Positive decaying solutions for differential equations with phi-Laplacian
- Zuzana Došlá^{1}Email author and
- Mauro Marini^{2}
Received: 4 February 2015
Accepted: 19 May 2015
Published: 12 June 2015
Abstract
We solve a nonlocal boundary value problem on the half-close interval \([1,\infty)\) associated to the differential equation \((a(t)\vert x^{\prime} \vert ^{\alpha} \operatorname {sgn}x^{\prime} )^{\prime }+b(t)\vert x\vert ^{\beta} \operatorname {sgn}x=0\), in the superlinear case \(\alpha<\beta\). By using a new approach, based on a special energy-type function E, the existence of slowly decaying solutions is examined too.
Keywords
MSC
1 Introduction
Equations (1) and (3) arise in the study of radially symmetric solutions of elliptic differential equations with phi-Laplacian operator in \(\mathbb{R}^{3}\); see, e.g., [1, 2].
By a solution of (3) we mean a differentiable function x on an interval \(I_{x}\subseteq{}[1,\infty)\), such that \(a(\cdot)\vert x^{\prime }(\cdot)\vert ^{\alpha}\) is continuously differentiable and satisfies (3) on \(I_{x}\). In addition, x is called local solution if \(I_{x}\) is bounded and proper solution if \(I_{x}\) is unbounded and \(\sup \{ \vert x(t)\vert :t\geq T \} >0\) for any large \(T\geq1\). As usual, a proper solution of (3) is said to be oscillatory if it has a sequence of zeros tending to infinity, otherwise it is said to be nonoscillatory. Equation (3) is said to be oscillatory if any its proper solution is oscillatory.
Let \(x, y\in\mathbb{P}\) satisfy (6), (7), respectively. Then x, y tend to zero as \(t\rightarrow\infty\) and \(0< y(t)< x(t)\) for large t. Hence, proper solutions of (3) satisfying (6) are called slowly decaying solutions, and proper solutions satisfying (7) strongly decaying solutions.
Using a suitable change of variable and certain monotonicity properties of a energy-type function E, we prove that (8) has infinitely many solutions. Consequently, we get also a global multiplicity existence result for slowly decaying solutions of (3), which are positive decreasing on the whole interval \([1,\infty)\). We recall that in the superlinear case \(\alpha<\beta\), sufficient conditions for existence of slowly decaying solutions are difficult to establish, due to the problem to find sharp upper and lower bounds; see, e.g., [4], p.241, [5], p.3.
Observe that necessary and sufficient conditions for existence of solutions of (3), which satisfy (5) or (7), can easily be produced; see, e.g., [3, 6] or [7], Section 14. Moreover, the same is true for slowly decaying solutions in the sublinear case \(\alpha>\beta\); see, e.g., [2, 8]. In the opposite situation, that is, in the superlinear case \(\alpha<\beta\), in spite of many examples of equations of type (3) having solutions of type (6), which can be easily produced, until now no general sufficient conditions for their existence are known.
The paper is completed by the solvability of a special BVP, in which also the initial starting point is fixed. Moreover, some examples and suggestions for future research complete the paper.
Recently, BVPs on infinite intervals, associated to equations with phi-Laplacian have been considered in [11, 12]. The case of nonlocal BVPs for the generalized Laplacian, has been studied, e.g., in [1, 13]. Finally, we refer the reader to [4, 14] for other references on this topic.
2 Preliminaries
We start with a change of the independent variable in (3), which will be useful.
Lemma 1
Proof
The following result is needed in the following.
Lemma 2
- (i)
- (ii)
Equation (1) has solutions \(x\in\mathbb{P}\) which satisfy \(\lim_{t\rightarrow\infty}tx(t)=\ell_{x}\), \(0<\ell_{x}<\infty\) if and only if \(Y<\infty\). Moreover, for any \(\ell_{x}\), \(0<\ell_{x}<\infty\), there exists \(x\in\mathbb{P}\) such that \(\lim_{t\rightarrow\infty}tx(t)=\ell_{x}\).
- (iii)
Equation (1) is oscillatory if and only if \(Y=\infty\).
3 The main result
Remark 1
In the superlinear case \(\alpha<\beta\), in virtue of (16), any local solution of (1) is a solution, i.e. it is continuable to infinity and is proper; see, e.g., [16], Theorem 3.2, or [15], Appendix A. Notice also that, under the weaker assumption \(b(t)\geq0\), \(\sup \{ b(t):t\geq T \} >0\) for any \(T\geq1\), there may exist equations of type (1) with uncontinuable solutions; see, e.g., [16], p.343.
The following holds.
Theorem 1
Lemma 3
Proof
Lemma 4
Proof
Proof
Proof of Theorem 1
From Lemma 4, the function \(E_{x}\) is nonincreasing on \([1,\infty)\) for any solution x of (1).
Let us show that x and \(x^{\prime}\) cannot have zeros for \(t\geq1\). By contradiction, if there exists \(t_{1}>1\) such that \(x(t_{1})=0\), then, in virtue of the uniqueness with respect to the initial data, we have \(x^{\prime }(t_{1})\neq0\). Hence \(E(t_{1})>0\), which contradicts (28). Similarly, if there exists \(t_{2}\geq1\) such that \(x^{\prime}(t_{2})=0\), we obtain \(E(t_{2})>0\), which is again a contradiction.
Hence, x is a solution of (BVP). Since there are infinitely many solutions which satisfy (26) with the choice of m taken with (27), the proof is now complete. □
From Theorem 1 and its proof, we get the following.
Corollary 1
Under assumptions of Theorem 1, (1) has infinitely many slowly decaying solutions, which are positive decreasing on the whole interval \([1,\infty)\). Moreover, (1) has also infinitely many strongly decaying solutions and every nonoscillatory solution of (1) tends to zero as \(t\rightarrow\infty\).
Proof
In virtue of Theorem 1 and its proof, the boundary value problem (BVP) is solvable by every solution x which satisfies (26) and (27). Clearly, these solutions are slowly decaying solutions.
Consequently, (1) has nonoscillatory solutions and, in view of Lemma 2(iii) we get \(Y<\infty\). Then the existence of infinitely many strongly decaying solutions follows from Lemma 2(ii). □
When the monotonicity condition (17) is valid only for large t, reasoning as in the proof of Theorem 1, we obtain the following.
Corollary 2
Assume \(Z=\infty\). If (16) is satisfied and the function G, given in (17) is nonincreasing for any large t, then (1) has infinitely many slowly decaying solutions, which are eventually positive decreasing. Moreover, (1) has also infinitely many strongly decaying solutions and every nonoscillatory solution of (1) tends to zero as \(t\rightarrow\infty\).
Finally, when also the initial starting point is fixed, we have the following.
Corollary 3
Proof
The assertion follows by a reasoning as in the proof of Theorem 1 and choosing \(m=x_{1}\) in (26). Taking into account that m satisfies (27) and \(c_{1}\) is given by (24), we get (29). The details are left to the reader. □
Remark 2
It is worth to note that the condition (17) may depend on the choice of the constant c in (15), i.e. on the choice of a primitive to b.
4 Oscillation and nonoscillation
In this section we discuss assumptions of Theorem 1, jointly with some consequences to the oscillation. Assumption (16) guarantees the continuability at infinity of any solution of (1) and its role is discussed in Remark 1. Concerning the condition \(Z=\infty\), a consequence of a result in [17] shows that it is a necessary condition for the solvability of (BVP). The following holds.
Theorem 2
Proof
Now, we discuss the monotonicity condition (17). We start by recalling the following nonoscillation result, which is an extension of a well-known Kiguradze criterion [18], Theorem 18.7.
Theorem 3
([7], Theorem 14.3)
A standard calculation shows that if (32) holds for \(t\geq t_{0}\geq1\), then (17) is valid on the same interval \([t_{0},\infty)\) as well. Thus, in view of Corollary 2, we can obtain an existence result for slowly decaying solutions x of (1). Nevertheless, condition (17) can be valid in a larger interval than \([t_{0},\infty)\). The next example illustrates this fact.
Example 1
Example 2
Furthermore, the function \(\bar{G}\) given in (32) is increasing for large t and any \(\varepsilon>0\). Hence, Theorem 3 cannot be used. Then it is a question whether (34) admits or does not admit oscillatory solutions.
When \(\alpha=1\), the coexistence between oscillatory solutions and nonoscillatory solutions can be obtained by using Lemma 2 and a result from [19], Theorem 1. The following holds.
Theorem 4
Proof
In view of [19], Theorem 1, any solution x of (35) which satisfies \(x(t_{0})x^{\prime}(t_{0})>0\) at some \(t_{0}\geq1\), is oscillatory. The remaining part of the statement follows from Lemma 2. □
The following example shows that both types of nonoscillatory decaying solutions can coexist with oscillatory solutions.
Example 3
Open problems
Example 3 suggests that for the existence of at least one slowly decaying solution, the assumption on monotonicity in (17) could be relaxed. Moreover, Example 3 (and Theorem 4) deal with the case \(\alpha=1\), that is, when the differential operator is the Sturm-Liouville disconjugate operator. Does the coexistence between oscillatory solutions and decaying solutions, illustrated in Example 3, occur also when \(\alpha\neq1\) (and \(\beta>\alpha\))?
Declarations
Acknowledgements
The first author is supported by Grant P201/11/0768 of the Czech Science Foundation.
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
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