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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1996/01\hfil On the number of solutions 
 \hfil\folio}
\def\leftheadline{\folio\hfil H. Dang, K. Schmitt \& R. Shivaji
 \hfil EJDE--1996/01}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1996}(1996), No.\ 01, pp.\ 1--9.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)
\hfill\break
telnet (login: ejde), ftp, and gopher access:
 ejde.math.swt.edu or ejde.math.unt.edu\bigskip} }

\topmatter
\title  On the number of solutions of boundary\\
value problems involving the $P$--Laplacian
\endtitle

\thanks \noindent
{\it 1991 Mathematics Subject Classifications:} 34B15, 35J15, 35J85.\hfil\break
{\it Key words and phrases:} p--Laplacian, radial solutions, 
boundary value problems.
\hfil\break
\copyright 1996 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted October 4, 1995. Published January 8, 1996. \hfil\break
Supported by the National Science Foundation. 
\endthanks
\author Hai Dang, Klaus Schmitt \& R. Shivaji  \endauthor
\address
Department of Mathematics and Statistics,
Mississippi State University, Mississippi State, MS  39762, USA
\endaddress
\email dang\@math.msstate.edu \endemail

\address Department of Mathematics, University of Utah,
 Salt Lake City, UT 84112, USA
\endaddress
\email schmitt\@math.utah.edu \endemail

\address
Department of Mathematics and Statistics,
Mississippi State University, Mississippi State, MS  39762, USA
\endaddress
\email shivaji\@math.msstate.edu \endemail

\abstract
This paper is concerned with multiplicity questions for solutions of 
the boundary value problem
$$\aligned(\varphi (u'))' + \lambda &f(t, u)=0,~ a<t<b \\
&u(a)=0=u(b) \endaligned \
$$
where $\varphi$ is an odd, increasing homeomorphism on ${\Bbb R}$, and
$\lambda$ is a positive parameter. The tools 
 employed are fixed point
and continuation methods.
\endabstract
\endtopmatter
\document

\heading 1. Introduction \endheading
In this paper, we are interested in the existence and multiplicities 
of positive
solutions of the boundary value problem.
$$\aligned(\varphi (u'))' + \lambda &f(t, u)=0,~ a<t<b \\
&u(a)=0=u(b) \endaligned \tag 1.1
$$
with $f$ continuous (but not necessarily locally Lipschitz continuous).  
We make the following assumptions: 
\newline
(A.1) $\varphi$ is an odd, increasing homeomorphism on ${\Bbb R}$ and
$$\limsup_{x \to \infty} \,\, \frac{\varphi (\sigma x)}{\varphi(x)} 
< \infty $$
for every $\sigma > 0.$

(A.2) $f:[a,b] \times [0, \infty) \to (0, \infty)$ is continuous and
there
 exists an
interval $[c,d] \subset (a,b),~c<d$ such that
$$\lim _{u \to \infty} \frac {f(t,u)}{\varphi (u)} = \infty $$
uniformly for $t \in [c,d].$

Our main result is:

\proclaim{Theorem 1}   Let {\rm (A.1)} and {\rm (A.2)} hold.  Then there exists a positive
number $\lambda ^*$ such that the problem $(1.1)_{\lambda}$ has at
 least two positive
solutions for $0< \lambda < \lambda ^*$, at least one for $\lambda =
 \lambda ^*$ and 
none for $\lambda > \lambda ^*$.
\endproclaim

Note that in the special case where $\varphi (u') = u'$, theorem 1 was
proved in [2,6] under the additional assumption that $f$ is of class
$C^2$.  Related results for the case $\varphi (u') = |u'|^{p-2} u'$
can be found in [4,5,7] and the references in these papers. As we shall see,
a more specific result, which establishes the existence of solution continua of
$(1.1)_{\lambda}$ which satisfy the above conditions, may be obtained also.

One of the primary motivations for studying problems of the above types
are boundary value problems for partial differential equations for
 perturbations of the
p-Laplacian of the form
$$
\aligned \text{ div} \left ( | \nabla u|^{p-2} \nabla u \right ) & +
 \lambda g (|x|, u) =0,
a < |x|<b, x \in {\Bbb R} ^N \\
& u = 0 \text { at } |x|\in \{a,b\} 
\endaligned
\tag 1.2
$$
where $N \neq p$ and $\lambda$ is a positive parameter.

Seeking the existence of  radial solutions,  one is led  to the equation
$$(|u'| ^{p-2} u')' + \lambda f (t, u) =0
$$
via the change of variables
 $$t= \left ( \frac {|N-p|}{(p-1)r} \right ) ^{\frac {N-p}{p-1}}
\, \, \,  ,r= |x|,$$
and
$$f(t,u) = \left ( \frac {(p-1)r}{|N-p|} \right ) ^{\frac {p(N-1)}{p-1}}
g \left ( \frac {|N-p|}{p-1}  t ^{-\frac {p-1}{N-p}},u \right ).
 $$

In proving theorem 1, we shall employ upper and lower solution methods. 
These methods are, of course, standard for semilinear equations;
they are also applicable in the nonlinear case and we present the
type of theorem needed for the nonlinear case. Such a theorem is
 established in section 2.
Section 3 is then devoted to the proof of theorem 1, and to some 
 more specific
information about the solution structure.


\heading 2. Upper and lower solutions
\endheading
Consider the problem
$$\aligned (\varphi (u'))' & + g (t,u) =0,~ a<t<b \\ 
& u(a)=0=u(b), \endaligned \tag 2.1
$$
where $\varphi$ is an odd, increasing homeomorphism on ${\Bbb R}$ and
$g:[a,b] \times {\Bbb R} \to {\Bbb R}$ is continuous. 
We say that a function $\alpha \in C ^0[a,b] $ is a lower solution of
 $(2.1),$ if
for each $t_0\in [a,b]$ there exists a neighborhood $I_{t_{0}}$ and 
 a finite set
of functions $ \{ \alpha _k \}_{1 \leq k \leq n}  \subset C ^1(I_{t_0})$
with $\varphi (\alpha ' _{k}) \in C ^1(I_{t_0})$ such that 
$$
\align
&\alpha (t) = \max_{1 \leq k \leq n}(\alpha _k (t)),~t\in (I_{t_0}) \\
&(\varphi (\alpha ' _{k}))' + g (t, \alpha _k) \geq 0,~t\in (I_{t_0}) 
 \tag 2.2 \endalign
$$ 
and
$$ \alpha (a) \leq 0,~ \alpha (b) \leq 0. \tag 2.3$$
An upper solution is defined similarly, replacing max by min and reversing the
inequalities in (2.2), (2.3).

 \proclaim{Theorem 2} Let $\alpha$ and $\beta$ be a pair of lower and an 
upper solutions to {\rm (2.1)}
respectively with $\alpha \leq \beta$.  Then {\rm (2.1)} has a minimum 
solution 
$u _{\text {min}}$ such that $\alpha \leq u _{\text {min}} \leq \beta$, and
if $u$ is any solution to {\rm(2.1)} with $u \geq \alpha$, then $u \geq 
u _{\text {min}}.$  An analogous result holds for the existence of a
 maximal solution.
\endproclaim

\demo{Proof} We first prove that there exists a solution $u$ to (2.1) with
$\alpha \leq u \leq \beta$. \newline
Define
$$\tilde g (t,v) = 
\left \{ \matrix 
g(t, \beta (t))+\frac{\beta -v}{1+v^2} & \text{ if } & v(t) \geq \beta (t)\\
g(t,v(t))       & \text{ if } & \alpha (t) \leq v(t) \leq \beta (t)\\
g(t, \alpha (t)) +\frac{\alpha -v}{1+v^2}& \text{ if } & v(t) \leq \alpha (t)
\endmatrix  \right .$$
For each $v \in C ^0[a,b],$ let $u=Av$ be the solution of
$$\align &(\varphi (u' ))' = - \tilde g (t,v),\\
& u(a) = 0 = u (b) \endalign$$
Note that
$$u(t) = \int ^t _a \varphi ^{-1} [ c - \int ^s _a \tilde
 g (\tau , v) d \tau] ds$$
where $c$ is such that $u(b)=0$.  Then $A: C^0[a,b]\to C^0[a,b]$ 
is a completely
continuous mapping.  Since $A$ is bounded, it follows from the 
Schauder fixed point
theorem that $A$ has a fixed point $u$.  We verify that $\alpha (t)
\leq u (t)\leq \beta
(t),~
a\leq t\leq b.$ Indeed, if there exists $t_0 \in [a,b]$ such that 
$u(t_0) < \alpha  (t_0)$, then the continuous function
$v=u-\alpha $ will have a negative minimum, say $d=v(\tilde t)$ and
 there exists an interval $(t_1, t_2)$ containing $\tilde t$ with
$v(t_1)=v(t_2)=0,~v(t)<0,~t\in(t_1,t_2)$ Furthermore $v$ is left and
right differentiable at every point in $(a,b).$ Thus $v'_+(\tilde
t)\geq
v'_-(\tilde t),$ where $v'_{\pm}$ denote the right and left
derivatives.
On the other hand, since $u$ is differentiable, the conditions on
$\alpha $
imply that  $v'_+(\tilde
t)=
v'_-(\tilde t),$ and $v$ is differentiable at $\tilde t.$ We next
employ the
differential equation satisfied by $u$ and the fact that $\alpha $
is a lower solution to conclude that $v$ has a local maximum at
 $\tilde t,$ which is 
in contradiction to the fact that $v$ assumes a global minimum 
at $\tilde t.$   Similarly, we have $u(t) \leq \beta (t),~a\leq
t\leq b$
 and thus $u$ is
a solution of (2.1).

Let $U = \{ v \in C^0[a,b] : v \geq \alpha$ and $v$ is an upper solution to
(2.1)$\}.$ 
Define $u_{ \text{min}} (t) = \text{ inf } \{ v(t): v \in U \}.$
Using the argument in [8, p.279] (also [1]), it can be verified that
$u_{\text{min}}$ is the  minimum solution of (2.1) whose
existence
 was asserted.
\enddemo
\heading 3 Existence results 
\endheading
In this section, we prove theorem 1.  Since we are interested in
nonnegative
 solutions
we shall make the convention that
$f(t,u)=f(t,0)$ if $u<0.$ We shall denote by $|\cdot |_k$ the norm
in the space $C^k[a,b].$

\proclaim{Lemma 3}  Let $v \in C^0[a,b]$ with $v \leq 0$ and let $u$ satisfy
$$(\varphi (u'))' = v$$
$$u(a)=0=u(b).$$
Then
$$u(t) \geq |u| _0 p(t), \quad t \in [a,b]$$
where
$$p(t)= \frac {\min (t-a, b-t)}{b-a}$$
\endproclaim
\demo{Proof} 
Since $\varphi (u')$ is nonincreasing and $\varphi ^{-1}$ is increasing it
follows that $u'$ is nonincreasing.  Hence, lemma 3 follows from lemma 2.2
in  [3]. For convenience to the readers, we give a direct proof. Let $|u|_0 =
 |u(T)|,~ T \in  [a,b]$. Since u is concave and $u(a) = 0$, it follows that
$$\aligned
&u(t) = u(cT + (1-c)a)\geq cu(T)\\
&\geq\frac{t-a}{b-a} |u|_0, \quad t \in [a,T] \endaligned$$
where $c= \frac{t-a}{T-a}$. Similarly,
$$ u(t)\geq\frac{b-t}{b-a} |u|_0, \quad t \in [T,b]$$
completing the proof of lemma 3.
\enddemo
\proclaim{Lemma 4}  Suppose that $g:[a,b] \times {\Bbb R}^+ \to {\Bbb R} ^+$
is  continuous and there exists a positive number $M$ and an interval
 $[a_1,b_1] \subset
(a,b)$ such that
$$g(t,u) \geq M(\varphi(u) +1),~ t \in [a_1,b_1],~ u \geq 0. \tag 3.1$$

There exists a positive number $M_0=M_0(\varphi, a_1,b_1)$ such that 
the problem
$$\aligned
& (\varphi (u'))' = -g (t,u)\\
& u(a)=0=u(b) \endaligned \tag 3.2$$
has no solution whenever $M \geq M_0.$
\endproclaim
\demo{Proof} Let $u$ be a solution of (3.2).  Then
$$u(t)= \int ^t_a \varphi ^{-1} [c - \int ^s _a g (\tau ,u) d \tau ]
ds
 \tag 3.3$$
where $c = \varphi (u'(a)).$
Let $|u|_0=u(t_0),~t_0 \in [a,b].$  Then $u'(t_0)=0$ and by (3.3),
$$u(t)= \int ^t _a \varphi ^{-1} [ \int ^{t_0} _s g(\tau ,u) d \tau ]
ds
 \tag 3.4$$
If $t_0 \geq \frac {a_1+b_1}{2}$, then
$$
\aligned
|u|_0 &\geq u(a_1) > \int ^{a_1} _a \varphi ^{-1} [M \int ^
{\frac {a_1+b_1}{2}} _{a_1}
(\varphi (u)+1)]\\
&>(a_1-a) \varphi ^{-1} [ M \frac {(b_1-a_1)}{2} [ \varphi
 ( |u| _0 \delta) +1]],
\endaligned
$$
where
$$  \delta = \min _{a_1 \leq t \leq b_1} p(t).
$$
This implies
$$\varphi ( \frac {|u|_0}{a_1-a} ) > M \frac {(b_1-a_1)}{2}
 [ \varphi ( |u| _0 \delta)
+1] \tag 3.5$$
If $t_0 \leq \frac {a_1+b_1}{2}$, then by rewriting (3.4) as
$$u(t) = \int ^b _t \varphi ^{-1} [ \int ^s _{t_0} g (\tau , u) d \tau ] ds$$
we deduce
$$\varphi \left ( \frac {|u|_0}{b-b_1} \right ) > \frac {M(b_1-a_1)}{2}
[ \varphi ( |u|_0 \delta) +1] \tag 3.6$$
Combining (3.5) and (3.6), we obtain
$$\varphi (\gamma |u|_0) > \frac {M(b_1-a_1)}{2} 
[ \varphi (|u| _0 \delta) +1]$$
where $\gamma = \max \left (\frac {1}{b-b_1}, \frac {1}{a_1-a} \right
).$
\newline
Consequently,
$$\frac {\varphi (\gamma |u| _0)}{\varphi ( \delta |u| _0)} > 
\frac M2 (b_1-a_1)$$
a contradiction to (A.1) if $M$ is sufficiently large.
\enddemo
\proclaim{Remark 5}  It follows from  the proof,  that problem 
{\rm (3.2)} has no solution $u$ satisfying
$$g(t, u(t)) \geq M (\varphi (u(t)) +1), \quad t \in [a_1,a_2],$$
if $M \geq M_0.$
\endproclaim

These considerations further imply the following result: 

\proclaim{Theorem 6}  There
exists a positive number $\bar {\lambda}$ such that problem $(1.1)_{\lambda}$
has no solution for $\lambda > \bar {\lambda}.$
\endproclaim
\demo{Proof}
It follows immediately from (A.2) that there exists a constant $\mu >0$ such 
that
$$
f(t,u)\geq \mu (\varphi (u)+1),~u\in {\Bbb R}^+,~c\leq t\leq d.
$$
Hence the result follows from the previous lemma.
\enddemo
\proclaim{Lemma 7} For each $\mu > 0$, there exists a positive constant 
$C _{\mu}$ such that the problem
$$\align( \varphi (u'))' = - \lambda & \theta f (t,u) - (1- \theta) M _0 
(| \varphi (u)| +1)\\
& u(a) = 0 = u(b) \tag 3.7 \endalign$$
with $\lambda \geq \mu, \, \theta \in [0,1]$ and $M _0$ given by remark 5, 
has no solution satisfying $|u|_0 > C _{\mu}.$
\endproclaim
\demo {Proof} Let $u$ be a solution of (3.7) with $\lambda \geq \mu$
and $\theta \in [0,1]$.  Then $u \geq 0$.  By (A.2), there exists $M_1
> 0$
 such
that 
$$f(t, u) > \frac {M_0}{\mu} (\varphi (u) +1) \tag 3.8$$
for $t \in [c,d]$ and $u \geq M _1.$
Let $\delta = \min _{c \leq t \leq d} p(t)$.  Then if $|u|_0 > \frac
{M_1}
{\delta}$ we
have by lemma 3
$$u(t) \geq |u|_0 \delta > M _1, \quad t \in [c,d]$$
which implies by (3.8) that
$$\align
& \lambda \theta f (t, u (t)) + (1 - \theta ) M _0 ( \varphi (u (t)) +1) \\
\geq & \frac {\lambda \theta M _0}{\mu} (\varphi (u(t)) +1) + (1-\theta) M _0
\varphi (u(t)) +1)\\
\geq & M _0 (\varphi (u(t))+1), \quad t \in [c,d] \endalign$$
a contradiction with remark 5, and the lemma is proved.
\enddemo
Now, let $\Lambda$ be the set of all $\lambda > 0$ such that $(1.1)_\lambda$
has a solution and let $\lambda ^* = \text{sup } \Lambda$.  Note that by
lemma 3, every solution of $(1.1) _\lambda$ is positive.

\proclaim{Lemma 8} $0 < \lambda ^* < \infty$ and $\lambda ^* \in \Lambda .$
\endproclaim
\demo{Proof} $u\in C^0[a,b]$ is a solution of (1.1)$_ {\lambda}$ if and only if
$u=F(\lambda ,u),$ where
$$
F:[0,\infty)\times C^0[a,b]\to C^0[a,b]
$$ is the completely continuous mapping given by
$$
u=F(\lambda, v),
$$
with $u$ the solution of
$$\align &(\varphi (u' ))' = - \lambda f(t,v),\\
& u(a) = 0 = u (b).
\endalign
$$
We note that $F(0,v)=0,~v\in C^0[a,b].$ Hence it follows from the
 continuation theorem
of Leray-Schauder that there exists a solution continuum $\Cal{C}
 \subset [0,\infty )\times
C^0[a,b]$ of
solutions of (1.1)$_{\lambda}$ which is unbounded in $[0,\infty )
\times C^0[a,b],$
and thus, (1.1)$_ {\lambda}$ has a solution for $\lambda > 0$ 
sufficiently small,
and hence $\lambda ^* > 0$. By theorem 6 , $\lambda ^* < \infty$. 
 We verify that $\lambda
^* \in \Lambda$. Let $\{\lambda _n\} _n \subset \Lambda$ be such 
that $\lambda _n \to \lambda
^*$ and let $\{u _n\}$ be the corresponding solutions of 
(1.1)$_{\lambda _n}$.  By 
lemma 7, $\{u _n\}$ is bounded in $C^1[a,b]$ and hence $\{u_n\}$ 
has a subsequence
converging to $u \in C ^0[a,b] $.  By standard limiting procedures, 
it follows
that $u$ is a solution of (1.1)$ _{\lambda ^*}.$
\enddemo
\proclaim{Lemma 9} Let $0 < \lambda < \lambda ^*$ and let $u _{\lambda ^*}$
be a solution of $(1.1) _{\lambda ^*}.$  Then there exists $\epsilon _0> 0$
such that $u _{\lambda ^*} + \epsilon, ~0\leq \epsilon
\leq \epsilon _0$ is an upper solution of 
$(1.1) _{\lambda}.$
\endproclaim

\demo{Proof} Let $c_1 > 0$ be such that $f(t, u _{\lambda ^*} (t)) \geq
c_1$ for every $t \in [a,b]$ and let $\epsilon _0>0 $ be such that
$$|f(t, u _{\lambda ^*} (t) + \epsilon) - f (t, u _{\lambda ^*} (t)) |
< \frac {c_1 (\lambda ^* - \lambda)}{\lambda}, \quad t \in [a,b],~
 0\leq \epsilon
\leq \epsilon _0.
$$
Then we have
$$\align
(\varphi (u' _{\lambda ^*}))' & = - \lambda ^* f (t, u _{\lambda ^*}) =
- \lambda f (t, u _{\lambda ^*} + \epsilon )+ \\
& + \lambda [ f (t, u _{\lambda ^*} + \epsilon) -f (t, u _{\lambda ^*})]
+ (\lambda - \lambda ^*) f (t, u _{\lambda ^*}) \\
& \leq - \lambda f (t, u _{\lambda ^*} + \epsilon) 
\endalign $$
i.e. $u _ \lambda ^* + \epsilon$ is an upper solution of $(1.1) _ \lambda.$
\enddemo

\demo{Proof of theorem 1}  Let $0 < \lambda < \lambda ^* .$  Since
$0$ is a lower solution and $u _{\lambda ^*}$ is an upper solution, there
exists a minimum solution $u _ \lambda $ of $(1.1) _ \lambda$ with
$0 \leq u _{\lambda} \leq u _{\lambda ^*}.$  We next establish the existence
of a second solution to $(1.1) _{\lambda}$. 

Let $F(\lambda, u)$ be defined as in the proof of lemma 8. Further
define
$$\tilde f (t,v(t)) = \left \{ \matrix
f(t, u _{\lambda ^*}(t) + \epsilon) & \text{if} & v(t) \geq u_{\lambda
^*}(t)
 + \epsilon \\
f(t, v(t)) & \text{if} & - \epsilon \leq v(t) \leq u _{\lambda ^*} (t)
+
 \epsilon \\
f(t, - \epsilon) & \text{if} & v(t) \leq - \epsilon \endmatrix \right .$$
where $\epsilon$ is given in lemma 9, and let
$\tilde F(\lambda ,u)$ be the operator analogous to $F$ defined by
$\tilde f.$
 Consider
$$B= \left \{ u \in C^0[a,b] : - \epsilon < u (t) < u _{\lambda ^*}
(t) + 
\epsilon ,
\quad t \in [a,b] \right \}.$$
Then $B$ is open and $u _{\lambda} \in B,~0\leq \lambda \leq \lambda ^*.$
 Since $\tilde F$ is bounded for $\lambda $ in compact intervals,
$$\text{deg } (I- \tilde F (\lambda ,\cdot ), B (u _{\lambda}, R), 0) =1$$
if $R$ is sufficiently large.  Here $B(u_{\lambda}, R)$ is the ball
centered at $u_{\lambda}$ with radius $R$ in $C^0[a,b].$
If there exists $u \in \partial B$ such that $u = \tilde F(\lambda ,
u)$ 
then $u$ is
a second solution of $(1.1) _{\lambda}$.  Suppose that
 $u \neq \tilde F(\lambda ,u)$
for every $u \in \partial B.$  Then $\text{deg }
(I- \tilde F(\lambda ,\cdot), B, 0) $ is
well defined and since  $\tilde F (\lambda ,\cdot )$ has no fixed point in
$B(u_{\lambda}, R)\setminus B$ ( see e.g [9]), we have by the excision property

$$\text{deg } (I -F(\lambda ,\cdot ), B, 0) = 
\text {deg } (I - \tilde F(\lambda ,\cdot ), B,
0) =1,~ 0\leq \lambda \leq \lambda ^*.
$$
On the other hand, it follows from lemma 7 that for $\mu >0 $ 
there exists $M>0 $ such that
for $\lambda \geq \mu,  $ $\lambda $ in compact intervals,
$$
\text{deg } (I -F(\lambda ,\cdot ), B(0,M), 0) = \text{constant},
$$
where $B(0,M)$ is the ball centered at 0 of radius $M$ in $C^0[a,b].$ 
The latter degree,
on the other hand, must equal 0, since for $\lambda >\lambda ^*$ no 
solutions exist.
Thus the existence of a second solution follows from the excision 
principle of the 
Leray-Schauder degree.
\enddemo

We remark that theorem 2, together with lemma 9 (appropriately
interpreted), 
implies that the
mapping
$$
\lambda \mapsto u_{\lambda},~ 0\leq \lambda \leq \lambda ^*,
$$
where $ u_{\lambda}$ is the minimal solution of (1.1)$_{\lambda }$
is a continuous mapping $[0,\lambda ^*]\to C^0[a,b].$ For it is the
case
that for any $\tilde \lambda \in (0, \lambda ^*]$
the minimal solutions $\{u_{\lambda }$ satisfy
$u_{\lambda}\leq u_{\tilde \lambda},~\lambda \leq \tilde \lambda .$
 Furthermore the limit $u=\lim_{\lambda \to \tilde \lambda}u_{\lambda}$
exists and is a solution of  (1.1)$_{\tilde \lambda }.$ Hence
 $u_{\tilde \lambda}=\lim_{\lambda \to \tilde \lambda}u_{\lambda}.$
It therefore follows that
$$
\{(\lambda , u_{\lambda}),~ 0\leq \lambda \leq \lambda ^*\}\subset \Cal C,
$$
where $\Cal C$ is the continuum in the proof of lemma 8. 
Using separation results on closed
sets in compact metric spaces 
(Whyburn's lemma), one may use the arguments used in the above proof
to verify that for each $\lambda \in (0,\lambda ^*)$ there are at
least 
two solutions
on the continuum $\Cal C.$



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\ref \no 9 \by M. Otani and T. Teshima \pages 8-10
\paper On the first eigenvalue of some quasilinear elliptic equations
\yr 1988 \vol 64
\jour Proc. Japan Acad. \endref

\endRefs


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