Open Access

Bargmann-Type Inequality for Half-Linear Differential Operators

Journal of Inequalities and Applications20092009:104043

https://doi.org/10.1155/2009/104043

Received: 7 May 2009

Accepted: 21 August 2009

Published: 28 September 2009

Abstract

We consider the perturbed half-linear Euler differential equation , , , with the subcritical coefficient . We establish a Bargmann-type necessary condition for the existence of a nontrivial solution of this equation with at least zero points in .

1. Introduction

The classical Bargmann inequality [1] originates from the nonrelativistic quantum mechanics and gives an upper bound for the number of bound states produced by a radially symmetric potential in the two-body system. In the subsequent papers, various proofs and reformulations of this inequality have been presented, we refer to [2, Chapter XIII], and to [35] for some details.

In the language of singular differential operators, Bargmann's inequality concerns the one-dimensional Schrödinger operator
(11)
It states that if the Friedrichs realization of has at least negative eigenvalues below the essential spectrum (what is equivalent to the existence of a nontrivial solution of the equation having at least zeros in ), then
(12)

where .

This inequality can be seen as follows. The Euler differential equation
(13)

with the subcritical coefficient is disconjugate in , that is, any nontrivial solution of (1.3) has at most one zero in this interval. Hence, if the equation , with given by (1.1), has a solution with at least positive zeros, the perturbation function must be "sufficiently positive" in view of the Sturmian comparison theorem. Inequality (1.2) specifies exactly what "sufficient positiveness" means.

In this paper, we treat a similar problem in the scope of the theory of half-linear differential equations:
(14)

In physical sciences, there are known phenomena which can be described by differential equations with the so-called -Laplacian , see, for example, [6]. If the potential in such an equation is radially symmetric, this equation can be reduced to a half-linear equation of the form (1.4).

There are many results of the linear oscillation theory, which concern the Sturm-Liouville differential equation:
(15)
which has been extended to (1.4). In particular, the linear Sturmian theory holds almost verbatim for (1.4), see, for example, [7, 8]. We will recall elements of the half-linear oscillation theory in the next section. Our main result concerns the perturbed half-linear Euler differential equation
(16)

where is a continuous function, and shows that if is the so-called subcritical coefficient, that is, , and there exists a solution of (1.6) with at least zeros in , then the integral satisfies an inequality which reduces to (1.2) in the linear case .

2. Preliminaries

In this short section, we present some elements of the half-linear oscillation theory which we need in the proof of our main result. As we have mentioned in the previous section, the linear and half-linear oscillation theories are in many aspects very similar, so (1.4) can be classified as oscillatory or nonoscillatory as in the linear case.

If is a solution of (1.4) such that is some interval , then is a solution of the Riccati-type differential equation
(21)

If (1.4) is nonoscillatory, that is, (2.1) possesses a solution which exists on some interval , among all such solutions of (2.1), there exists the minimal one , minimal in the sense that any other solution of (2.1) which exists on some interval satisfies in this interval, see [9, 10] for details.

In our treatment, the so-called half-linear Euler differential equation
(22)
appears. If we look for a solution of this equation in the form , then is a root of the algebraic equation
(23)
By a simple calculation (see, e.g., [8, Section  1.3]), one finds that (2.3) has a real root if and only if is less than or equal to the so-called critical constant , and hence (2.2) is nonoscillatory if and only if . In this case, the associated Riccati equation is of the form
(24)
and its minimal solution is , where is the smaller of (the two real) roots of (2.3). If , then is a solution of the equation
(25)

and is the minimal solution of this equation. A detailed study of half-linear Euler equation and of its perturbations can be found in [11].

3. Bargmann's Type Inequality

In this section, we present our main results, the half-linear version of Bargmann's inequality. We are motivated by the work in [4] where a short proof of this inequality based on the Riccati technique is presented. Here we show that this method, properly modified, can also be applied to (1.6).

Theorem 3.1.

Suppose that (1.6) with has a nontrivial solution with at least zeros in Then
(31)
where is the absolute value of the difference of the real roots of
(32)
and is the conjugate number to Moreover, the constant is strict in the sense that for every , there exists a continuous function such that (1.6) possesses a solution with zeros in and
(33)

Proof.

Let be a solution of (1.6) with zeros in denote these zeros by , and let Then by a direct computation we see that is a solution of the Riccati-type differential equation
(34)
(35)
Let be the roots of (3.2). Such pair of roots exists and it is unique since the function is convex, , , and According to (3.5), there exist such that , , and for , which means that for Then, we have
(36)
Now we prove that the constant is exact. Let be arbitrary and be sequences of positive real numbers constructed in the following way. Let be arbitrary and consider the differential equation
(37)

Denote by its nontrivial solution satisfying , (such solution exists and it is unique, see, e.g., [8, Section  1.1]) and let Since , see [8, page 39], there exists such that .

Now, let
(38)
and define for the function
(39)
Consider the solution of the equation
(310)
given by the initial conditions . Then for
(311)
Hence,
(312)
Now consider again (3.7) and the associated Riccati-type differential equation
(313)
(which is related to (3.7) by the substitution ). This equation has a constant solution and this solution is the minimal one (see the end of Section 2). This means that any solution of (3.13) which starts with the initial condition blows down to at a finite time , which is a zero point of the associated solution of (3.7). Now, let
(314)
In summary, we have constructed a solution of the equation
(315)
for which and
(316)
The construction of and is now analogical. As a result we obtain the function defined as for and , and for , for which
(317)
and the equation
(318)

has a solution with zeros at

Finally, we change the discontinuous function to a continuous one such that Such a modification is an easy technical construction which can be described explicitly, but for us is only important its existence. According to the Sturmian comparison theorem, the equation possesses a nontrivial solution with at least zeros and
(319)

which we needed to prove.

Remark 3.2.

If , then and the roots of (3.2) are
(320)

Hence, and (3.1) reduces to (1.2).

Declarations

Acknowledgment

The authors thank the referees for their valuable remarks and suggestions which contributed substantially to the present version of the paper. The first author is supported by the Grant OTKA CK80228 and the second author is supported by the Research Project MSM0021622409 of the Ministry of Education of the Czech Republic and the Grant 201/08/0469 of the Grant Agency of the Czech Republic.

Authors’ Affiliations

(1)
Department of Analysis, University of Miskolc
(2)
Department of Mathematics and Statistics, Masaryk University

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Copyright

© G. Bognár and O. Došlý. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.