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Halflinear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval
Boundary Value Problemsvolume 2013, Article number: 221 (2013)
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 $t\in (a,b)$, $\mathrm{\infty}<a<b<\mathrm{\infty}$, ${r}^{1q},c,w\in {L}^{1}(a,b)$, $q=\frac{p}{p1}$ being the conjugate exponent of p, and $r(t)>0$, $w(t)>0$. Equation (1) can be written as the first order system
${\mathrm{\Phi}}^{1}(u)={u}^{q2}u$ being the inverse function of Φ, and the integrability assumption on the functions ${r}^{1q}$, c, w implies the unique solvability of this system. The original paper of Elbert [1], where the existence and uniqueness results are proved via the halflinear version of the Prüfer transformation, deals with continuous functions in (2), but the idea of the proof applies without change to integrable coefficients when, as a solution x, u, absolutely continuous functions are considered (which satisfy (2) a.e. in $(a,b)$).
Along with (1), we consider the separated boundary conditions
where ${sin}_{p}$, ${cos}_{p}$ are the halflinear goniometric functions, which will be recalled in the next section. Motivated by the paper [2], where the linear SturmLiouville differential equation (which is the special case of (1)) is considered, we investigate the limit behavior (as $b\to a+$) of the first eigenvalue of (1), (3) in dependence on α, β. We show that this limit behavior is, in a certain sense, the same as for an eigenvalue problem when boundary conditions (3) are associated with an equation with constant coefficients.
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 $p=2$ corresponds to the Newtonian fluid)
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 $p=2$, the problem of dependence of eigenvalues of (1), (3) on the functions r, c, w and the boundary data a, b, α, β was a subject of the investigation in several recent papers. We refer to [3–6] and the references therein.
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 ${sin}_{p}$ is defined as the solution of the differential equation
given by the initial condition $x(0)=0$, ${x}^{\prime}(0)=1$. The function ${sin}_{p}$ is ${\pi}_{p}$ antiperiodic (and hence $2{\pi}_{p}$ periodic) with ${\pi}_{p}:=\frac{2\pi}{psin(\pi /p)}$. The derivative ${({sin}_{p}t)}^{\prime}=:{cos}_{p}t$ defines the halflinear cosine function. These functions satisfy the halflinear Pythagorean identity
The halflinear functions ${tan}_{p}$ and ${cot}_{p}$ are defined in a natural way as
The inverse functions to these functions on $({\pi}_{p}/2,{\pi}_{p}/2)$ resp. $(0,{\pi}_{p})$ are denoted by ${arctan}_{p}$ and ${arccot}_{p}$. By a direct computation, using (5) and the fact that (4) can be written as
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 $v=r\mathrm{\Phi}({x}^{\prime}/x)$. The fact that we have in disposal a Riccatitype differential equation and the generalized Prüfer transformation implies that the linear oscillation theory extends almost verbatim to (1). In particular, similarly to the linear case, the eigenvalues of (1), (3) form an increasing sequence ${\lambda}_{n}\to \mathrm{\infty}$, and the n th eigenfunction has exactly $n1$ zeros in $(a,b)$, see [8] and also [[9], Section 5.7]. For some recent references in this area, we refer to [3] and the references therein.
3 Equation with constant coefficients
In this section, as a motivation, we consider the equation
i.e., $r\equiv 1$, $c\equiv 0$, $w\equiv p1$ in (1). The Riccati equation associated with (9) is
First, consider the case $\alpha =\beta \in (0,{\pi}_{p})$ in (3). Then the first eigenfunction of (9), (3) corresponds to the solution of (10) satisfying
To underline the dependence of eigenvalues of our eigenvalue problem on b, α, β, we denote the first eigenvalue by ${\lambda}_{1}(b,\alpha ,\beta )$. Also, when $\alpha =\beta $, we use the notation ${\lambda}_{1}(b,\alpha ,\alpha )=:{\lambda}_{1}(b,\alpha )$.
Obviously, the solution of (10) satisfying (11) is a constant solution when
If $0<\alpha <\beta <{\pi}_{p}$, i.e., $v(a)=\mathrm{\Phi}({cot}_{p}\alpha )>\mathrm{\Phi}({cot}_{p}\beta )=v(b)$, we need v to be decreasing. When the length of the interval $(a,b)$ tends to zero, ${v}^{\prime}(t)$ cannot be bounded from below in $(a,b)$, and hence
In the opposite case, when $0<\beta <\alpha <{\pi}_{p}$, i.e., $v(a)=\mathrm{\Phi}({cot}_{p}\alpha )<\mathrm{\Phi}({cot}_{p}\beta )=v(b)$, we need v to be increasing, and using the same argument as before, we have ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}$. Finally, if $\alpha =0$ and $\beta \in (0,{\pi}_{p})$, we have $v(a+)=\mathrm{\infty}$, and hence ${lim}_{b\to a+}{\lambda}_{1}(b,0,\beta )=\mathrm{\infty}$, similarly, if $\beta ={\pi}_{p}$ and $\alpha \in (0,{\pi}_{p})$, we have $v(b)=\mathrm{\infty}$, and hence also ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha ,{\pi}_{p})=\mathrm{\infty}$.
The previous simple considerations are summarized in the next theorem.
Theorem 1 Let ${\lambda}_{1}(b,\alpha ,\beta )$ denote the first eigenvalue of (9), (3) with $\alpha \in [0,{\pi}_{p})$, $\beta \in (0,{\pi}_{p}]$. Then
Remark 1 In this section, for the sake of simplicity, we have considered a constant coefficients equation in the form (9), i.e., with $c=0$, $r=1$, and $w=p1$. If the functions r, c, w in (1) are constants equal to $r,c,w\in \mathbb{R}$, a slight modification of the previous considerations shows that for $\alpha =\beta $ we have
(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 $t\to a+$, which implies various limit behavior of the first eigenvalue. The behavior of the higher eigenvalues is described at the end of this section.
In the proofs of the next theorems, given $\mathrm{\Lambda}\in \mathbb{R}$, $\phi (t)=\phi (t,\mathrm{\Lambda})$, denotes the Prüfer angle of a solution x of (1) with $\lambda =\mathrm{\Lambda}$ satisfying (3) at a, i.e., $\phi (a,\mathrm{\Lambda})=\alpha $.
We start with the most interesting case $\alpha =\beta $ in (3).
Theorem 2 Suppose that ${lim}_{t\to a+}\frac{C(t)}{W(t)}=\mathrm{\infty}$. Then for any $\alpha \in (0,{\pi}_{p})$, we have
Proof We will show that ${\lambda}_{1}(b,\alpha )<\mathrm{\Lambda}$ for any $\mathrm{\Lambda}\in \mathbb{R}$ if $b>a$ is sufficiently close to a. Since the eigenfunction corresponding to the first eigenvalue ${\lambda}_{1}$ has no zero on $(a,b)$ (see, e.g., [10]), we can use the Riccati equation (8) for $v=\mathrm{\Phi}({cot}_{p}\phi )$ instead of equation (7) for φ. Using the meanvalue theorem for Lebesgue integrals in computing the integral ${\int}_{a}^{b}{r}^{1q}{cot}_{p}\phi {}^{p}$, integration of (8) gives
where ${t}_{b}\in (a,b)$. Hence, for b sufficiently close to a,
i.e., $\phi (b)>\alpha =\beta $. Thus, since φ was the Prüfer angle corresponding to a solution of (1) with $\lambda =\mathrm{\Lambda}$ satisfying (3) at $t=a$, we have ${\lambda}_{1}(b,\alpha )<\mathrm{\Lambda}$ for the first eigenvalue when b is in a sufficiently small right neighborhood of a (since we need $\phi (b)=\alpha $ for $\lambda ={\lambda}_{1}(b,\alpha )$). Therefore, (12) holds. □
Theorem 3 Suppose that
and let
Then we have ${\lambda}_{1}(b,\alpha )\to \mathrm{\infty}$ as $b\to a+$ for any $\alpha \in (0,{\pi}_{p})\setminus [{\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast}]$.
Proof Let $\alpha \in (0,{\alpha}^{\ast})\cup ({\pi}_{p}{\alpha}^{\ast},{\pi}_{p})$ be fixed, and take $\delta >0$ so small that
Such a positive δ exists according to the definition of the number ${\alpha}^{\ast}$. Formula (13) implies that for τ sufficiently close to α, we have the inequality
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 $\phi (b)>\alpha $. This implies that ${\lambda}_{1}(b,\alpha )<\mathrm{\Lambda}$ for every $\mathrm{\Lambda}\in \mathbb{R}$, and thus, ${\lambda}_{1}(b,\alpha )\to \mathrm{\infty}$ as $b\to a+$. □
Theorem 4 Assume that
Let
Then we have ${\lambda}_{1}(b,\alpha )\to \mathrm{\infty}$ as $b\to a+$ for any $\alpha \in ({\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast})$.
Proof The first formula in (14) implies that $C(t)>0$ for t in a right neighborhood of a. Take $\alpha \in ({\alpha}^{\ast},{\pi}_{p}{\alpha}^{\ast})$, and let $\delta >0$ be so small that
Again, such $\delta >0$ exists according to the definition of ${\alpha}^{\ast}$. Hence, for τ sufficiently close to α, from (16), we have for t close to a that
Let $\mathrm{\Lambda}\in \mathbb{R}$ be arbitrary. Similarly as in the proof of the previous theorem,
if b is sufficiently close to a, i.e., $\phi (b)<\alpha $, and hence ${\lambda}_{1}(b,\alpha )>\mathrm{\Lambda}$. Therefore, ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}$. □
Theorem 5 Assume that
If
then
Proof Let $\mathrm{\Lambda}\in \mathbb{R}$ be arbitrary. Similarly as in the previous theorems,
If $L=\mathrm{\infty}$, the expression in line (17) tends to −∞ as $b\to a+$ while remaining terms on the righthand side of the previous formula are bounded. Hence the expression on the righthand side is negative for b close to a, which means that $\phi (b)>\alpha $ for these b. However, this implies that ${\lambda}_{1}(b,\alpha )<\mathrm{\Lambda}$ in a right neighborhood of a, and since Λ was arbitrary, we have ${lim}_{b\to a}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}$. The same arguments imply that ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha )=\mathrm{\infty}$ if $L=+\mathrm{\infty}$.
Finally, if $L\in \mathbb{R}$, take first $\mathrm{\Lambda}=L+\epsilon $. Since the last term in (18) tends to zero as $b\to a+$, we obtain using the same argument as in the previous part of the proof that ${\lambda}_{1}(b,\alpha )<L+\epsilon $ for b sufficiently close to a. Taking $\mathrm{\Lambda}=L\epsilon $, we obtain ${\lambda}_{1}(b,\alpha )>L\epsilon $ for b in a right neighborhood of a, and this completes the proof. □
Theorem 6 Suppose that
and $\alpha \in (0,{\pi}_{p})$. If
then
Proof Denote $\lambda (b):={\lambda}_{1}(b,\alpha )$, and let $\phi (t,\lambda (b))$ be the Prüfer angle of the solution x of (1) with $\lambda =\lambda (b)={\lambda}_{1}(b,\alpha )$ satisfying (3) with $\alpha =\beta $. Then
for some ${t}_{b}\in (a,b)$. Therefore, from (21), (22), we obtain
Letting $b\to a+$ and using that
(which means that $\lambda (b)$ is bounded in a neighborhood of a), we see that
and (20) is proved. □
Theorem 7 If ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha )$ has a finite value for two different values of ${\alpha}_{1}$ and ${\alpha}_{2}$ with ${\alpha}_{1},{\alpha}_{2}\in (0,{\pi}_{p})$ and ${\alpha}_{1}\ne {\pi}_{p}{\alpha}_{2}$, then
exist finite, and for each $\alpha \in (0,{\pi}_{p})$, we have
Proof First of all, we have
Denote ${\lambda}_{1}(b,{\alpha}_{i})=:{\mu}_{i}(b)$ for $i=1,2$, and let $\phi (t,{\mu}_{i}(b))$ be the Prüfer angle of the solution of (1), (3) with $\lambda ={\mu}_{i}(b)$ and $\alpha =\beta $. Since $\phi (b,{\mu}_{i}(b))=\phi (a,{\mu}_{i}(b))$, we have from (8)
where ${t}_{i,b}\in (a,b)$. Hence, one gets
Subtracting these two equations and using the fact that
for b sufficiently close to a (this follows from (24)), we have
as $b\to a+$, so ${lim}_{b\to a+}\frac{R(b)}{W(b)}$ exists finite, and hence from (25), the same holds for ${lim}_{b\to a+}\frac{C(b)}{W(b)}$, and the conclusion follows from Theorem 5. □
We finish the paper with a brief treatment of the case $\alpha \ne \beta $ in (3). We show that the situation is similar as in the case of a constant coefficients equation treated in Section 3.
Theorem 8 Let $\alpha \in [0,{\pi}_{p})$, $\beta \in (0,{\pi}_{p}]$.

(i)
If $\alpha <\beta $, then
$$\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}.$$(26) 
(ii)
If $\alpha >\beta $, then
$$\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}.$$
Proof We will prove the part (i) only, the proof of (ii) is analogical. Let $\mathrm{\Lambda}\in \mathbb{R}$ be arbitrary, and let $\phi (t,\mathrm{\Lambda})$ be the Prüfer angle of the solution x of (1) with $\lambda =\mathrm{\Lambda}$ satisfying (3) at $t=a$. Since φ is a continuous function of t, and $\phi (a,\mathrm{\Lambda})=\alpha <\beta $, we have $\phi (b,\mathrm{\Lambda})<\beta $ if b is sufficiently close to a. Hence, using the same argument as before, we have ${\lambda}_{1}(b,\alpha ,\beta )>\mathrm{\Lambda}$, which implies (26). □
Remark 2 Until now, we have considered the first eigenvalue ${\lambda}_{1}(b,\alpha ,\beta )$ only. Concerning the asymptotic behavior of higher eigenvalues ${\lambda}_{n}(b,\alpha ,\beta )$, we have
for any $\alpha \in [0,{\pi}_{p})$, $\beta \in (0,{\pi}_{p}]$. This formula follows in the case $\alpha <\beta $ from the general theory of halflinear eigenvalues problem (see [8, 10]), which says that the eigenvalues form an increasing sequence tending to ∞, and from the fact that ${lim}_{b\to a+}{\lambda}_{1}(b,\alpha ,\beta )=\mathrm{\infty}$. If $\alpha >\beta $, then ${cot}_{p}\alpha <{cot}_{p}\beta $, but for $n\in \mathbb{N}$, we have $\alpha <\beta +n{\pi}_{p}$ and higher eigenvalues correspond to the situation when the Prüfer angle of a solution of (1) satisfying the first condition in (3) satisfies $\phi (b)=\beta +n{\pi}_{p}$. Hence, the growth of φ must be unbounded when $b\to a+$, and hence (27) holds also for $\alpha >\beta $.
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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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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.
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Keywords
 Eigenvalue Problem
 Riccati Equation
 Constant Coefficient
 Linear Case
 Order Differential Equation