Half-linear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval

${\mathrm{\Phi }}^{-1}(u)={|u|}^{q-2}u$ being the inverse function of Φ, and the integrability assumption on the functions $r^{1-q}$, 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 half-linear 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 $(a,b)$).

where ${sin}_p$, ${cos}_p$ are the half-linear goniometric functions, which will be recalled in the next section. Motivated by the paper [2], where the linear Sturm-Liouville 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.

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 half-linear 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.

First, we recall the concepts of half-linear 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 right-hand 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 right-hand side of (2) is not Lipschitzian, so this theory cannot be directly applied to (2).

which is related to (1) by the Riccati substitution $v=r\mathrm{\Phi }(x^{\prime }/x)$. The fact that we have in disposal a Riccati-type 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 $n-1$ 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.

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 ${\lambda }_1(b,\alpha ,\alpha )=:{\lambda }_1(b,\alpha )$.

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.

(actually, this least eigenvalue does not depend on the endpoints a, b).

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).

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. □

Then we have ${\lambda }_1(b,\alpha )\to -\mathrm{\infty }$ as $b\to a+$ for any $\alpha \in (0,{\pi }_p)\setminus [{\alpha }^{\ast },{\pi }_p-{\alpha }^{\ast }]$.

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 $b\to a+$. □

Then we have ${\lambda }_1(b,\alpha )\to \mathrm{\infty }$ as $b\to a+$ for any $\alpha \in ({\alpha }^{\ast },{\pi }_p-{\alpha }^{\ast })$.

if b is sufficiently close to a, i.e., $\phi (b)<\alpha$, and hence ${\lambda }_1(b,\alpha )>\mathrm{\Lambda }$. Therefore, ${lim}_{b\to a+}{\lambda }_1(b,\alpha )=\mathrm{\infty }$. □

If $L=-\mathrm{\infty }$, the expression in line (17) tends to −∞ as $b\to a+$ while remaining terms on the right-hand side of the previous formula are bounded. Hence the expression on the right-hand 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. □

and (20) is proved. □

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.

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). □

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 half-linear 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 $\alpha >\beta$.