- Research
- Open Access
Boundary value problems for singular second order equations
- Alessandro Calamai^{1}Email author,
- Cristina Marcelli^{2} and
- Francesca Papalini^{2}
https://doi.org/10.1186/s13663-018-0645-0
© The Author(s) 2018
- Received: 7 May 2018
- Accepted: 31 July 2018
- Published: 3 September 2018
Abstract
Keywords
- Boundary value problems
- Nonlinear differential operators
- Φ-Laplacian operator
- Singular equation
- Nagumo condition
MSC
- 34B15
- 34B24
- 34C25
- 34L30
1 Introduction
In this framework, the existence results are usually obtained by means of a fixed point technique combined with the upper and lower solutions method. Another important tool to get a priori bounds for the derivatives of the solutions is a Nagumo-type growth condition on the function f. Let us observe that, when the nonlinear term a is present in the differential operator, some assumptions are required to the differential operator Φ, which in general is assumed to be homogeneous, or having at most linear growth at infinity.
According to our knowledge, very few papers have been devoted to this type of equations, just for a restricted class of nonlinearities f (see [11, 12]).
Our goal is to obtain existence results for the Dirichlet problem associated with (4), as well as for other boundary value problems with different boundary conditions, including, as particular cases, the classical periodic, Neumann, and Sturm–Liouville problems, but involving the (possibly vanishing) function k.
In order to obtain the existence result, we adopt a suitable combination of fixed point techniques applied to an auxiliary functional Dirichlet problem, and the method of lower and upper solutions (see Sect. 2). Our main growth assumption on the right-hand side f is a weak form of the Wintner–Nagumo condition similar to the one in (3).
The last part of the paper (see Sect. 4) is devoted to various types of boundary value problems, including the periodic problem, Neumann problem, and Sturm–Liouville problem, for which we derive the existence of a solution by applying the existence result for some auxiliary Dirichlet problems.
2 Auxiliary results
By a solution of problem (6) we mean a function \(u \in W^{1,p}(I)\), with \(u(0)=a\), \(u(T)=b\), such that \(\Phi\circ(k \cdot u')\in W^{1,1}(I)\) and \((\Phi(k(t)u'(t)))'=F_{u}(t)\) a.e. on I.
The following lemma will be used in the next existence result.
Lemma 2.1
Proof
The following existence result holds.
Proof
Let now \((x_{n})_{n}\) be a sequence in \(W^{1,p}(I)\) converging to \(x \in W^{1,p}(I)\). By the continuity of the operator F, we get that \((F_{x_{n}})_{n}\) converges to \(F_{x}\) in \(L^{1}(I)\) and, by (13), \((I_{x_{n}})_{n}\) converges to \(I_{x}\).
Claim 3: G is a compact operator. Let us fix a bounded set \(D\subset W^{1,p}(I)\). We have to show that \(G(D)\) is relatively compact, that is, for any sequence \((x_{n})_{n}\subset D\), the sequence \((G_{x_{n}})_{n}\) admits a subsequence converging in \(W^{1,p}(I)\).
This shows that \(G(D)\) is relatively compact in \(W^{1,p}(I)\).
By virtue of what we proved in Claims 1–3, we can apply the Schauder fixed point theorem to achieve the existence of a fixed point for the operator G, and this concludes the proof. □
3 Dirichlet problem
In this section we consider problem (P), where \(\Phi :\mathbb{R}\to\mathbb{R}\) is a generic strictly increasing homeomorphism, \(k:I\to\mathbb{R}\) is a continuous nonnegative function satisfying (5). Finally, \(f:I\times\mathbb{R}^{2}\to \mathbb{R}\) is a Carathéodory function, that is, the map \(t\mapsto f(t,x,y)\) is measurable on I for every \((x,y)\in\mathbb{R}^{2}\), and the map \((x,y)\mapsto f(t,x,y)\) is continuous on \(\mathbb{R}^{2}\) for a.e. \(t\in I\).
The main result of the paper is the following existence theorem.
Theorem 3.1
Assume the existence of a pair of lower and upper solutions \(\sigma, \tau\in\mathcal {W}_{p}\) of the equation in (P), satisfying \(\sigma(t) \le\tau(t)\) for every \(t \in\mathbb{R}\).
Then, for every \(a,b\) such that \(\sigma(0)\le a\le\tau(0)\), \(\sigma (T)\le b\le\tau(T)\), problem (P) has a solution \(u_{a,b} \in\mathcal {W}_{p}\) such that \(\sigma(t)\le u_{a,b}(t)\le\tau(t)\) for every \(t \in I\).
Proof
Let us now fix a point \(t \notin\partial(I^{+}) \cup\partial(I^{-})\) such that the derivatives \(\sigma'(t)\), \(\tau'(t)\), \(x'(t)\), and \(x_{n}'(t)\), for all \(n\in\mathbb{N}\), exist. If \(t\in I^{+}\), then for n sufficiently large \(x_{n}(t)>\tau(t)\) too. Hence, in this case \(\mathcal{U}_{x_{n}}'(t)=\mathcal{U}_{x}'(t)\). Similarly, if \(t \in I^{-}\) again, \(\mathcal{U}_{x_{n}}'(t)=\mathcal {U}_{x}'(t)\) for n sufficiently large. Moreover, if \(t\in I^{\circ}\), then for large n we have also \(\sigma(t)< x_{n}(t)<\tau(t)\), and so \(\mathcal{U}_{x_{n}}'(t)=x_{n}'(t)\) and \(\mathcal{U}_{x}'(t)=x'(t)\). Therefore, \(\mathcal{U}_{x_{n}}'(t)\to\mathcal{U}_{x}'(t)\) for a.e. \(t\in I^{+}\cup I^{-}\cup I^{\circ}\).
Finally, if \(t\notin I^{+}\cup I^{-}\cup I^{\circ}\cup\partial(I^{+})\cup \partial(I^{-})\), then \(x(\theta)=\sigma(\theta)\) (or \(x(\theta)=\tau(\theta)\)) for θ in a neighborhood J of t. If \(x(\theta)=\sigma(\theta )\) in J, then \(x'(\theta)=\sigma'(\theta)\) in J and since \(\mathcal{U}_{x_{n}}'(t)\in\{ x_{n}'(t), \sigma'(t)\}\), we have \(\mathcal{U}_{x_{n}}'(t)\to x'(t)=\mathcal{U}_{x}'(t)\). One can reason similarly when \(x(\theta)=\tau(\theta)\) in J.
Consequently, Theorem 2.2 applies yielding a solution \(u \in \mathcal {W}_{p}\) of problem (32).
Claim 2: The solution u of problem ( 32 ) verifies \(\sigma(t) \le u(t) \le\tau(t)\) for all \(t \in I\). Let us show that \(\sigma(t) \le u(t)\) for all \(t \in I\), the other inequality being analogous.
Claim 4: The solution u of problem ( 32 ) verifies \(|k(t) u'(t)| \le L\) for all \(t \in I\). Assume by contradiction that this does not hold; then one of the following is true: either \(\max\{k(t) u'(t): t\in I\} >L\) or \(\min\{k(t) u'(t): t\in I\} <-L\).
Similarly, one can prove that the case \(\min\{k(t) u'(t): t\in I\} <-L\) leads to a contradiction, and Claim 4 follows.
Finally, by what we have proved in Claims 2 and 4, we deduce (28). □
Remark 3.2
Let us observe that, if \(k(t) >0\) for all \(t \in I\), then the solution u of problem (P) is actually a \(C^{1}\) function. This follows from the fact that, if \(k(t) >0\), all the fixed points of the operator G defined in (12) are of class \(C^{1}(I)\).
Remark 3.3
Notice that in the Wintner–Nagumo condition (27) the function ψ could be chosen as a constant. When this is possible (that is, when the growth of the right-hand side f with respect the variable y is, at most, linear), then condition (27) does not require any relation among the differential operator Φ, the function \(k(t)\) appearing inside Φ, and the function f. Instead, when f has a superlinear growth in the variable y, then condition (27) implies a link between the rates of growth of Φ, f (with respect to y), and the exponent p. This is illustrated in the following examples.
Example 3.4
Therefore, for every \(a,b\in[-N,N]\), there exists a solution of problem (44).
We provide now an application of Theorem 3.1 for a rather general right-hand side, with possible superlinear growth with respect to \(u'\).
Corollary 3.5
Then problem (45) admits solutions for every \(a,b\in \mathbb{R}\).
Proof
Remark 3.6
4 General nonlinear boundary conditions
The result established for Dirichlet problems can be applied to obtain existence results also for more general boundary conditions, as already showed in [3] and then in [14, Sect. 4].
The key ingredient is a compactness-type result for the solutions of Dirichlet problems (see [14, Lemma 1]) that in the present more general framework of weak solutions, belonging to \(W^{1,p}\), has to be reformulated as follows.
Lemma 4.1
Proof
We can assume without loss of generality, by passing to subsequences, that \(a_{n}\to a_{0}\), \(b_{n}\to b_{0}\). Set now \(z_{n}(t):= (\Phi(k(t)u_{n}'(t)))' \); from (26), we have \(|z_{n}(t)|\le h_{M,\gamma_{0}}(t)\) a.e. \(t \in I\). Hence, the two sequences \((u'_{n})_{n}\) and \((z_{n})_{n}\) are both uniformly integrable. Thus, by applying the Dunford–Pettis theorem, we deduce the existence of two subsequences \((u'_{n_{k}})_{k}\) and \((z_{n_{k}})_{k}\) such that \(u'_{n_{k}}\rightharpoonup g\) and \(z_{n_{k}}\rightharpoonup h\) weakly in \(L^{1}(I)\) for some \(g,h\in L^{1}\). Moreover, since \(|k(0) \cdot u_{n}'(0)|< L\) for every \(n\in\mathbb{N}\), we can also assume that \(k(0) \cdot u_{n}'(0)\to y_{0}\) for some \(y_{0}\in \mathbb{R}\).
Remark 4.2
Let us observe that we consider “weighted” boundary conditions involving \(k(0)u'(0)\), \(k(T)u'(T)\) since we look for solutions in the set \(\mathcal {W}_{p}\), that is, functions \(u \in W^{1,p}(I)\) with \(k \cdot u' \in C(I)\).
As in [14, Theorem 3] one can prove an existence result for the general problem (51).
Theorem 4.3
As an immediate consequence of Theorem 4.3, the following existence result follows.
Theorem 4.4
Theorem 4.5
Then problem (55) has a solution \(u \in\mathcal {W}_{p}\) such that \(\sigma(t)\le u(t)\le\tau(t)\) for every \(t \in I\).
5 Results and discussion
The main novelty consists in the introduction of the function \(k(t)\) inside the operator Φ, which can vanish in such a way that the equation becomes singular. To handle this kind of problem, we widen the space of solutions, choosing the more appropriate class of Sobolev functions. The proof of our results is based on the Schauder fixed point theorem.
6 Conclusions
7 Methods
Not applicable.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Authors’ information
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Funding
Not applicable.
Authors’ contributions
All authors read and approved the final manuscript. The contribution of the authors is equal.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Bereanu, C., Mawhin, J.: Periodic solutions of nonlinear perturbations of Φ-Laplacians with possibly bounded Φ. Nonlinear Anal., Theory Methods Appl. 68, 1668–1681 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Cabada, A., O’Regan, D., Pouso, R.L.: Second order problems with functional conditions including Sturm–Liouville and multipoint conditions. Math. Nachr. 281, 1254–1263 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Cabada, A., Pouso, R.L.: Existence results for the problem \((\phi(u^{\prime})) ^{\prime}=f(t,u,u^{\prime})\) with periodic and Neumann boundary conditions. Nonlinear Anal., Theory Methods Appl. 30, 1733–1742 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Cabada, A., Pouso, R.L.: Existence results for the problem \((\phi(u^{\prime}))^{\prime}=f(t,u,u^{\prime})\) with nonlinear boundary conditions. Nonlinear Anal., Theory Methods Appl. 35, 221–231 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Calamai, A.: Heteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators. J. Math. Anal. Appl. 378(2), 667–679 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Cupini, G., Marcelli, C., Papalini, F.: Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators. Differ. Integral Equ. 24(7–8), 619–644 (2011) MathSciNetMATHGoogle Scholar
- El Khattabi, N., Frigon, M., Ayyadi, N.: Multiple solutions of boundary value problems with ϕ-Laplacian operators and under a Wintner–Nagumo growth condition. Bound. Value Probl. 2013, 236 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Ferracuti, L., Marcelli, C., Papalini, F.: Boundary value problems for highly nonlinear inclusions governed by non-surjective Φ-Laplacians. Set-Valued Var. Anal. 19(1), 1–21 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Ferracuti, L., Papalini, F.: Boundary value problems for strongly nonlinear multivalued equations involving different Φ-Laplacians. Adv. Differ. Equ. 14, 541–566 (2009) MATHGoogle Scholar
- Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9. Chapman & Hall/CRC, Boca Raton (2006) MATHGoogle Scholar
- Liu, Y.: Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line. Nonlinear Anal., Model. Control 17, 460–480 (2012) MathSciNetMATHGoogle Scholar
- Liu, Y., Yang, P.: Existence and non-existence of positive solutions of BVPs for singular ODEs on whole lines. Kyungpook Math. J. 55, 997–1030 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Marcelli, C.: The role of boundary data on the solvability of some equations involving non-autonomous, nonlinear differential operators. Bound. Value Probl. 2013, 252 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Marcelli, C., Papalini, F.: Boundary value problems for strongly nonlinear equations under a Wintner–Nagumo growth condition. Bound. Value Probl. 2017, 183 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Minhós, F.: Sufficient conditions for the existence of heteroclinic solutions for ϕ-Laplacian differential equations. Complex Var. Elliptic Equ. 62, 123–134 (2017) MathSciNetView ArticleMATHGoogle Scholar