In this section, we consider general half-linear eigenvalue problem (1), (3). We denote

$R(t)={\int}_{a}^{t}{r}^{1-q}(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{2em}{0ex}}C(t)={\int}_{a}^{t}c(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{2em}{0ex}}W(t)={\int}_{a}^{t}w(s)\phantom{\rule{0.2em}{0ex}}ds.$

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* $\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=-\mathrm{\infty}.$

(12)

*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 mean-value theorem for Lebesgue integrals in computing the integral

${\int}_{a}^{b}{r}^{1-q}|{cot}_{p}\phi {|}^{p}$, integration of (8) gives

$\begin{array}{rl}v(b)-v(a)& =\mathrm{\Phi}({cot}_{p}\phi (b))-\mathrm{\Phi}({cot}_{p}\alpha )\\ =[\frac{C(b)}{W(b)}-\mathrm{\Lambda}-(p-1)\frac{R(b)}{W(b)}|{cot}_{p}\phi ({t}_{b}){|}^{p}]W(b),\end{array}$

where

${t}_{b}\in (a,b)$. Hence, for

*b* sufficiently close to

*a*,

$\mathrm{\Phi}({cot}_{p}\phi (b))-\mathrm{\Phi}({cot}_{p}\alpha )\le (\frac{C(b)}{W(b)}-\mathrm{\Lambda})W(b)<0,$

*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*
$\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}<\mathrm{\infty},\phantom{\rule{2em}{0ex}}\underset{t\to a+}{lim}\frac{R(t)}{W(t)}=\mathrm{\infty},$

*and let*
${\alpha}^{\ast}={arccot}_{p}{(max\{0,\frac{1}{p-1}\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}\})}^{1/p}\in (0,\frac{{\pi}_{p}}{2}].$

*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

$(p-1)|{cot}_{p}\alpha {|}^{p}>\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{C(t)}{R(t)}+2\delta .$

(13)

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

$(p-1)|{cot}_{p}\tau {|}^{p}>\frac{C(t)}{R(t)}+\delta $

when

*t* is sufficiently close to

*a*. Again, integration of (8), together with the mean value theorem applied to the last integral on the right-hand side of (8), gives

$\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))-\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{R(b)}{W(b)}(\frac{C(b)}{R(b)}-(p-1)|{cot}_{p}\phi ({t}_{b}){|}^{p})-\mathrm{\Lambda}]W(b)<0.\end{array}$

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*
$\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{t\to a+}{lim}\frac{C(t)}{W(t)}=\mathrm{\infty}.$

(14)

*Let*
${\alpha}^{\ast}={arccot}_{p}{\left(\frac{1}{p-1}\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}\right)}^{1/p}\in [0,\frac{{\pi}_{p}}{2}).$

(15)

*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

$(1-2\delta )\underset{t\to a+}{lim\hspace{0.17em}inf}\frac{C(t)}{R(t)}>(p-1)|{cot}_{p}\alpha {|}^{p}.$

(16)

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

$(1-\delta )\frac{C(t)}{R(t)}>(p-1)|{cot}_{p}\tau {|}^{p}.$

Let

$\mathrm{\Lambda}\in \mathbb{R}$ be arbitrary. Similarly as in the proof of the previous theorem,

$\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))-\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{C(b)}{W(b)}(1-(p-1)\frac{R(b)}{C(b)}{|{cot}_{p}\phi ({t}_{b})|}^{p})-\mathrm{\Lambda}]W(b)>0\end{array}$

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*
$\underset{t\to a+}{lim\hspace{0.17em}sup}\frac{R(t)}{W(t)}<\mathrm{\infty}.$

*If*
$\underset{t\to a+}{lim}(\frac{C(t)}{W(t)}-(p-1)\frac{R(t)}{W(t)}{|{cot}_{p}\phi |}^{p})=L\in \mathbb{R}\cup \{\pm \mathrm{\infty}\},$

*then*
$\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L.$

*Proof* Let

$\mathrm{\Lambda}\in \mathbb{R}$ be arbitrary. Similarly as in the previous theorems,

$\begin{array}{r}\mathrm{\Phi}({cot}_{p}\phi (b))-\mathrm{\Phi}({cot}_{p}\alpha )\\ \phantom{\rule{1em}{0ex}}=[\frac{C(b)}{W(b)}-(p-1)\frac{R(b)}{W(b)}{|{cot}_{p}\alpha |}^{p}\end{array}$

(17)

$\phantom{\rule{2em}{0ex}}-\mathrm{\Lambda}+(p-1)\frac{R(b)}{W(b)}({|{cot}_{p}\alpha |}^{p}-{|{cot}_{p}\phi ({t}_{b})|}^{p})]W(b).$

(18)

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. □

**Theorem 6**
*Suppose that*
$\underset{t\to a+}{lim}\frac{R(t)}{W(t)}<\mathrm{\infty}$

(19)

*and* $\alpha \in (0,{\pi}_{p})$.

*If* $\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L\phantom{\rule{1em}{0ex}}\mathit{\text{exists finite}},$

*then*
$\underset{t\to a+}{lim}(\frac{C(t)}{W(t)}-(p-1)\frac{R(t)}{W(t)}{|{cot}_{p}\alpha |}^{p})=L.$

(20)

*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

$\begin{array}{rl}0=& \mathrm{\Phi}({cot}_{p}\beta )-\mathrm{\Phi}({cot}_{p}\alpha )\\ =& \mathrm{\Phi}\left({cot}_{p}\phi (b,\lambda (b))\right)-\mathrm{\Phi}\left({cot}_{p}\phi (a,\lambda (b))\right)\\ =& C(b)-\lambda (b)W(b)-(p-1)R(b){|{cot}_{p}\alpha |}^{p}\\ +(p-1)R(b)({|{cot}_{p}\alpha |}^{p}-{\left|{cot}_{p}\phi ({t}_{b},\lambda (b))\right|}^{p})\\ =& [\frac{C(b)}{W(b)}-(p-1)\frac{R(b)}{W(b)}{|{cot}_{p}\alpha |}^{p}\end{array}$

(21)

$-\lambda (b)+(p-1)\frac{R(b)}{W(b)}({|{cot}_{p}\alpha |}^{p}-{\left|{cot}_{p}\phi ({t}_{b},\lambda (b))\right|}^{p})]W(b)=0$

(22)

for some

${t}_{b}\in (a,b)$. Therefore, from (21), (22), we obtain

$\frac{C(b)}{W(b)}-(p-1)\frac{R(b)}{W(b)}{|{cot}_{p}\alpha |}^{p}=\lambda (b)-(p-1)\frac{R(b)}{W(b)}({|{cot}_{p}\alpha |}^{p}-{\left|{cot}_{p}\phi ({t}_{b},\lambda (b))\right|}^{p}).$

Letting

$b\to a+$ and using that

$\underset{t\to a+}{lim}\lambda (b)=\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=L\in \mathbb{R}$

(which means that

$\lambda (b)$ is bounded in a neighborhood of

*a*), we see that

$\underset{b\to a+}{lim}\phi ({t}_{b},\lambda (b))=\alpha ,$

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* $\underset{t\to a+}{lim}\frac{R(t)}{W(t)}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{t\to a+}{lim}\frac{C(t)}{W(t)}$

*exist finite*,

*and for each* $\alpha \in (0,{\pi}_{p})$,

*we have* $\underset{b\to a+}{lim}{\lambda}_{1}(b,\alpha )=\underset{t\to a+}{lim}\frac{C(t)}{W(t)}-(p-1){|{cot}_{p}\alpha |}^{p}\underset{t\to a+}{lim}\frac{R(t)}{W(t)}.$

(23)

*Proof* First of all, we have

${|{cot}_{p}{\alpha}_{1}|}^{p}\ne {|{cot}_{p}{\alpha}_{2}|}^{p}.$

(24)

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)

$C(b)-{\mu}_{i}(b)W(b)-(p-1)R(b){\left|{cot}_{p}\phi ({t}_{i,b},{\mu}_{i}(b))\right|}^{p}=0,\phantom{\rule{1em}{0ex}}i=1,2,$

where

${t}_{i,b}\in (a,b)$. Hence, one gets

$\frac{C(b)}{W(b)}-(p-1)\frac{R(b)}{W(b)}{\left|{cot}_{p}\phi ({t}_{i,b},{\mu}_{i}(b))\right|}^{p}={\mu}_{i}(b),\phantom{\rule{1em}{0ex}}i=1,2.$

(25)

Subtracting these two equations and using the fact that

$\left|{cot}_{p}\phi ({t}_{1,b},{\mu}_{1}(b))\right|\ne \left|{cot}_{p}\phi ({t}_{2,b},{\mu}_{2}(b))\right|$

for

*b* sufficiently close to

*a* (this follows from (24)), we have

$\frac{R(b)}{W(b)}=\frac{{\mu}_{1}(b)-{\mu}_{2}(b)}{{|{cot}_{p}\phi ({t}_{1,b},{\mu}_{1}(b))|}^{p}-{|{cot}_{p}\phi ({t}_{2,b},{\mu}_{2}(b))|}^{p}}\to \frac{{\lambda}_{1}(a+,{\alpha}_{1})-{\lambda}_{1}(a+,{\alpha}_{2})}{{|{cot}_{p}{\alpha}_{1}|}^{p}-{|{cot}_{p}{\alpha}_{2}|}^{p}}$

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

$\underset{b\to a+}{lim}{\lambda}_{n}(b,\alpha ,\beta )=\mathrm{\infty}$

(27)

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 $.