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Bernsteintype theorem for ϕLaplacian
Fixed Point Theory and Applicationsvolume 2019, Article number: 1 (2019)
Abstract
In this paper we obtain a solution to the secondorder boundary value problem of the form \(\frac{d}{dt}\varPhi'(\dot{u})=f(t,u,\dot{u})\), \(t\in [0,1]\), \(u\colon \mathbb {R}\to \mathbb {R}\) with Sturm–Liouville boundary conditions, where \(\varPhi\colon \mathbb {R}\to \mathbb {R}\) is a strictly convex, differentiable function and \(f\colon[0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) is continuous and satisfies a suitable growth condition. Our result is based on a priori bounds for the solution and homotopical invariance of the Leray–Schauder degree.
Introduction
In this paper we study the existence of solutions to the boundary value problems (BVPs)
where \(\varPhi '\) is an increasing homeomorphism, the scalar function f is continuous, \(\alpha ,a >0\) and \(\beta , b\geq 0\).
The solvability of various secondorder twopoint BVPs with p or ΦLaplacian has been discussed extensively in the literature, see the recent works [1,2,3,4,5,6,7,8,9] for results, methods, and references.
In 1912, Bernstein [10] proved that the BVP
has a unique \(C^{2}\)solution if \(f(t,u,v)\) is continuous, has continuous partial derivatives \(f_{u}\) and \(f_{v}\) on \([0,1]\times \mathbb {R}^{2}\), there is a constant \(K>0\) such that
and
where A, B are functions bounded on each compact subset of \([0,1]\times \mathbb {R}\).
In 1978, Granas et al. [11] proved similar results for (1) with either Dirichlet, Neumann, or periodic boundary conditions. The authors have established the existence of solutions to the considered problems by replacing (3) with the following assumption: There is a constant \(M>0\) such that
The uniqueness of the solution to (1.4), (BC) follows from the assumption that the partial derivatives \(f_{u}\) and \(f_{v}\) exist, are bounded, and \(f_{u}\geq 0\) on \([0,1]\times \mathbb {R}^{2}\).
In 1983, Baxley [12] proved Bernsteintype theorems for boundary value problems for (1) with nonlinear boundary conditions. In 1988, Frigon and O’Regan [13] established existence results of this type for (1), (2) and (1), (BC).
The aim of this paper is to give Bernsteintype existence theorems for BVPs with ΦLaplacian. Throughout this paper we assume that \(\varPhi \colon \mathbb {R}\to \mathbb {R}\) satisfies the following conditions:
 (\(\varPhi _{1}\)):

Φ is strictly convex, differentiable and \(\varPhi (x)/\vert x\vert \to \infty \) as \(\vert x\vert \to \infty \);
 (\(\varPhi _{2}\)):

\(\varPhi (0) = \varPhi '(0) = 0\);
 (\(\varPhi _{3}\)):

\((\varPhi ')^{1}\) is continuously differentiable;
 (\(\varPhi _{4}\)):

there exists a constant \(K_{\varPhi }>1\) such that
$$ K_{\varPhi }\varPhi (x)\leq \varPhi '(x) x\quad \text{for all $x\in \mathbb {R}$.} $$
We assume also that \(f\colon [0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) is continuous and satisfies the following:
 (\(f_{1}\)):

There exists a constant \(M>0\) such that
$$ x f(t,x,0) > 0 \quad \text{for $ \vert x \vert >M$,} $$  (\(f_{2}\)):

There exist positive functions S, T bounded on bounded sets such that
$$ \bigl\vert f(t,x,v) \bigr\vert \leq S(t,x) \bigl(\varPhi '(v) \cdot v\varPhi (v)\bigr) + T(t,x). $$
Now, we can state our main result.
Main Theorem
Suppose that Φ and f satisfy (\(\varPhi _{1}\))–(\(\varPhi _{4}\)) and (\(f_{1}\)), (\(f_{2}\)), respectively. Then problem (P), (BC) has at least one solution in \(C^{2}([0,1],\mathbb {R})\).
To establish the validity of the above result, we apply the Leray–Schauder degree theory on a suitable constructed map. To define its domain, we use a priori bounds.
To prove the existence, we use topological methods. This approach has already been used by many authors. In [11] and [13] the authors considered the case of a Laplace operator with various boundary conditions. Generalizations to the pLaplacian and to the operator defined by an arbitrary increasing homeomorphism were developed in [3] and [5], respectively. The main idea in the paper [11] was to use the topological transversality theorem. This is a fixed point type theorem (see [15]). We decided to use an approach via Leray–Schauder degree theory instead, since it is essentially equivalent but the degree theory is familiar to a broader audience.
However, in [3] and [5] authors subject the equation to very specific boundary conditions, namely \(u(0) = A\), \(\dot{u}(1) = B\). In order to show the existence for general Sturm–Liouville conditions, more effort has to be put in as can be seen below.
Auxiliary results
Lemma 2.1
Let X be a metric space, and let \(G\colon X \times \mathbb {R}\to \mathbb {R}\) be continuous. Suppose that

(1)
for every \(v \in X\), function \(g_{v}\colon \mathbb {R}\to \mathbb {R}\), defined by \(g_{v}(c) = G(v,c)\), is an increasing homeomorphism;

(2)
if \(\{v_{n}\}\) is bounded and \(b_{n} \to \pm \infty \), then \(G(v_{n},b_{n}) \to \pm \infty \).
Then, for each fixed constant \(C \in \mathbb {R}\), the function \(c\colon X \to \mathbb {R}\) defined by \(G(v,c(v)) = C\) is continuous.
Proof
Suppose that function c is not continuous, i.e., there exist \(\epsilon > 0\) and a sequence \(v_{n}\) converging to some \(v_{0}\) such that \(\vert c(v_{n})  c(v_{0})\vert > \epsilon \). By the definition of c, \(G(v_{n},c(v_{n})) = C\). In particular, both \(v_{n}\) and \(G(v_{n},c(v _{n}))\) are bounded. This, together with (2), implies that \(c(v_{n})\) is bounded. Take a subsequence \(c(v_{n_{k}})\) which converges to some \(c'\). Note that \(c' \neq c(v_{0})\) because \(\vert c(v_{n})  c(v_{0})\vert > \epsilon \). By the continuity of G, we have \(G(v_{n_{k}}, c(v_{n _{k}})) \to G(v_{0},c')\). But \(G(v_{n_{k}}, c(v_{n_{k}})) = C\) and \(G(v_{0},c') \neq G(v_{0},c(v_{0})) = C\) by (1). A contradiction. □
If \(g_{v}\) is differentiable and \(g_{v}'\) is positive, then the conclusion follows from implicit function theorem. However, in the problem that we consider, \(g'_{v}\) is only nonnegative.
Remark 2.2
Note that this trivializes in [3, 5]. For boundary conditions considered therein \(c_{1}\) and \(c_{2}\) are constants independent of v. We cannot proceed in such a way here.
Now introduce the map \(\hat{K}\colon C^{0}([0,1]) \times \mathbb {R}\times \mathbb {R}\to C^{1}([0,1])\) defined by
and by \(C^{1}_{\mathrm{BC}}([0,1])\) denote the set of the functions in \(C^{1}([0,1])\) which satisfies (BC).
For every v, we would like to choose \(c_{1}\) and \(c_{2}\) in such a way that \(u = \hat{K}(v,c_{1},c_{2})\) is an element of \(C^{1}_{\mathrm{BC}}\). Moreover, we need that \(c_{1}\) and \(c_{2}\) depend continuously on v.
Lemma 2.3
Let (\(\varPhi _{1}\)) and (\(\varPhi _{3}\)) hold. Then, for every fixed \(v \in C^{0}([0,1])\), there exists a unique pair of constants \(c_{1}(v)\), \(c_{2}(v)\) such that \(\hat{K}(v,c_{1}(v),c_{2}(v)) \in C^{1}_{\mathrm{BC}}([0,1])\). Moreover, the functions \(c_{1},c_{2}\colon C ^{0}([0,1]) \to \mathbb {R}\) are continuous.
Proof
Put \(u = K(v,c_{1},c_{2})\). Then
and
Clearly, u will satisfy the boundary conditions (BC) if \(c_{1}\) and \(c_{2}\) are such that \(\alpha c_{1}+\beta (\varPhi ')^{1}(c _{2})=A \) and
from where we get
Since \((\varPhi ')^{1}\) is increasing, the function
is increasing with respect to c. We can apply Lemma 2.1 for \(C=B\) to conclude that (4) defines a unique constant \(c_{2}\) depending continuously on v, and so \(c_{1}\) is also unique and depends continuously on v. □
Now, for \(\lambda \in [0,1]\), consider the family of differential equations
Note that if u is a \(C^{1}\) solution to problem \((P_{\lambda })\), then \(u \in C^{2}\). Indeed, u̇ reads
and by assumption (\(\varPhi _{3}\)) and the continuity of f, it is continuously differentiable.
The next lemma is a variant of [13, Theorem 3.3].
Lemma 2.4
Assume that (\(\varPhi _{1}\))–(\(\varPhi _{3}\)) and (\(f_{1}\)) hold. Let \(u\in C^{1}([0,1])\) be a solution to \((P_{\lambda })\), (BC) for \(\lambda \in [0,1]\). If \(\vert u\vert \) achieves its maximum at \(t_{0}\in (0,1)\), then
Proof
Suppose on the contrary that \(\vert u\vert \) achieves its maximum at \(t_{0} \in (0,1)\). We can assume that \(u(t_{0})>M\). In the case \(u(t_{0}) \leq M\) the proof is similar. It is clear that \(\dot{u}(t_{0})=0\). For \(t\in [0,1]\) we have
Since \(\varPhi '(\dot{u}(t_{0}))=\varPhi '(0)=0\),
and
Combining the above, we get
Hence, using \((P_{\lambda })\), we have
Note that, for \(0<\lambda \leq 1\), \(xf(t,x,0)>0\), \(\vert x\vert >M\) implies \(\lambda xf(t,x,0)>0\), \(\vert x\vert >M\). Thus, by assumption (\(f_{1}\)), \(\lambda u(t_{0})f(t_{0},u(t_{0}),0)>0\). The continuity of f, u, and u̇ implies that there exists a neighborhood \(N\subset (0,1)\) of \(t_{0}\) such that
Since \(u\in C^{1}\) and achieves its maximum at \(t_{0}\), there exist \(t_{0}^{}\) and \(t_{0}^{+}\) such that

\(u(t)>M\) for \(t\in (t_{0}^{},t_{0}^{+})\),

\(\dot{u}(t)\geq 0\) on \((t_{0}^{},t_{0}]\),

\(\dot{u}(t)\leq 0\) on \([t_{0},t_{0}^{+})\).
Hence \(\varPhi '(\dot{u}(t))\geq 0\) for \(t\in (t_{0}^{},t_{0}]\) and \(\varPhi '(\dot{u}(t))\leq 0\) for \(t\in [t_{0},t_{0}^{+})\), since \(\varPhi '\) is increasing. This implies that
and
It follows that for t close to \(t_{0}\)
a contradiction. Thus \(u(t_{0})\leq M\). □
Lemma 2.5
Assume that (\(\varPhi _{1}\))–(\(\varPhi _{3}\)) and (\(f_{1}\)) hold. Let \(u\in C^{1}([0,1])\) be a solution to \((P_{\lambda })\), (BC) for \(\lambda \in [0,1]\). There exists a constant \(M_{0}>0\) independent of λ and u such that
Proof
For \(\lambda =0\), problem \((P_{\lambda })\) has a unique linear solution, so there is a constant \(C>0\) such that \(\vert u(t)\vert \leq C\) for \(t\in [0,1]\). Let \(0<\lambda \leq 1\). If \(\vert u\vert \) achieves its maximum at \(t=0\), then \(u(0)\dot{u}(0)\leq 0\). The boundary conditions give
and consequently \(\vert u(0)\vert \leq \vert A/\alpha \vert \). Similarly, \(\vert u(1)\vert \leq \vert B/a\vert \). If the maximum is at any \(t_{0}\in (0,1)\), then by Lemma 2.4 we get \(\vert u(t)\vert \leq M\). As a result, for \(\lambda \in [0,1]\), we have
□
Now we provide bounds for u̇. The proof of the following theorem is based on [13].
Lemma 2.6
Assume that (\(\varPhi _{1}\))–(\(\varPhi _{4}\)), (\(f_{1}\)), and (\(f_{2}\)) hold. Let \(u\in C^{1}([0,1])\) be a solution to \((P_{\lambda })\), (BC) for \(\lambda \in [0,1]\). There exists a constant \(M_{1}>0\), independent of λ and u, such that
Proof
Since we have obtained a priori bounds \(\vert u(t)\vert \leq M_{0}\), it is easy to observe that there exists a constant \(C\geq 0\) independent of λ and u such that
for some \(t_{0}\in [0,1]\). The point \(t_{0}\) belongs to an interval \([\mu ,\nu ]\subset [0,1]\) such that the sign of \(\dot{u}(t)\) does not change in \([\mu ,\nu ]\) and \(\dot{u}(\mu )=\dot{u}(t_{0})\) and/or \(\dot{u}(\nu )=\dot{u}(t_{0})\).
Assume that \(\dot{u}(\mu )=\dot{u}(t_{0})\) and \(\dot{u}(t)\geq 0\) for every \(t\in [\mu ,\nu ]\). The other cases are treated similarly and the same bound is obtained.
Denote by \(S_{0}\), \(T_{0}\) the upper bounds of S and T, respectively, on \([0,1]\times [M_{0},M_{0}]\). Since
we have
For \(\mu \leq \tau \leq t\), we have
Note that since Φ is a convex differentiable function and \(\varPhi (0)=0\), we have \(\varPhi (x)\leq \varPhi '(x)x\) for every \(x\in \mathbb {R}\). Thus, \(0 \leq S_{0} (\varPhi '(\dot{u}(\mu ))\dot{u}(\mu )\varPhi ( \dot{u}(\mu )) )\). On the other hand, there exists \(C_{0}\geq 0\) such that \(S_{0} (\varPhi '(\dot{u}(\mu ))\dot{u}(\mu )\varPhi ( \dot{u}(\mu )) )+T_{0} \leq C_{0}\). Hence,
Set \(g(\tau ) = S_{0} \int _{\mu }^{\tau }\dot{u} \vert \frac{d}{dt} \varPhi '(\dot{u}) \vert \,d\sigma + C_{0}\), then integration by substitution yields
Thus
and by (\(\varPhi _{4}\))
The last inequality gives \(\vert \dot{u}(t)\vert \leq M_{1}\) for all \(t\in [0,1]\). □
Proof of the main theorem
Introduce the map \(N\colon C^{1}_{\mathrm{BC}}([0,1]) \to C^{0}([0,1])\) defined by
and for \(\lambda \in [0,1]\) consider the composition \(\widehat{K} \circ \lambda N\), where the map \(\widehat{K}\colon C([0,1])\to C^{1}([0,1])\) is well defined by Lemma 2.3. Moreover, by the Arzela–Ascoli theorem, K̂ is compact. Since N is continuous, the composition \(\hat{K} \circ \lambda N\) is also compact.
The fixed points of \(\widehat{K}\circ \lambda N\) are of interest to us. Instead of looking for fixed points of \(K\circ N\), one can look for zeros of \(\mathit{Id}  K \circ N\). For this we will use the Leray–Schauder degree and its homotopical invariance. Consider the homotopy \(H\colon [0,1] \times C^{1}_{\mathrm{BC}}([0,1]) \to C^{1}_{\mathrm{BC}}([0,1])\) given by
Observe first that \(H(0,l)=0\), where \(l=\widehat{K}(0)\) is unique. Thus, if \(\overline{B}_{r}(0)\) is a closed ball with center 0 and radius r with the property \(l\in B_{r}(0)\), then
It is well known that if r is such that
then also
It is not hard to check that \(u\in C^{1}_{\mathrm{BC}}([0,1])\) is a zero of \(\mathit{Id}\widehat{K}\circ \lambda N\) if and only if u is a solution to BVP \((P_{\lambda })\), (BC). Thus, each zero \(u\in C^{1} _{\mathrm{BC}}([0,1])\) of \(\mathit{Id}\widehat{K}\circ \lambda N\) satisfies the bound
where \(K=\max \{M_{0},M_{1}\}+1\), where \(M_{0}\) and \(M_{1}\) are the constants from Lemmas 2.5 and 2.6. Clearly, (5) holds for \(r=K\) and so, since in particular \(l\in B_{K}(0)\), we have
This means that \(\mathit{Id}\widehat{K}\circ \lambda N\) has at least one zero \(u_{0}\in B_{K}(0)\), which is a \(C^{1}([0,1])\)solution to BVP of family \((P_{\lambda })\) arisen when \(\lambda =1\), that is, \(u_{0}\) is a \(C^{1}([0,1])\)solution to (P), (BC). However, as a solution of (P), \(u_{0}\) is such that, for some constant c, we have
from where, keeping in mind (\(\varPhi _{3}\)) and the continuity of f, we get \(u_{0}\in C^{2}([0,1])\).
Examples
Example 4.1
Let \(\varPhi \colon \mathbb {R}\to \mathbb {R}\), \(\varPhi (x)=\frac{1}{p}\vert x\vert ^{p}\), \(1< p\leq 2\). It is easy to see that the function Φ satisfies assumptions (\(\varPhi _{1}\))–(\(\varPhi _{3}\)). Moreover, since \(\frac{1}{p}\vert x\vert ^{p}\leq x^{2}\vert x\vert ^{p2}\), one can take \(K_{\varPhi }=p\) in (\(\varPhi _{4}\)).
Define the function \(f:\mathbb {R}\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) by the formula
One can easily check that
for \(\vert x\vert >1\). Since \(\varPhi '(v)v\varPhi (v)=\frac{1p}{p}\vert v\vert ^{p}\) and
assumption (\(f_{2}\)) is satisfied with \(S(t,x)=\frac{p\vert x^{3}x}{(p1)(1+t ^{2})}\) and \(T(t,x)=\frac{\vert x^{3}x\vert }{1+t^{2}}\).
Assume that Φ satisfies assumptions (\(\varPhi _{1}\))–(\(\varPhi _{4}\)) and that the functions \(S,T\colon \mathbb {R}\times \mathbb {R}\to (0,\infty )\) are continuous and such that \(x T(t,x)>0\) for \(\vert x\vert >M\). Then the function
satisfies our assumptions.
Example 4.2
Let \(\varPhi (x)=\sum_{i=1}^{n}\frac{1}{p_{i}}\vert x\vert ^{p_{i}}\), \(1< p_{i} \leq 2\), for \(i=1,2,\ldots,n\). Then Φ satisfies assumptions (\(\varPhi _{1}\))–(\(\varPhi _{3}\)). Assumption (\(\varPhi _{4}\)) is satisfied with \(K_{\varPhi }=\min \{p_{1},\ldots,p _{n}\}\).
Example 4.3
Let \(\varPhi (x)=\frac{1}{p}\vert x\vert ^{p} \log (1+x^{2})\). The function Φ satisfies all assumptions. In particular, as \(\varPhi '(x)=x\vert x\vert ^{p2} \log (1+x^{2})+\frac{1}{p}\vert x\vert ^{p}\frac{2x}{1+x^{2}}\) and
we can take \(K_{\varPhi }=p\). One can also consider functions of the form \(\varPhi (x)=\frac{1}{p}\vert x\vert ^{p}\log ^{r}(1+\vert x\vert ^{s})\) for suitable choice of p, r, s.
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Acknowledgements
The authors would like to thank Professor Andrzej Granas for suggesting this problem during the seminar in Gdańsk in 2016.
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Funding
M. Starostka was partially supported by Grants Beethoven2 and Preludium9 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081 and no. 2015/17/N/ST1/02527.
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Correspondence to Jakub Maksymiuk.
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Keywords
 ΦLaplacian
 Boundary value problem
 Fixed point
 A priori bounds
 Leray–Schauder degree