \(B(\mathbf{0}, \rho):=\{\mathbf{u}\in\mathbb{R}^{N}\mid \Vert \mathbf{u} \Vert _{\infty}<\rho\}\) (\(\rho>0\)), and for shortness, we shall write \(B_{\rho}\) instead of \(B(\mathbf{0},\rho)\).
Lemma 2.1
Let \(f:[2,N1]_{\mathbb{Z}}\to\mathbb{R}\) be a function and consider the discrete NeumannSteklov problem
A function u is a solution of (2.2) if and only if
i.e., if and only if
in which case the solutions of (2.2) are given by the operator equation
Proof
Problem (2.2) can be written as
which implies that
This together with (2.3) concludes that
Moreover, a function u is a solution of (2.2) if and only if (2.3) holds, i.e., if and only if (2.4) is true.
By summing from \(s=1\) to k in (2.6), it follows that
Thus, we have that the solutions of (2.2) are given by (2.5). □
Remark 2.2
Lemma 2.1 means that \((f,A,B)\) belongs to the range of the nonlinear mapping \(\mathbf{u}\rightarrow[\nabla(\phi(\Delta\mathbf{u})), \phi (\Delta u_{1}), \phi(\Delta u_{N1})]\) if and only if \(\tilde{Q}(f,A,B)=0\).
Lemma 2.3
Let \(F:\mathbb{R}^{N}\to\mathbb{R}^{N2}\) be a continuous operator which takes bounded sets into bounded sets, and consider the abstract discrete NeumannSteklov problem
A function u is a solution of (2.7) if and only if \(\mathbf{u}\in\mathbb{R}^{N}\) is a fixed point of the continuous operator \(\mathcal{A}_{F}:\mathbb{R}^{N}\to \mathbb{R}^{N}\) defined by \(\mathcal{A}_{F}(\mathbf{u}):=P\mathbf {u}+QF(\mathbf{u}) \frac{1}{N2} [h_{N}(u_{N})h_{1}(u_{1}) ]+H_{1}\circ\phi^{1}\circ [H_{2}(IQ)F(\mathbf{u})+G(\mathbf{u}) ]\), where \(G(\mathbf{u}):=(G(u_{1}), G(u_{2}), \ldots, G(u_{N1}))\in\mathbb {R}^{N1}\) satisfying
Furthermore, \(\Vert \Delta(\mathcal{A}_{F}(\mathbf{u})) \Vert _{\infty}< a\) for all \(\mathbf{u}\in\mathbb{R}^{N}\) and the operator \(\mathcal{A}_{F}(\mathbf {u})\) is completely continuous on \(\mathbb{R}^{N}\).
Proof
It follows from Lemma 2.1 that problem (2.7) is equivalent to
Let \(\mathbf{u}\in\mathbb{R}^{N}\), then
By Lemma 2.1, the first equation in (2.9) can be rewritten as
This together with the second equation in (2.9) implies that (2.9) can be written as the single equation
that is,
□
The existence of solutions for nonlinear NeumannSteklov problems
In this section, we will show the existence of solutions for the nonlinear NeumannSteklov problems
where \(f:[2,N1]_{\mathbb{Z}}\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is continuous with respect to the second and third variables, \(h_{1}:\mathbb {R}\rightarrow\mathbb{R}\), \(h_{N}:\mathbb{R}\rightarrow\mathbb{R}\) are continuous.
Clearly, the following NeumannSteklov problem
has no solution, which means that the existence of NeumannSteklov problem is nontrivial. We will show that some rather general sign conditions upon f, \(h_{1}\), \(h_{N}\) suffice to get existence.
To this end, for \(\lambda\in[0,1]\), we introduce the family of abstract nonlinear NeumannSteklov problems
where Q̃ is defined in (2.1), (3.2) can be rewritten in a more explicit form
It is worthwhile pointing out that (3.2) coincide with (3.1) for \(\lambda=1\), and if u is a solution of (3.2), then, using the definition of Q̃ and Remark 2.2 to (3.2), we get that
Therefore, (3.2) is equivalent to
or, in a more explicit form,
For any \(\lambda\in[0,1]\), the nonlinear operator \(\mathcal{A}\) on \(\mathbb{R}^{N}\) associated to (3.2) by Lemma 2.3 is the operator \(\mathcal{A}(\lambda,\cdot)\), where \(\mathcal{A}\) is defined on \([0,1]\times\mathbb{R}^{N} \) as follows:
where G is defined by (2.8). Clearly, \(\mathcal{A}\) is a continuous operator. Moreover, it is not difficult to verify that \(\mathcal {A}:[0,1]\times\mathbb{R}^{N}\to\mathbb{R}^{N}\) is completely continuous.
Now, we give the following technical results to obtain the main result.
Lemma 3.1
Suppose that there exists \(R>0\) such that when \(\mathbf{u}_{L}\geq R\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), and when \(\mathbf {u}_{M}\leqR\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), it follows that
If \((\lambda,\mathbf{u})\in[0,1]\times\mathbb{R}^{N}\) is such that \(\mathbf{u}=\mathcal{A}(\lambda,\mathbf{u})\), then \(\Vert \mathbf{u} \Vert < R+aN\).
Proof
Let \((\lambda, \mathbf{u})\in[0,1]\times\mathbb{R}^{N}\) be such that \(\mathbf{u}=\mathcal{A}(\lambda,\mathbf{u})\). Choosing \(\lambda=0\), we get
This means that
Since
it yields that
If \(\mathbf{u}_{M}\leqR\) (resp. \(\mathbf{u}_{L}\geq R\) ), then it follows from (3.6), (3.8) that
By using (3.7), we conclude that
Obviously, we get \(\mathbf{u}_{M}\leq\mathbf{u}_{L}+\sum_{k=1}^{N1} \vert \Delta u_{k} \vert \), which together with (3.8), (3.9) leads to
Thus, (3.8) and (3.10) conclude that \(\Vert \mathbf{u} \Vert < R+aN\). □
Lemma 3.2
Suppose that there exist \(R>0\) and \(\varepsilon\in\{1,1\}\) such that when \(\mathbf{u}_{L}\geq R\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), it follows that
and when \(\mathbf{u}_{M}\leqR\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), it yields that
Then, for all sufficiently large \(\rho>0\),
and problem (3.1) has at least one solution.
Proof
Let \(\rho>R+aN\) be a constant, and let \(\mathcal{A}\) be the operator given by (3.5). By using Lemma 3.1 and the homotopy invariance of the Brouwer degree [19], we deduce that
On the other hand, we get that
However, the range of the mapping \(\mathbf{u}\to P\mathbf{u}+QN_{f}(\mathbf{u})\frac{h_{N}(u_{N})h_{1}(u_{1})}{N2}\) is contained in the subspace of constant functions, isomorphic to \(\mathbb{R}\), hence, applying a reduction property of Brouwer degree, we get
However, it follows from (3.11), (3.12) and \(\rho>R\) that
have opposite sign, and using (3.11), (3.12), we deduce that
Thus, \(\operatorname{deg}(I\mathcal{A}(1,\cdot),B_{\rho},0)=\varepsilon \), that is to say, there exists \(\mathbf{u}\in B_{\rho}\) such that \(\mathbf{u}=\mathcal {A}(1,\mathbf{u})\), which is a solution of problem (3.1). □
Lemma 3.2 may imply the existence of a positive solution of (3.1). A limiting argument allows to weaken the sign condition; however, this generalization can also be proved directly using another way based on the following lemma, see Section 4.
For any \(\mathbf{u}\in\mathbb{R}^{N}\), decompose it in the form
and let \(W=\{\mathbf{u}\in\mathbb{R}^{N}\mid u_{1}=0\}\).
Lemma 3.3
The set \(\mathcal{S}\) of the solutions \((\bar{u},\tilde{\mathbf{u}})\in\mathbb{R}\times W\) of the following problem
contain a continuum \(\mathcal{C}\) whose projection on \(\mathbb{R}\) is \(\mathbb{R}\) and whose projection on W is contained in the ball \(B_{aN}\).
Proof
From Lemma 2.3 and (3.15), for any fixed \(\bar{u}\in\mathbb {R}\), problem (3.16) is equivalent to the fixed point problem in \(\mathbb{R}^{N}\) as follows:
It is easy to see that \(\tilde{\mathcal{A}}\) is completely continuous in \(\mathbb{R}^{N}\), and for each \((\bar{u},\tilde{\mathbf{u}})\in\mathbb{R}\times\mathbb {R}^{N1}\), we get
Moreover, for any fixed \(\bar{u}\in\mathbb{R}\), any possible fixed point \(\tilde{\mathbf{u}}\) of \(\tilde{\mathcal{A}}(\bar{u},\cdot)\) satisfies
On the other hand, by the same reasons, for any fixed \(\lambda\in [0,1]\), each possible fixed point \(\tilde{\mathbf{u}}\) of
satisfies (3.17), which yields that
Therefore, from (3.18), (3.19) and [8], Lemma 5, there exists a continuum \(\mathcal{C}\) whose projection on \(\mathbb{R}\) is \(\mathbb {R}\) and whose projection on W is contained in the ball \(B_{aN}\). □
Theorem 3.4
Suppose that there exist \(R>0\) and \(\varepsilon\in\{1,1\}\) such that when \(\mathbf{u}_{L}\geq R\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), it follows that
and when \(\mathbf{u}_{M}\leqR\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\), it yields that
Then problem (3.1) has at least one solution.
Proof
From Lemma 3.3, let us consider the continuum \(\mathcal{C}\). If \((R+aN,\tilde{\mathbf{u}})\in\mathcal{C}\), then \(R+aN+\tilde{u}>R\) and it follows from (3.20) that
If \((RaN,\tilde{\mathbf{u}})\in\mathcal{C}\), then \(RaN+\tilde {u}<R\) and it follows from (3.21) that
From the intermediate value theorem for a continuous function on a connected set, there exists \((\bar{u},\tilde{\mathbf{u}})\in\mathcal {C}\) such that
which means that \(\mathbf{u}=\bar{u}+\tilde{\mathbf{u}}\) is a solution of (3.1). □
Corollary 3.5
Let \(g:[2,N1]_{\mathbb{Z}}\times\mathbb {R}^{2}\to\mathbb{R}\), \(p:[2,N1]_{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}\), \(h_{1}:\mathbb {R}\to\mathbb{R}, h_{N}:\mathbb{R}\to\mathbb{R}\) be continuous, which satisfies g is bounded on \([2,N1]_{\mathbb{Z}}\times\mathbb{R}\times(a,a)\) and \(h_{1}\), \(h_{N}\) are bounded on \(\mathbb{R}\). If
or
uniformly in \(k\in[2,N1]_{\mathbb{Z}}\), then the problem
has at least one solution.
Corollary 3.6
Let \(g:[2,N1]_{\mathbb{Z}}\times\mathbb {R}^{2}\to\mathbb{R}\) be continuous and bounded on \([2,N1]_{\mathbb{Z}}\times\mathbb{R}\times(a,a)\), \(h_{1}:\mathbb{R}\to\mathbb{R}\), \(h_{N}:\mathbb{R}\to\mathbb{R}\) be continuous and satisfy
Then, for any fixed \(\mu\neq0\), the problem
has at least one solution.
Example 3.7
Let \(\mathbf{e}=(e_{2},e_{3},\ldots,e_{N1})\in\mathbb{R}^{N2}\), \(m\in\mathbb{R}\), \(n\in\mathbb{R}\backslash\{0\}\), \(p>1\) and \(q\geq0\). Then the problem
has at least one solution for each continuous bounded function \(h_{1},h_{N}:\mathbb{R}\to\mathbb{R}\), where \(\phi(s)=\frac{s}{\sqrt {1\kappa s^{2}}}\) and \(\kappa>0\) is constant.
Upper and lower solutions for the nonlinear NeumannSteklov problem
In this section, we give the method of upper and lower solutions to the nonlinear NeumannSteklov boundary value problem
which extends the method of upper and lower solutions for the linear Neumann boundary value problem [8], Theorem 3, Theorem 4 and Remark 8.
Definition 4.1
A function \(\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{N})\) (resp. \(\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{N})\)) is called a lower solution (resp. an upper solution) for (4.1) if \(\Vert \Delta\alpha \Vert _{\infty}< a\) (resp. \(\Vert \Delta\beta \Vert _{\infty}< a\) ) and
Such a lower (resp. an upper) solution is called strict if inequality (4.2) is strict.
Theorem 4.2
If (4.1) has a lower solution \(\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{N})\) and an upper solution \(\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{N})\) such that \(\boldsymbol{\alpha}\leq\boldsymbol{\beta}\), then (4.1) has a solution u such that \(\boldsymbol{\alpha}\leq\mathbf{u}\leq \boldsymbol{\beta}\). Moreover, if α and β are strict, then \(\boldsymbol{\alpha}<\mathbf{u}<\boldsymbol{\beta}\), and
where \(\Omega_{\boldsymbol{\alpha},\boldsymbol{\beta}}= \{\mathbf{u}\in \mathbb{R}^{N}\mid \boldsymbol{\alpha}<\mathbf{u}<\boldsymbol{\beta}, \Vert \Delta \mathbf{u}\Vert_{\infty}<a\}\), and \(\mathcal{A}_{f}\) is the fixed point operator associated to (4.1).
Proof
Let \(p:[2,N1]_{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}\) be a continuous function defined by
and define \(F:[2,N1]_{\mathbb{Z}}\times\mathbb{R}^{2}\to\mathbb{R}\) by \(F(k,u,v)=f(k,p(k,u),v)\). Let us consider the modified problem
We claim that if u is a solution of (4.4), then \(\boldsymbol{\alpha}\leq\mathbf{u}\leq\boldsymbol{\beta}\), so that u is a solution of (4.1).
We first prove that \(\alpha\leq u\). Suppose on the contrary that there exists \(k_{0}\in[1,N]_{\mathbb{Z}}\) such that \(\max_{k\in[1,N]_{\mathbb{Z}}}(\boldsymbol{\alpha}\mathbf{u})=\alpha_{k_{0}}u_{k_{0}}>0\). If \(k_{0}\in [2,N1]_{\mathbb{Z}}\), then \(\Delta\alpha_{k_{0}1}\geq\Delta u _{k_{0}1}\) and \(\Delta\alpha _{k_{0}}\leq\Delta u _{k_{0}}\). Since ϕ is an increasing homeomorphism, we have that \(\nabla(\phi (\Delta\alpha_{k_{0}}))\leq\nabla(\phi(\Delta u _{k_{0}}))\). Moreover, it follows from α is a lower solution of (4.1) that
which is a contradiction. If \(\max_{k\in[1,N]_{\mathbb{Z}}}(\boldsymbol{\alpha}\mathbf{u})=\alpha_{1}u_{1}>0\), then \(\Delta\alpha_{1}\leq\Delta u _{1}\), and hence
which contradicts the definition of a lower solution. If \(\max_{k\in [1,N]_{\mathbb{Z}}}(\boldsymbol{\alpha}\mathbf{u})=\alpha_{N}u_{N}>0\), then \(\Delta\alpha_{N1}\geq\Delta u _{N1}\), and we get that
which contradicts the definition of a lower solution.
Similarly, we also obtain that \(u_{k}\leq\beta_{k}\), \(k\in[1,N]_{\mathbb{Z}}\). Notice that if α, β are strict, then, by the same reasoning, we get that \(\alpha<\mathbf{u}<\beta\) with \(\alpha<\beta\). Moreover, from the definition of strict lower and upper solution, neither α nor β can be a solution of (4.4). So (4.4) has no solution on the boundary of \(\Omega_{\alpha, \beta}\).
From Corollary 3.6, we obtain the existence of a solution for (4.4) and the relation Brouwer degree
for all large enough \(\rho>0\), and the fixed point operator \(\tilde {\mathcal{A}}\) associated to (4.4). Furthermore, if α, β are strict and \(\rho>0\) sufficiently large, then it follows from (4.5) and the additivityexcision property of the Brouwer degree[19] that
It is easy to see that the completely continuous operator \(\mathcal {A}_{f}\) with (4.1) is equal to \(\tilde{\mathcal{A}}\) on \(\bar{\Omega }_{\boldsymbol{\alpha},\boldsymbol{\beta}}\), which means that \(\operatorname {deg}(I\mathcal{A}_{f},\Omega_{\boldsymbol{\alpha},\boldsymbol{\beta}},\mathbf{0})=1\). □
By a similar argument in [8], Theorem 4, we can conclude that the existence result in Theorem 4.2 also is true when the lower and upper solutions are not ordered.
Theorem 4.3
Assume that (4.1) has a lower solution α and an upper solution β, then (4.1) has at least one solution.
Proof
Let the continuum \(\mathcal{C}\) be given in Lemma 3.3. Suppose that there exists some \((\bar{u},\tilde{\mathbf{u}})\in\mathcal{C}\) such that
then \(\bar{u}+\tilde{\mathbf{u}}\) is a solution of (4.1). If
then, for any fixed \((\bar{u},\tilde{\mathbf{u}})\in\mathcal{C}\), \(\bar {u}+\tilde{\mathbf{u}}\) is an upper solution for (4.1) by applying (3.16). Moreover, \((\alpha_{M}+aN,\tilde{\mathbf{u}})\in\mathcal{C}\) is an upper solution for (4.1) with \(\alpha_{M}+aN+\tilde{u}_{k}\geq\alpha_{k}\), \(k\in[1,N]_{\mathbb{Z}}\). That is, the existence of a solution for (4.1) follows from Theorem 4.2. By a similar way, if
then \((\beta_{L}aN, \tilde{\mathbf{u}})\in\mathcal{C}\) is a lower solution for (4.1) with \(\beta_{L}aN+\tilde{u}_{k}\leq\beta_{k}, k\in [1,N]_{\mathbb{Z}}\), which means that (4.1) has at least one solution follows again from Theorem 4.2. □
If we choose the constant lower and upper solutions for (4.1) in Theorem 4.2 and Theorem 4.3, then we get the following simple existence condition.
Corollary 4.4
Suppose that there exist constants a and b such that
Then problem (4.1) has at least one solution.
Notice that Theorem 4.3 can deal with the case \(a=+\infty\), the key point is the following a priori estimation result.
Lemma 4.5
Let \(\varphi:(a,a)\to\mathbb{R}\) (\(a\leq +\infty\)) be an increasing homeomorphism with \(\varphi(0)=0\). Set \(\vert h_{1} \vert \) is bounded by \(M_{1}\) and \(\vert h_{N} \vert \) is bounded by \(M_{N}\), there exists \(q:[2,N1]_{\mathbb{Z}}\to\mathbb{R}\) such that
If u is a solution of the nonlinear NeumannSteklov boundary value problem
then \(\Vert \mathbf{u} \Vert _{\infty}\leq c\), where \(c:=\max\{ \vert \varphi^{1}[\pm(M_{N}+2M_{1}+2N \Vert q^{} \Vert _{\infty})] \vert \}\).
Proof
Assume that u is a solution of (4.7). Then it follows that
and
From (4.6), we have that there is a function q such that f is bounded from below, and
(4.9) together with (4.8) implies that
From this inequality and (3.7), the conclusion is true. □
Theorem 4.6
Let \(\varphi:\mathbb{R}\to\mathbb{R}\) be an increasing homeomorphism with \(\varphi(0)=0\). Suppose that all conditions of Lemma 4.5 hold and problem (4.7) has a lower solution α and an upper solution β. Then (4.7) has at least one solution.
Proof
Let c be given in Lemma 4.5, \(a_{1}=\max\{ \Vert \Delta \alpha \Vert _{\infty}, \Vert \beta \Vert _{\infty}, c\}+1\) and \(a=a_{1}+1\). Set \(\phi:(a,a)\to\mathbb{R}\) be an increasing homeomorphism such that \(\phi=\varphi\) on \([a_{1},a_{1}]\). It is easy to verify that α is a lower solution of (4.1) and β is an upper solution of (4.1). From Theorem 4.3, (4.1) has a solution u, which is also a solution of (4.7) by Lemma 4.5. □
Remark 4.7
This result is new even in the case φ is identity operator, i.e., \(\varphi=\mathit{id}_{\mathbb{R}}\).
AmbrosettiProdi type results for the nonlinear NeumannSteklov problem
In this section, let us consider the following nonlinear NeumannSteklov boundary value problem:
where \(s\in\mathbb{R}\), \(f:[2,N1]_{\mathbb{Z}}\times\mathbb{R}^{2}\to \mathbb{R}\) is continuous with respect to the second and third variables and satisfies the coercivity condition
\(h_{1}, h_{N}:\mathbb{R}\to\mathbb{R}\) are continuous and satisfy the following conditions:
Now we shall obtain the existence and multiplicity of the solutions of (5.1) in terms of the value of the parameter s.
Lemma 5.1
Suppose that f, \(h_{1}\), \(h_{N}\) satisfy conditions (5.2), (5.3), (5.4), respectively. Then, for each \(b\in \mathbb{R}\), there exists \(\rho=\rho(b)>0\) such that any possible solution u of (5.1) with \(s\geq b\) belongs to the open ball \(B_{\rho}\).
Proof
Let u be a solution of (5.1) and \(s\geq b\). Then it follows that u satisfies
From conditions (5.2), (5.3) and (5.4), there is a constant \(R>0\) such that if \(\vert u \vert \geq R\), \(k\in[2,N1]_{\mathbb{Z}}\), \(v\in(a,a)\),
Therefore, if \(\mathbf{u}_{L}\geq R\), \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\) or if \(\mathbf{u}_{M}\leqR\), \(\Vert \Delta\mathbf {u} \Vert _{\infty}< a\), then
The conclusion follows from Lemma 3.1. □
Theorem 5.2
Suppose that f, \(h_{1}\), \(h_{N}\) satisfy conditions (5.2), (5.3), (5.4), respectively. Then there exists \(s_{1}\in\mathbb{R}\) such that problem (5.1) has no solution with \(s>s_{1}\), at least one solution with \(s=s_{1}\), or at least two solutions with \(s< s_{1}\).
Proof
Let \(S_{i}=\{s\in\mathbb{R}\mid(5.1) \text{ has at least } i \text{ solutions}\}\) (\(i\geq1\)). We shall divide the proof into five steps to obtain the conclusion.
Step 1. We show that \(S_{1}\neq\emptyset\).
Set \(s^{\star}<\min_{k\in[2,N1]_{\mathbb{Z}}}f(k,0,0)\). From (5.2), there exists \(R^{\star}<0\) such that \(\max_{k\in[2,N1]_{\mathbb{Z}}}f(k, R^{\star},0)< s^{\star}\). Hence, \(\alpha\equiv R^{\star}\) is a strict lower solution and \(\beta\equiv0\) is a strict upper solution for (5.1) with \(s=s^{\star}\). \(s^{\star}\in S_{1}\) follows from Theorem 4.2.
Step 2. We prove that
Let \(\hat{\mathbf{u}}\) be a solution of (5.1) with \(s=s_{0}\) and set \(s< s_{0}\). Then it is easy to see that \(\hat{\mathbf{u}}\) is a strict upper solution for (5.1). Take \(R_{0}<\hat{\mathbf{u}}^{}_{L}\) such that \(\max_{k\in[2,N1]} f(k,R_{0},0)< s\) and \(\alpha\equiv R_{0}\) is a strict lower solution for (5.1). Hence, \(s\in S_{1}\) by using Theorem 4.2.
Step 3. We claim that
Set \(s\in\mathbb{R}\) and suppose that (5.1) has a solution u. Then \(\Vert \Delta\mathbf{u} \Vert _{\infty}< a\) and (5.6) hold, which implies that \(s\leq c\) with \(c=\sup_{[2,N1]_{\mathbb{Z}}\times\mathbb{R}\times(a,a)}f\frac{\inf_{\mathbb{R}}h_{N}\sup_{\mathbb{R}}h_{1}}{N2}\). That is, \(s_{1}=\sup S_{1}\) is finite. Clearly, (5.7) yields that \(S_{1}\supset(\infty, s_{1})\).
Step 4. \(S_{2}\supset(\infty, s_{1})\).
For any \(s\in\mathbb{R}\), let \(\mathcal{A}(s,\cdot)\) be the fixed point operator in \(\mathbb{R}^{N}\) associated to (5.1). Take \(s_{2}< s_{1}< s_{3}\). It follows from Lemma 5.1 that there is a ρ such that each possible zero of \(I\mathcal{A}(s,\cdot)\) with \(s\in [s_{2},s_{3}]\) is \(\mathbf{u}\in B_{\rho}\). Furthermore, the Brouwer degree \(\operatorname{deg}(I\mathcal{A}(s,\cdot),B_{\rho},\mathbf{0})\) is well defined and does not depend upon \(s\in[s_{2},s_{3}]\). Notice that \(\mathbf{u}\mathcal{A}(s,\mathbf{u})\neq0\) for all \(\mathbf{u}\in\mathbb{R}^{N}\) by using (5.8), which means that \(\operatorname{deg}(I\mathcal{A}(s_{3},\cdot),B_{\rho},\mathbf{0})=0\), so that \(\operatorname{deg}(I\mathcal{A}(s_{2},\cdot),B_{\rho},\mathbf{0})=0\). By the excision property of the Brouwer degree, \(\operatorname {deg}(I\mathcal{A}(s_{2},\cdot),B_{\rho}',\mathbf{0})=0\) with \(\rho'>\rho\). Let \(s\in(s_{2},s_{3})\) and \(\hat{\mathbf{u}}\) be a solution of (5.1). Then \(\hat{\mathbf{u}}\) is a strict upper solution of (5.1) with \(s=s_{2}\). Choose \(R<\hat{\mathbf{u}}^{}_{L}\) such that \(\max_{k\in[2,N1]} f(k,R,0)< s_{2}\). Then R is a strict lower solution of (5.1) with \(s=s_{2}\). Subsequently, from Theorem 4.2, (5.1) has a solution in \(\Omega _{\hat{\mathbf{u}},R}\) with \(s=s_{2}\) and \(\operatorname{deg}(I\mathcal {A}(s_{2},\cdot),\Omega_{\hat{\mathbf{u}},R},\mathbf{0})=1\). Take \(\rho '\) large enough, it follows from the additivity property of the Brouwer degree [19] that
This implies that (5.1) has the second solution in \(B_{\rho'}\backslash \Omega_{\hat{\mathbf{u}},R}\) with \(s=s_{2}\).
Step 5. We claim that \(s_{1}\in S_{1}\).
Let \(\{\eta_{j}\}\) be a sequence in \(( \infty,s_{1})\) satisfying \(\lim_{j\to\infty}\eta_{j}=s_{1}\), and let \(\mathbf{u}^{j}\) be a solution of (5.1) with \(s=\eta_{j}\) given by Step 3. Then we deduce from Lemma 2.3 that
Applying Lemma 5.1, there exists \(\rho>0\) such that \(\Vert \mathbf {u}^{j} \Vert <\rho\) (\(j\geq1\)). Since \(\mathcal{A}\) is completely continuous, there is a subsequence of \(\mathbf{u}^{j}\), relabeling if necessary such that \(\mathbf{u}^{j}\to\mathbf{u}\in\mathbb{R}^{N}\) and satisfies \(\mathbf {u}=\mathcal{A}(s_{1},\mathbf{u})\). That is, u is a solution of (5.1) with \(s=s_{1}\). □
Last, we shall give a similar result for the following dual AmbrosettiProdi condition.
Theorem 5.3
Suppose that a continuous function f satisfies the coercivity condition
\(h_{1}, h_{N}:\mathbb{R}\to\mathbb{R}\) are continuous and satisfy the following conditions:
Then there exists \(s_{1}\in\mathbb{R}\) such that problem (5.1) has no solution with \(s< s_{1}\), at least one solution with \(s=s_{1}\), or at least two solutions with \(s>s_{1}\).
Corollary 5.4
Let \(g:[2,N1]_{\mathbb{Z}}\times\mathbb {R}^{2}\to\mathbb{R}\), \(p:[2,N1]_{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}\), \(h_{1}:\mathbb {R}\to\mathbb{R}, h_{N}:\mathbb{R}\to\mathbb{R}\) be continuous. If g is bounded on \([2,N1]_{\mathbb{Z}}\times\mathbb{R}\times(a,a)\) \(h_{1}\), \(h_{N}\) satisfy (5.3), (5.4) (resp. (5.11), (5.12)) and p satisfies
Then there exists \(s_{1}\in\mathbb{R}\) such that the following problem
has no solution with \(s>s_{1}\) (resp. \(s< s_{1}\)), at least one solution with \(s=s_{1}\), and at least two solutions with \(s< s_{1}\) (resp. \(s>s_{1}\)).
Example 5.5
Let \(\mathbf{e}=(e_{2},e_{3},\ldots,e_{N1})\in\mathbb{R}^{N2}\), \(m\in\mathbb{R}\), \(n<0\) (resp. \(n>0\)), \(p>0\) and \(q\geq0\). Then there exists \(s_{1}\in \mathbb{R}\) such that the problem
has no solution with \(s>s_{1}\) (resp. \(s< s_{1}\)), at least one solution with \(s=s_{1}\), and at least two solutions with \(s< s_{1}\) (resp. \(s>s_{1}\)), where \(h_{1},h_{N}:\mathbb{R}\to\mathbb{R}\) satisfy (5.3), (5.4) (resp.(5.11), (5.12)), \(\phi(s)=\frac{s}{\sqrt{1s^{2}}}\).
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by NSFC (No.11626188, No.11671322, No.11501451), Gansu provincial National Science Foundation of China (No.1606RJYA232) and NWNULKQN1516.
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LY and MR completed the main study, carried out the results of this article and drafted the manuscript, LB checked the proofs and verified the calculation. All the authors read and approved the manuscript.
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Lu, Y., Ma, R. & Lu, B. Existence and multiplicity of solutions for discrete NeumannSteklov problems with singular ϕLaplacian. Adv Differ Equ 2017, 312 (2017). https://doi.org/10.1186/s1366201713731
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DOI: https://doi.org/10.1186/s1366201713731
MSC
 34B15
 39A12
Keywords
 NeumannSteklov conditions
 discrete
 singular ϕLaplacian
 Brouwer degree