Abstract
We investigate the limit behavior of the first eigenvalue of the halflinear eigenvalue problem when the length of the interval tends to zero. We show that the important role is played by the limit behavior of ratios of primitive functions of coefficients in the investigated halflinear equation.
MSC: 34C10.
1 Introduction
We consider the eigenvalue problem associated with the halflinear second order differential equation
with
Along with (1), we consider the separated boundary conditions
where
The investigation of halflinear eigenvalue problems is motivated, among others, by
the fact that the partial differential equation with the pLaplacian (which models, e.g., the flow of nonNewtonian fluids, while the linear case
and the spherically symmetric potential c, can be reduced to an equation of the form (1). For this reason, motivated also by
the linear case
The paper is organized as follows. In the next section, we recall essentials of the qualitative theory of halflinear differential equations. Section 3 deals with the eigenvalue problem for an equation with constant coefficients. The main results of the paper, limit formulas for the first eigenvalue of (1), (3), are given in Section 4.
2 Preliminaries
First, we recall the concepts of halflinear goniometric functions. These functions, in the form presented here, appeared for the first time in [1]. In a modified form, they can also be found in other papers, e.g., in [7].
The halflinear sine function
given by the initial condition
The halflinear functions
The inverse functions to these functions on
one can verify the formulas
The original proof of the unique solvability of (1) (and hence of (2), see [1]) is based on the halflinear version of the Prüfer transformation. Let x be a nontrivial solution of (1), put
Then the Prüfer angle φ solves the first order differential equation
The righthand side of (7) is a Lipschitzian function with respect to φ, hence the standard existence, uniqueness, and continuous dependence on the initial data theory applies to this equation, and these results carry over via (6) to (2) and (1). Observe at this place that the righthand side of (2) is not Lipschitzian, so this theory cannot be directly applied to (2).
The Prüfer transformation is closely associated with the Riccatitype differential equation
which is related to (1) by the Riccati substitution
3 Equation with constant coefficients
In this section, as a motivation, we consider the equation
i.e.,
First, consider the case
To underline the dependence of eigenvalues of our eigenvalue problem on b, α, β, we denote the first eigenvalue by
Obviously, the solution of (10) satisfying (11) is a constant solution when
If
In the opposite case, when
The previous simple considerations are summarized in the next theorem.
Theorem 1Let
Remark 1 In this section, for the sake of simplicity, we have considered a constant coefficients
equation in the form (9), i.e., with
(actually, this least eigenvalue does not depend on the endpoints a, b).
4 Limit behavior of the first eigenvalue
In this section, we consider general halflinear eigenvalue problem (1), (3). We denote
In the next theorems, we discuss various asymptotic behavior of ratios of the functions
C, R, W for
In the proofs of the next theorems, given
We start with the most interesting case
Theorem 2Suppose that
Proof We will show that
where
i.e.,
Theorem 3Suppose that
and let
Then we have
Proof Let
Such a positive δ exists according to the definition of the number
when t is sufficiently close to a. Again, integration of (8), together with the mean value theorem applied to the last integral on the righthand side of (8), gives
Hence
Theorem 4Assume that
Let
Then we have
Proof The first formula in (14) implies that
Again, such
Let
if b is sufficiently close to a, i.e.,
Theorem 5Assume that
If
then
Proof Let
If
Finally, if
Theorem 6Suppose that
and
then
Proof Denote
for some
Letting
(which means that
and (20) is proved. □
Theorem 7If
exist finite, and for each
Proof First of all, we have
Denote
where
Subtracting these two equations and using the fact that
for b sufficiently close to a (this follows from (24)), we have
as
We finish the paper with a brief treatment of the case
Theorem 8Let
(i) If
(ii) If
Proof We will prove the part (i) only, the proof of (ii) is analogical. Let
Remark 2 Until now, we have considered the first eigenvalue
for any
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.
Acknowledgements
The research of the first author was carried out as a part of the TAMOP4.2.2/B10/120100008 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union and cofinanced by the European Social Fund. The second author was supported by the Grant GAP 201/11/0768 of the Czech Grant Agency.
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