# Bargmann-Type Inequality for Half-Linear Differential Operators

- Gabriella Bognár
^{1}and - Ondřej Došlý
^{2}Email author

**2009**:104043

**DOI: **10.1155/2009/104043

© G. Bognár and O. Došlý. 2009

**Received: **7 May 2009

**Accepted: **21 August 2009

**Published: **28 September 2009

## Abstract

## 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 [3–5] for some details.

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 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).

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 (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.

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.

Proof.

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 .

which we needed to prove.

Remark 3.2.

## 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

## References

- Bargmann V:
**On the number of bound states in a central field of force.***Proceedings of the National Academy of Sciences of the United States of America*1952,**38:**961–966. 10.1073/pnas.38.11.961MathSciNetView ArticleMATHGoogle Scholar - Reed M, Simon B:
*Methods of Modern Mathematical Physics, Vol. IV. Analysis of Operators*. Academic Press, Boston, Mass, USA; 1978.MATHGoogle Scholar - Blanchard Ph, Stubbe J:
**Bound states for Schrödinger Hamiltonians: phase space methods and applications.***Reviews in Mathematical Physics*1996,**8**(4):503–547. 10.1142/S0129055X96000172MathSciNetView ArticleMATHGoogle Scholar - Schmidt KM:
**A short proof for Bargmann-type inequalities.***The Royal Society of London*2002,**458**(2027):2829–2832. 10.1098/rspa.2002.1021MathSciNetView ArticleMATHGoogle Scholar - Setô N:
**Bargmann's inequalities in spaces of arbitrary dimension.***Publications of the Research Institute for Mathematical Sciences. Kyoto University*1974,**9:**429–461.View ArticleMATHGoogle Scholar - Díaz JI:
*Nonlinear Partial Differential Equations and Free Boundaries. Vol. I: Elliptic Equations, Research Notes in Mathematics*.*Volume 106*. Pitman, Boston, Mass, USA; 1985:vii+323.Google Scholar - Agarwal RP, Grace SR, O'Regan D:
*Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+672.View ArticleMATHGoogle Scholar - Došlý O, Řehák P:
*Half-Linear Differential Equations, North-Holland Mathematics Studies*.*Volume 202*. Elsevier, Amsterdam, The Netherlands; 2005:xiv+517.MATHGoogle Scholar - Elbert Á, Kusano T:
**Principal solutions of non-oscillatory half-linear differential equations.***Advances in Mathematical Sciences and Applications*1998,**18:**745–759.MathSciNetMATHGoogle Scholar - Mirzov JD:
**Principal and nonprincipal solutions of a nonlinear system.***Tbilisskiĭ Gosudarstvennyĭ Universitet. Institut Prikladnoĭ Matematiki. Trudy*1988,**31:**100–117.MathSciNetMATHGoogle Scholar - Elbert Á, Schneider A:
**Perturbations of the half-linear Euler differential equation.***Results in Mathematics*2000,**37**(1–2):56–83.MathSciNetView ArticleMATHGoogle Scholar

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