Journal of Inequalities and Applications Volume 2009, Article ID 104043,7pages doi:10.1155/2009/104043
Research Article
Bargmann-Type Inequality for Half-Linear
Differential Operators
Gabriella Bogn ´ar
1and Ondˇrej Do ˇsl ´y
21Department of Analysis, University of Miskolc, 3515 Miskolc-Egytemv´aros, Hungary 2Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
611 37 Brno, Czech Republic
Correspondence should be addressed to Ondˇrej Doˇsl ´y,[email protected]
Received 7 May 2009; Revised 29 July 2009; Accepted 21 August 2009
Recommended by Martin J. Bohner
We consider the perturbed half-linear Euler differential equationΦx γ/tpctΦx 0,
Φx : |x|p−2x,p > 1, with the subcritical coefficient γ < γp : p−1/pp. We establish a Bargmann-type necessary condition for the existence of a nontrivial solution of this equation with at leastn1zero points in0,∞.
Copyrightq2009 G. Bogn´ar and O. Doˇsl ´y. 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.
1. Introduction
The classical Bargmann inequality1originates 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 to2, Chapter XIII, and to 3–5for some details.
In the language of singular differential operators, Bargmann’s inequality concerns the one-dimensional Schr ¨odinger operator
τy: y
γ
t2 ct
y, γ < 1
4, t∈0,∞. 1.1
the equationτy 0 having at leastn1zeros in0,∞, then
∞
0
tctdt > n
1−4γ, 1.2
wherect max{ct,0}.
This inequality can be seen as follows. The Euler differential equation
x γ
t2x 0 1.3
with the subcritical coefficientγ <1/4 is disconjugate in0,∞, that is, any nontrivial solution of1.3has at most one zero in this interval. Hence, if the equationτy 0, withτgiven by 1.1, has a solution with at leastn1positive zeros, the perturbation functioncmust be “sufficiently positive” in view of the Sturmian comparison theorem. Inequality1.2specifies exactly what “sufficient positiveness” means.
In this paper, we treat a similar problem in the scope of the theory of half-linear differential equations:
rtΦxctΦx 0, Φx: |x|p−2x, p >1. 1.4
In physical sciences, there are known phenomena which can be described by differential equations with the so-calledp-LaplacianΔpu : div∇up−2∇u, 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 form1.4.
There are many results of the linear oscillation theory, which concern the Sturm-Liouville differential equation:
rtxctx 0, 1.5
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
Φxγ
tp ct Φx 0, t∈0,∞, 1.6
wherecis a continuous function, and shows that ifγ is the so-called subcritical coefficient, that is,γ < γp : p/p−1p, and there exists a solution of1.6with at leastn1zeros
in0,∞, then the integral∞0tp−1c
tdtsatisfies an inequality which reduces to1.2in the
linear casep 2.
2. Preliminaries
and half-linear oscillation theories are in many aspects very similar, so1.4can be classified as oscillatory or nonoscillatory as in the linear case.
Ifxis a solution of1.4such thatxt/0 is some intervalI, thenw : rΦx/xis a solution of the Riccati-type differential equation
wct p−1r1−q|w|q 0, q: p
p−1. 2.1
If 1.4 is nonoscillatory, that is, 2.1 possesses a solution which exists on some interval T,∞, among all such solutions of2.1, there exists the minimalonew, minimal in the sense that any other solutionwof2.1which exists on some intervaltw,∞satisfieswt>wt
in this interval, see9,10for details.
In our treatment, the so-called half-linear Euler differential equation
Φx γ
tpΦx 0 2.2
appears. If we look for a solution of this equation in the formxt tλ, thenλis a root of the
algebraic equation
|λ|p−Φλ γ
p−1 0. 2.3
By a simple calculationsee, e.g.,8, Section 1.3, one finds that2.3has a real root if and only ifγ is less than or equal to the so-called critical constantγp : p−1/pp, and hence
2.2is nonoscillatory if and only ifγ ≤ γp. In this case, the associated Riccati equation is of
the form
w γ
tp
p−1|w|q 0, 2.4
and its minimal solution iswt Φλ1t1−p, whereλ1is the smaller ofthe two realroots of 2.3. Ifvt tp−1w, thenvis a solution of the equation
v p−1
t −
p−1
t |v|
q−γ
t, 2.5
andvt≡Φλ1is the minimal solution of this equation. A detailed study of half-linear Euler equation and of its perturbations can be found in11.
3. Bargmann’s Type Inequality
Theorem 3.1. Suppose that1.6withγ < γp p−1/pphas a nontrivial solution with at least
n1zeros in0,∞.Then
∞
0
tp−1ctdt > nkγ, q, 3.1
wherekγ, qis the absolute value of the difference of the real roots of
Fγλ: |λ|q−λ
q−1γ 0 3.2
andq p/p−1 is the conjugate number top.Moreover, the constantkγ, qis strict in the sense that for everyε > 0, there exists a continuous functioncsuch that1.6possesses a solution with
n1zeros in0,∞and
∞
0
tp−1ctdt≤nkγ, qε. 3.3
Proof. Letxbe a solution of1.6withn1zeros in0,∞,denote these zeros byt0 < t1 < · · ·< tn, and letvt tp−1Φx/x.Then by a direct computation we see thatvis a solution
of the Riccati-type differential equation
v p−1
t v−
γ
t −
p−1|v|q−tp−1ct
−p−1Fγv−tp−1ct, t∈ti, ti1, i 0, . . . , n−1,
3.4
vti− −∞, vti ∞. 3.5
Letλ1 < λ2 be the roots of3.2. Such pair of roots exists and it is unique since the function
Fγλ is convex, Fγ±∞ ∞, Fγ1/Φq 0, and Fγ1/Φq γ −γp/p −1 < 0.
According to3.5, there existξi, ηi ∈ti, ti1such thatvξi λ2,vηi λ1, andλ1< vt<
λ2fort∈ξi, ηi, which means thatFγvt<0 fort∈ξi, ηi.Then, we have
∞
0
tp−1c
tdt≥ n
i 0
ηi
ξi tp−1c
tdt≥ n
i 0
ηi
ξi
tp−1ctdt
n
i 1
ηi
ξi
−vt−p−1Fγvt
dt > n
i 1
vt
ξi
ηi
n
i 1
vξi−v
ηi
nλ2−λ1 nk
γ, q.
Now we prove that the constant kγ, q is exact. Let ε > 0 be arbitrary and αi, βi, Ti be
sequences of positive real numbers constructed in the following way. Let t0 ∈ 0,∞ be arbitrary and consider the differential equation
Φx γ
tpΦx 0. 3.7
Denote byx0its nontrivial solution satisfyingx0t0 0,x0t0 1such solution exists and it is unique, see, e.g.,8, Section 1.1and letv0: tp−1Φx0/x0.Since lim
t→ ∞v0t v2, see8,
page 39, there existsT1> t0such thatv0T1. Now, let
α1:
γp−γ
T1 , β1
: εT1 4nγp−γ
, 3.8
and define fort∈T1, T1β1the function
c1t: 1
β1tp−1
kγ, q ε
4nα1
. 3.9
Consider the solutionvof the equation
v −p−1|v|
q
t
p−1v
t −
γ
t −t
p−1c
1t, t∈
T1,T1β1
, 3.10
given by the initial conditionsvT1 v0T1. Then fort∈T1,T1β1
v −p−1
t
|v|q−v γp
p−1
γp−γ
t −t
p−1c 1t
≤ γp−γ
t −
1
βi
kγ, q ε
4n
−γp−γ
T1
≤ −1
βi
kγ, q ε
4n
.
3.11
Hence,
vT1β1
vT1
T1β1
T1
vtdt < v24ε
n−
kγ, q ε
4n
v2−v2−v1 v1.
3.12
Now consider again3.7and the associated Riccati-type differential equation
v −γ
tp
which is related to3.7by the substitutionv tp−1Φx/x . This equation has a constant solutionv v1 and this solution is the minimal onesee the end ofSection 2. This means that any solution of3.13which starts with the initial conditionvT1β1< v1blows down to−∞at a finite timet1 > T1β1, which is a zero point of the associated solutionxof3.7. Now, let
c1t
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
0, t∈t0,T1,
c1t, t∈
T1,T1β1
,
0, t∈T1β1, t1
.
3.14
In summary, we have constructed a solution of the equation
Φxγ
tpc1t Φx 0 3.15
for whichxt0 0 xt1and
t1
t0
tp−1c1tdt
T1β1
T1
tp−1c1tdt
kγ, q ε
4nα1β1
kγ, q ε
4n ε
4n
kγ, q ε
2n.
3.16
The construction ofTi, βi, αi,citand cit, i 2, . . . , n,is now analogical. As a result we
obtain the functionc:0,∞ → 0,∞defined asct 0 fort∈0, t0andt∈tn,∞, and
ct citfort∈ti−1, ti, for which
∞
0
tp−1ctdt nkγ, qε
2, 3.17
and the equation
Φxγ
tpct Φx 0 3.18
has a solution with zeros att ti, i 0, . . . , n.
Finally, we change the discontinuous functionctto a continuous one ct ≥ ct
such thattnt 0t
p−1ct−ctdt < ε/2.Such a modification is an easy technical construction
the Sturmian comparison theorem, the equationΦx γ/tpctΦx 0 possesses a
nontrivial solution with at leastn1zeros and
∞
0
tp−1ctdt≤nkγ, qε, 3.19
which we needed to prove.
Remark 3.2. Ifp 2, thenFγλ λ2−λγand the roots of3.2are
λ1,2 1
2
1±
1−4γ
. 3.20
Hence,kγ,2 |λ1−λ2|
1−4γand3.1reduces to1.2.
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.
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