Exact multiplicity for degenerate two-point boundary value problems with p-convex nonlinearity

Nonlinear Analysis 52 (2003) 1569–1590www.elsevier.com/locate/na

Exact multiplicity for degenerate two-pointboundary value problems with p-convex

nonlinearity�

J. Kar)atson, P.L. Simon∗

Department of Applied Analysis, E�otv�os Lor�and University, Budapest, Hungary

Received 27 March 2001; accepted 10 April 2002

Abstract

The exact number of positive solutions of a degenerate quasilinear two-point boundary valueproblem is investigated. For the generalization of earlier results concerning the non-degeneratecase with convex nonlinearity, suitably de6ned p-convex nonlinearities are considered. Strictlyp-convex C2 functions having a non-negative root are classi6ed according to the shape of thebifurcation diagram of positive solutions versus the length of the interval. We have uniquenesswhen f(0)6 0 and 0, 1 or 2 solutions when f(0)¿ 0 (similarly to the non-degenerate case),provided that the number of solutions is 6nite. However, now there may also occur a continuumof solutions, connected to a dead core type phenomenon. The proof of our results relies onthe shooting method for the characterization of the shape of the time-map. In contrast to thenon-degenerate case, the shooting method does not determine directly the number of solutions,owing to the lack of uniqueness of the corresponding IVP. Exact conditions on the uniquenessof the IVP and, in the case of non-uniqueness, the number and types of its local solutions aregiven. Based on this, all the positive solutions of the BVP can be compiled.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Degenerate two-point boundary value problem; Time-map; Bifurcation diagram of positivesolutions; Dead core; Non-uniqueness of IVP

� This research was supported by the Hungarian National Research Funds AMFK under Magyary Zolt)anScholarship, OTKA under grant no. F034840 and FKFP under grant no. 0049/2001.

∗ Corresponding author.E-mail addresses: [email protected] (J. Kar)atson), [email protected] (P.L. Simon).

0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(02)00268 -7

mailto:[email protected]

mailto:[email protected]

1570 J. Kar�atson, P.L. Simon /Nonlinear Analysis 52 (2003) 1569–1590

1. Introduction

We investigate the exact number of positive solutions of the degenerate quasilinearboundary value problem

(|u′|p−2u′)′ + f(u) = 0; (1)

u(−R) = u(R) = 0; u¿ 0 in (−R; R); (2)

where p¿ 2 and f is a C1 function on [0;∞) whose roots are isolated. We will alsoassume that f satis6es a so-called p-convexity property, a natural generalization ofconvexity. A solution (de6ned precisely later) is not required to be C2, but only tohave the smoothness demanded by the equation.We note that the relation p¿ 2 is signi6cant both in our de6nition of p-convexity

and in the treatment of the degeneracy in (1). (Namely, De6nition 3 cannot be re-peated for p¡ 2 unless the condition f∈C1[0;∞) is violated, further, for p¡ 2 thecoeJcient of u′′ in the formal equation (5) would become ∞ instead of 0.) Hence thecase p¡ 2 is not covered by the presented results and may be the subject of furtherresearch.The non-degenerate case p = 2 is a background for our investigations. Multiplicity

of solutions has been extensively studied, but most results give an estimate for thenumber of solutions, and exact multiplicity is a much more diJcult issue deserving at-tention even in one dimension [19,28,29]. Especially, it has turned out that for (strictly)concave and convex functions f the exact multiplicity can often be obtained. For con-cave nonlinearity the sign of f(0) essentially determines the number of solutions, andthe obtained classi6cation extends to semilinear equations in several dimensions [3–5].In general words, we have uniqueness when f(0)¿ 0 and 0, 1 or 2 solutions whenf(0)¡ 0. For convex nonlinearity the sign of f(0) also plays an important role, but thenumerous results in several dimensions essentially concern special cases of functionsf [15,20,25,26,31]. However, in one dimension the convex case can also be classi6ed[16,21]: as a counterpart of the concave case, we have uniqueness when f(0)6 0and 0, 1 or 2 solutions when f(0)¿ 0. In several dimensions the known results forf(0)6 0 suggest that uniqueness is preserved, but for f(0)¿ 0 the behaviour is muchmore complicated, even including the existence of countably many solutions. We notehere that linear functions f are a limiting case between concave and convex ones(yielding uncountably many or no solutions).In the degenerate case p¿ 2 multiplicity of solutions has been widely investi-

gated, also including the higher dimensional case. Important investigations in one di-mension are found in [14,23]; in the case of a ball we refer to the summary [22]and the references there. The case of combined p-convex–p-concave nonlinearitieson a general domain has been studied using a sophisticated combination of vari-ational and topological methods in [1,12]. The analogue of the linear case is the(p-linear) function f(u) = |u|p−2u, yielding similarly no solution or uncountablymany eigenfunctions [11]. This suggests that the suitably de6ned classes of p-convexand p-concave functions are separated by the p-linear one, and the results corre-sponding to convex and concave nonlinearity in the non-degenerate case hold for

J. Kar�atson, P.L. Simon /Nonlinear Analysis 52 (2003) 1569–1590 1571

these classes. Indeed, in the case f(0)¿ 0, de6ning p-concavity with f(u)=up−1

decreasing, the uniqueness result is preserved [9] even in the n-dimensional case.However, according to our knowledge, no p-convex case has beeninvestigated.We remark that the distinction between p-convex and p-concave functions is also

relevant concerning the problem of stability for the corresponding parabolic problem.Namely, the positive solutions are unstable (resp. stable) in the case of p-convex (resp.p-concave) nonlinearity under a suitable sign condition for f(0) [18]. The same resulthad been proved for p= 2 in a more general setting in [17,24].In this paper our aim is 6rst to give a natural de6nition of p-convexity, then to

determine the bifurcation diagram of positive solutions versus the radius R for strictlyp-convex functions.There is no unique straightforward de6nition of p-convexity. We follow the idea

of [9,14], but compare f′(u) with growth of power p − 2 and avoid the restric-tion f(0)6 0. Our de6nition essentially requires that f is convex and its growthexceeds power p − 1. We do not consider decreasing functions since in this caseuniqueness immediately follows for (1) and (2) from the monotone operatorproperty.The result of this paper is the classi6cation of strictly p-convex C2 functions having

a non-negative root according to the number of positive solutions. Essentially, we haveuniqueness when f(0)6 0 and 0, 1 or 2 solutions when f(0)¿ 0 (similarly to thenon-degenerate case), provided that the number of solutions is 6nite. However, nowthere may also occur a continuum of solutions, and this phenomenon arises when thesolution may contain a dead core, i.e. it is constant on a subinterval.These main results are contained in Section 5. Besides the above distinction ac-

cording to the sign of f(0), we consider the growth of f at in6nity as a secondaspect of classi6cation. The latter has two types from our point of view (super-p-linearor asymptotically p-linear), thus we obtain four cases, dealt with inTheorems 1–4.Our main tool is the shooting method, which enables us to study multiplicity us-

ing the time-map. In the non-degenerate case it has been widely applied e.g. in[2,28,29]. Related arguments in the degenerate case are found e.g. in [7,14,23,27].In contrast to the non-degenerate case, the uniqueness of the corresponding initialvalue problems may be violated, hence the solutions of (1)–(2) cannot be obtaineddirectly from them. Therefore, we give exact conditions on the uniqueness of theIVP and, in the case of non-uniqueness, on the number and types of its local so-lutions. Based on this setting, we introduce the notion of minimal solutions whichmakes it possible to de6ne the time-map. Then the shape of the time-map is foundvia its implicit equation, using the following three characteristic properties: the do-main of the time-map, the limit of the time-map at the boundary points of its domain,and the monotonicity of the time-map on the maximal subintervals of its domain.The investigation of these properties relies strongly on the integral formula of thetime-map and Sturm-type theorems. The shape of the time-map determines the num-ber of minimal solutions, from which all the positive solutions of (1) and (2) can becompiled.


2. Preliminaries

In this section we summarize some basic results required for our investigations. Mostof these results are known, see e.g. [14,23,27], and are only presented for the reader’sconvenience.

2.1. First-order formulations

Let us introduce the function

’(x) = |x|p−2x (x∈R): (3)

De�nition 1. The function u∈C1(a; b) is called a solution of (1) on the interval(a; b) if

(i) ’ ◦ u′ is C1 on (a; b),(ii) (’ ◦ u′)′ + f ◦ u= 0 on (a; b).

The following properties of solutions are found e.g. in [14,23,27]. Namely, thesepropositions formulate the solution via an integro-diNerential equation, a kind of strongform and a Hamiltonian function, respectively.

Proposition 1. The function u∈C1(a; b) is a solution of (1) on the interval (a; b) ifand only if for some r0 ∈ (a; b)

’(u′(r)) = ’(u′(r0))−∫ r

r0f(u(s)) ds (r ∈ (a; b)): (4)

Proposition 2. Let u : (a; b) → R be a solution of (1) on the interval (a; b). If forsome r ∈ (a; b) u′(r) �= 0, then u is C2 in a neighbourhood of r and satis6es theequation

(p− 1)|u′|p−2u′′ + f(u) = 0: (5)

Let

F(x) =∫ x

0f(s) ds: (6)

Proposition 3 (See Guedda and Veron [14] and Otani [27]). Let u∈C1(a; b) be notconstant on any open subinterval of (a; b). Then the following two statements areequivalent:

(i) u is a solution of (1) on (a; b)


(ii) The Hamiltonian

H (r) =p− 1p

|u′(r)|p + F(u(r)) (r ∈ (a; b)) (7)

is constant on (a; b).

2.2. The initial value problem

As we shall see later, the solutions of (1) and (2) can be obtained as solutions (orjunction of 6nite number of solutions) of the following initial value problem (IVP) forsome c; r0 ∈R:

(|u′|p−2u′)′ + f(u) = 0; (8)

u(r0) = c; u′(r0) = 0: (9)

In this subsection we investigate the local solutions of the above IVP. For the casep¿ 2 even the local behaviour of solutions is non-trivial since we have no uniquenessin general.As follows from the proof of Proposition 3, the roots of u′ are isolated for any

locally non-constant solution of (8). Therefore the sign of u′ is constant in a left andright neighbourhood of r0:

Proposition 4. If u is a solution of (8) and (9), then there exists �¿ 0 such that oneof the following three statements holds:

u′|(r0 ; r0+�) ≡ 0; u′|(r0 ; r0+�)¿ 0; u′|(r0 ; r0+�)¡ 0:

De�nition 2. The functions satisfying the above sign conditions are called right locallyconstant/increasing/decreasing, Presp.

Remark 1. The statement of Proposition 4 is valid on (r0 − �; r0), too. The solutionssatisfying the sign conditions in a left neighbourhood of r0 are called left locallyconstant/increasing/decreasing, resp.

In order to establish the existence of solutions satisfying these lower and upper signconditions, we introduce the following sets:

Dlow =

{c¿ 0: ∃�¿ 0; f|(c−�; c)¿ 0;

∫ c

c−�

1p√F(c)− F(s)

ds¡∞};

Dup =

{c¿ 0: ∃�¿ 0; f|(c;c+�)¡ 0;

∫ c+�

c

1p√F(c)− F(s)

ds¡∞}:


Now we establish how the exact number of local solutions of (8) and (9) depends onwhether c is in Dlow or Dup.

Lemma 1. (i) System (8) and (9) has at most one locally constant solution. It existsif and only if f(c) = 0.(ii) System (8) and (9) has at most one right locally increasing (left locally

decreasing) solution. It exists if and only if c∈Dup.(iii) System (8) and (9) has at most one right locally decreasing (left locally

increasing) solution. It exists if and only if c∈Dlow.

Proof. (i) is obvious.(ii) Let u be a right locally increasing solution of (8) and (9) on [r0; r0 + �). Then

Proposition 3 implies that

p− 1p

|u′(r)|p + F(u(r)) = F(c) (r ∈ [r0; r0 + �)):

Since u′(r)¿ 0 (for r ∈ (r0; r0 + �)), therefore

�pu′(r)

p√F(c)− F(u(r))

= 1; (10)

where

�p =(p− 1p

)1=p

: (11)

Integration from r0 to r implies c∈Dup and

�up(u(r)) =r − r0�p

;

where the function �up is de6ned by

�up(x) =∫ x

c

1p√F(c)− F(s)

ds (12)

on a right neighbourhood of c. This solution u is unique since it is given by the formula

u(r) = �−1up

(r − r0�p

): (13)

The proof of the opposite direction is similar. If c∈Dup, then it is easy to see that ude6ned by (13) is a right locally increasing solution.The proof of the existence and uniqueness of left locally decreasing solution follows

the same way where in (10) we have −1 on the right side and we integrate from r tor0. Then one obtains the following formula for the solution:

u(r) = �−1up

(r0 − r�p

):


(iii) The proof of this part is similar to that of (ii), now using the function

�low(x) =∫ c

x

1p√F(c)− F(s)

ds: (14)

(Then the right locally decreasing solution is determined by

�low(u(r)) =r − r0�p

; (15)

the left locally increasing one by the same with r0 − r.)

Remark 2. It is easy to check the following statements:

1. If f(c)¿ 0, then c∈Dlow and c �∈ Dup.2. If f(c)¡ 0, then c∈Dup and c �∈ Dlow.3. If f(c) = 0 and f is locally negative from the left-hand side, then c �∈ Dlow.4. If f(c) = 0 and f is locally positive from the right-hand side, then c �∈ Dup.

According to Proposition 4, the restriction of any solution of problem (8) and (9)to a suJciently small right neighbourhood of r0 belongs to one of the following types:right locally constant, right locally increasing or right locally decreasing. Moreover,according to Lemma 1 the problem can have at most one solution of each type. Hence,the problem can have at most three solutions on the given right neighbourhood of r0,and similarly for a suJciently small left neighbourhood of r0. Concerning the possiblenumber of solutions we may rely on Remark 2, and altogether we get the followingstatements.

Corollary 1. (i) If f(c) �= 0, then the local solution of (8) and (9) is unique. (In casef(c)¿ 0 it is locally less than c, in case f(c)¡ 0 it is locally greater than c.) Iff(c) = 0 and c �∈ Dlow ∪Dup, then the local solution of (8) and (9) is unique (and itis constant).(ii) If f(c) = 0 and f changes its sign from negative to positive, then the local

solution of (8) and (9) is unique, and it is constant.(iii) Local non-uniqueness may occur only in the case when f(c) = 0 and f is

either locally positive from left or locally negative from right. Altogether there maybe at most 3 local solutions on the left and at most 3 local solutions on the right.

Proposition 5. If c∈Dup, then (8) cannot have a solution for which u(r)¿c , u′(r)¡ 0 on some interval (r0;∞) and lim∞ u= c.

Proof. Assume for contrary that there exists a solution with the above properties. Thenaccording to (5) u′′(r)¿ 0. Hence lim∞ u′ = 0 and Proposition 3 imply

p− 1p

|u′(r)|p + F(u(r)) = F(c):


Rearranging and integrating from r0 to ∞ we get∫ u(r0)

c

1p√F(c)− F(s)

ds=∞;

which contradicts c∈Dup.

3. p-convex functions

Based on the considerations in the Introduction, we will use the following notion of(strict) p-convexity.

De�nition 3. Let p¿ 2, and let f∈C1[0;∞) have an absolute minimum.f is called (strictly) p-convex if for any absolute minimizer �∈ [0;∞) the function

k(u) :=f′(u)

|u− �|p−2 (u∈ [0;∞); u �= �)

is (strictly) increasing.

Remark 3. (Strict) 2-convexity coincides with (strict) convexity within the consideredclass.

Now some properties of (strictly) p-convex functions are proved.

Proposition 6. If f is strictly p-convex then f is strictly convex.

Proof. Let � be an absolute minimizer of f. Then f′(�) = 0. (The only possibleexceptional case �= 0, f′(0)¿ 0 is excluded since then lim0+ k =+∞ would hold.)Owing to

f′(u) = k(u)|u− �|p−2; (16)

it suJces that

k(u)¿ 0(u¿�); k(u)¡ 0 (u¡�) (17)

in order to prove that f′ strictly increases. Let 6rst �¿ 0. Since k may have at mostone root, therefore f′ does not change sign locally on each side of �, which impliesf′(u)¡ 0 (u¡�) and f′(u)¿ 0 (u¿�) locally (since � is a minimizer). Using (16),this implies (17) locally, and thus (since k increases) also globally. Finally, for �= 0the above argument is needed to the right of � only.

Remark 4. Similarly, if f is p-convex then f is convex. (Then in the proof the rootsof k may form an interval on which f is constant.)

Example. Let �¿ 0, b∈R, f(u) := |u − �|q + b. Then f is (strictly) p-convex forq¿p− 1 (q¿p− 1).


Remark 5. If f is strictly p-convex, then the minimizer �∈ [0;∞) is unique.

Proposition 7. Let f∈C2 be (strictly) p-convex, � a minimizer,

l(u) := f′(u)(u− �)− (p− 1)f(u) (u¿ 0): (18)

Then

(i) l is (strictly) decreasing on [0; �] and (strictly) increasing on [�;∞);(ii) if f(�)6 0 then l(u)¿ 0 (l(u)¿ 0) for u∈ [0;∞); u �= �.

Proof. (i) Using (16), we have

l(u) = k(u)|u− �|p−2(u− �)− (p− 1)f(u);

l′(u) = k ′(u)|u− �|p−2(u− �)

for all u∈ [0;∞); u �= �. Since k is (strictly) increasing, (i) is proved.(ii) This follows from (i) and l(�) =−(p− 1)f(�)¿ 0.

Proposition 8. Let f∈C2 be (strictly) p-convex, � a minimizer. Then the function

K(u) :=f(u)− f(�)

|u− �|p−2(u− �)(u∈ [0;∞); u �= �)

is (strictly) increasing on [0;∞).

Proof. There holds

K ′(u) =l(u) + (p− 1)f(�)

|u− �|p :

Let n(u) := l(u)+(p−1)f(�). Then n(�)=0, hence for u �= � Proposition 7(i) impliesn(u)¿ 0 (n(u)¿ 0), i.e. K ′(u)¿ 0 (K ′(u)¿ 0) for all u �= �.

Remark 6. In the special case �=0 and f(�)6 0, Proposition 8 implies that f(u)=up−1

is increasing, which is the p-convex analogue of the function class studied in [9] (whichmight be called p-concave).

4. The time-map

4.1. De6nition and basic equations

In this subsection we consider the global solutions of the IVP

(|u′|p−2u′)′ + f(u) = 0; (19)

u(0) = c; u′(0) = 0: (20)

According to Corollary 1 local uniqueness can be violated only at the roots of f,which are assumed to be isolated, and according to Lemma 1 there are at most three


local solutions of (8) at the roots of f. Therefore, on any interval [0; R] where (19)and (20) has a positive solution it also has a unique minimal global positive solution.Let us denote this solution by u(r; c).Now we can de6ne the time-map in the usual way [16,28,30]:

De�nition 4. The time-map associated to the above initial value problem is the fol-lowing function T :

D(T ) = {c¿ 0: ∃r ¿ 0 u(r; c) = 0}; T (c) = min{r ¿ 0: u(r; c) = 0}:

Proposition 9. For all c∈D(T ), r → u(r; c) is strictly decreasing.

Proof. Let u(r) = u(r; c). Arguing by contradiction, let d be the minimal value forwhich there exist r1¡r2 in [0; T (c)) such that u(r1) = u(r2) = d. Then one of thederivatives u′(r1) and u′(r2) is 0, hence according to Proposition 3 the other one is 0,too. Using this we can de6ne the following solution of (19) and (20):

Ru(r) =

{u(r) if r ¡ r1;

u(r + r2 − r1) if r¿ r1:

Then Ru(T (c)− r2 + r1) = 0, which contradicts the minimality of u.

The function T is determined by the implicit equation

u(T (c); c) ≡ 0 (21)

and the assumption u(r; c)¿ 0 if r ∈ [0; T (c)).DiNerentiating (21) one gets the following equations for the derivative of T :

@ru(T (c); c)T ′(c) + @cu(T (c); c) ≡ 0: (22)

DiNerentiating (19) and (20) with respect to c and introducing the notation h(r; c)=@cu(r; c), we get for all c∈D(T )

(p− 1)(|u′|p−2h′)′ + f′(u)h= 0; (23)

h(0; c) = 1; h′(0; c) = 0: (24)

Namely, when u′(r) �= 0 then the required equation follows from the strong form (5),whose derivative w.r.t. c is

0 = (p− 1)[(p− 2)|u′|p−4u′u′′h′ + |u′|p−2h′′] + f′(u)h

= (p− 1)(|u′|p−2h′)′ + f′(u)h;

using the continuity of @cu which follows from that of f′. Then the equality extendsby continuity to the (isolated) roots of u′.DiNerentiating (19) with respect to r where u′(r) �= 0 and introducing notation

v(r; c) = @ru(r; c), we similarly get

(p− 1)(|u′|p−2v′)′ + f′(u)v= 0: (25)


Using the above notations, (22) is written as

v(T (c); c)T ′(c) + h(T (c); c) ≡ 0: (26)

Besides the implicit equation (21), there holds an integral formula for the time-map:using Proposition 9, we may substitute r0=0 and r=T (c) into (14) and (15) to obtain

T (c) = �p

∫ c

0

1p√F(c)− F(s)

ds; (27)

where �p is de6ned in (11).

4.2. The shooting method

The essence of the shooting method is to obtain the solutions of the boundary valueproblem (1) and (2) as solutions of the initial value problem (19) and (20). This canbe done in the case when all solutions of the boundary value problem (1) and (2)are symmetric to 0 (e.g. in the non-degenerate case p = 2). As we will see, in thedegenerate case p¿ 2 the boundary value problem may have solutions not symmetricto 0. In this subsection we give a suJcient condition for f to preserve the symmetryof solutions for p¿ 2.In order to prove symmetry, usually a reSection is combined with uniqueness. (This

can be used also to prove radial symmetry of solutions on balls [13].) In the degeneratecase uniqueness may be violated, therefore an extra condition is needed for f to havesymmetry.

Lemma 2. Assume that any root % of f (if it exists) satis6es % �∈ Dlow ∪ Dup. Thenthe following two statements are equivalent:

(i) u is a positive solution of (1) and (2).(ii) u is symmetric to 0 and it is a minimal positive solution of (19) and (20) for

some c∈D(T ) satisfying T (c) = R.

Proof. It is obvious that (ii) implies (i). Let us assume that u is a positive solution of(1) and (2). Then there exists r0 ∈ (−R; R) such that u′(r0) = 0. Introducing c= u(r0),u is a solution of (8) and (9). Let Ru(r)= u(2r0− r). This is also a solution of (8) and(9). According to Corollary 1(i), now there holds local uniqueness for all IVPs. Hencethe global solution of (8) and (9) is unique, that is Ru(r) = u(r) for all r. Arguing bycontradiction we obtain r0 = 0. Namely, let us assume r0¡ 0. Then

0 = u(−R) = Ru(−R) = u(2r0 + R);

which implies that u has a root (2r0+R) in (−R; R) as a contradiction. The assumptionr0¿ 0 leads to contradiction in a similar way. Since r0 = 0 and Ru(r) = u(r), thereforeu is symmetric to 0. The minimality of u follows from global uniqueness.

Corollary 2. Under the assumptions of Lemma 2, the number of solutions of theboundary value problem (1) and (2) is equal to the number of elements of the inverseimage T−1({R}).


Corollary 3. Each of the following conditions is su;cient for Lemma 2 (and thusCorollary 2) to hold.

(i) If f has one positive root % and changes its sign at % from negative to positive.(Then Remarks 2–4, guarantees % �∈ Dlow ∪ Dup.)

(ii) If f is p-convex and f(0)6 0. (This is a special case of (i).)

Remark 7. The solutions of the initial value problem can be used also in the case whenthe solution of the boundary value problem is not symmetric to 0 (i.e. uniqueness isviolated). If u is a solution of the boundary value problem (1) and (2) and (a; b) ⊂(−R; R) is a maximal interval where u′ does not vanish, then u is a solution of the initialvalue problem (8) and (9) on (a; b) with r0 =a or r0 =b. Hence, every solution of theboundary value problem can be obtained as the junction of 6nite number of solutionsof the initial value problem. The behaviour of the solutions at the junction points isdescribed by Lemma 1. The exact realization of this idea for p-convex nonlinearitiesis carried out in Theorem 3.

4.3. The domain D(T ) of the time-map

Determining D(T ) is based on the equation

p− 1p

|u′(r)|p + F(u(r)) = F(c) (28)

which follows from Proposition 3 for every solution of (19) and (20). This equationimplies that for any c∈D(T ) F(s)6F(c) if s6 c (by substituting s = u(r)), and incase F(u(r0)) = F(c) we have u′(r0) = 0, hence u is a solution of (8) and (9) withu(r0) = s. If u is a minimal solution, then u is left locally decreasing and right locallydecreasing at r0, hence from Lemma 1 s∈Dlow∩Dup. Therefore, we have the followingProposition for the domain of the time-map:

Lemma 3. The positive number c is in D(T ) if and only if for every s∈ [0; c] one ofthe following two statements holds:

(i) F(s)¡F(c),(ii) F(s) = F(c) and s∈ (Dlow ∪ {0}) ∩ (Dup ∪ {c}).

Proof. (1) Let c∈D(T ) and s∈ [0; c]. Then there exists r ∈ [0; T (c)] such that u(r) =s. Hence, from (28) there follows F(s)6F(c). We prove that F(s) = F(c) impliess∈ (Dlow ∪ {0}) ∩ (Dup ∪ {c}). Again by (28) we have u′(r) = 0.Let r1 be the 6rst point in [0; T (c)] where u(r1) = s. Then in case s �= c u is left

locally decreasing at r1, and satis6es (8) and (9) with r0 = r1, u(r0) = s. Lemma 1implies s∈Dup. Thus s∈Dup ∪ {c}.Let r2 be the last point in [0; T (c)] where u(r2) = s. Then in case s �= 0 u is right

locally decreasing at r2, and satis6es (8) and (9) with r0 = r2, u(r0) = s. Lemma 1implies s∈Dlow. Thus s∈Dlow ∪ {0}.


(2) Let c¿ 0 such that for every s∈ [0; c] (i) or (ii) holds. (ii) implies that c∈Dlow,hence there exists a right locally decreasing solution u of (19) and (20). We show thatthis solution has an extension which has a root.Let us assume that for some r0¿ 0, u′(r0)=0. Then from (28) F(u(r0))=F(c) and

by (ii) u(r0)∈Dlow ∪ Dup (if u(r0) �= 0). According to Lemma 1 u can be extendedafter r0 as a right locally decreasing solution. Moreover, f(u(r0))=0, namely (arguingby contradiction), f(u(r0)) �= 0 implies that F is strictly monotone at u(r0), hencethere exists s in a neighbourhood of u(r0) for which F(s)¿F(u(r0)) = F(c), whichcontradicts (i) and (ii).Since u′(r)=0 implies f(u(r))=0, and f has only a 6nite number of roots, therefore

there exists r0¿ 0 such that u′(r)¡ 0 if r ¿ r0.Now we prove that u has a root in (r0;∞). Arguing by contradiction let us assume

that u′(r)¡ 0 and u(r)¿ 0 in (r0;∞). Hence there exists the limit lim∞ u=�, further,there exists u′′ and u satis6es (5) in (r0;∞). Consequently, u′′(r)¿ 0 for r large,therefore u′ is increasing, hence the limit lim∞ u′ exists and it is zero. From (28)there follows F(�) = F(c), and (ii) implies �∈Dup, which leads to a contradictionwith Proposition 5.

4.4. The limit of the time-map at the boundary points of D(T )

The following results involve the number &p de6ned by:

&p = 2∫ 1

0

1p√1− tp

dt =2&

p sin &=p

(see [6,10]. Then &2 = &. Further, &p=2 is the 6rst root of the solution of the p-linearequation (|u′|p−2u′)′ + (p − 1)|u|p−2u = 0 with u′(0) = 0. (Consequently, if we setL|u|p−2u instead of (p − 1)|u|p−2u with some L¿ 0, then the 6rst root changes to((p− 1)=L)1=p&p=2.)

Lemma 4. Let 0∈ @D(T ).

(a) If limu→0 f(u)=up−1 = +∞ then lim0 T = 0.(b) If limu→0 f(u)=up−1 = L∈ (0;+∞) then lim0 T = ((p− 1)=L)1=p&p=2:(c) If limu→0 f(u)=up−1 = 0 then lim0 T =+∞.

Proof. We 6rst prove part (b).Let �¿ 0 be 6xed. For suJciently small c there holds (L − �)sp−16f(s)6 (L +

�)sp−1 (s6 c), hence, using formulas (27) and (11), we have

T (c) = �p

∫ c

0

1p√F(c)− F(s)

ds6 �p

(p

L− �

)1=p ∫ c

0

1p√cp − sp

ds

=(p− 1L− �

)1=p &p2;


and similarly

T (c)¿(p− 1L+ �

)1=p &p2:

The proof of parts (a) and (c) is similar, now estimating f(s) by Ls from below(above) with arbitrarily large (small) L¿ 0, respectively.

For the following result see [23], Corollary 2.6.

Lemma 5. Let +∞∈ @D(T ).

(a) If limu→+∞ f(u)=up−1 = +∞ then lim+∞ T = 0.(b) If limu→+∞ f(u)=up−1 = L∈ (0;+∞) then lim+∞ T = ((p− 1)=L)1=p&p=2:(c) If limu→+∞ f(u)=up−1 = 0 then lim+∞ T =+∞.

Lemma 6. Let c0¿ 0 belong to @D(T ) \ D(T ). Then limc0 T =+∞.

Proof. Let R¿ 0. Let �¿ 0 such that u(r; c0)¿ � (r ∈ [0; R]). The continuous depen-dence of u(r; c) on c is uniform on compact intervals, hence for small enough �¿ 0we have u(r; c)¿ 0 (r ∈ [0; R]) for all |c − c0|¡�.

4.5. The monotonicity of the time-map

Proposition 10. Let f∈C2[0;∞) be a strictly p-convex function having a root and�¿ 0 be its minimizer.

If c∈D(T ), c¿�, then h= @cu has a root r1 ∈ (0; T (c)).(Moreover, if r2 ∈ (0; T (c)] such that u(r2) = �, then r1¡r2.)

Proof. Let

l(u) = f′(u)(u− �)− (p− 1)f(u) (u¿ 0)

as in (18). Since f(�)6 0, Proposition 7 yields l(u)¿ 0 (u¿ 0).Let

A(r) := (p− 1)|u′(r)|p−2(u′(r)h(r)− (u(r)− �)h′(r)) (r ∈ [0; T (c)]):

Then A(0) = 0. Further, using (19) and (23), we have

A′(r) = (p− 1)(|u′|p−2u′)′(r)h(r)− (p− 1)(|u′|p−2h′)′(r)(u(r)− �)

= h(r)[− (p− 1)f(u(r)) + f′(u(r))(u(r)− �)] = h(r)l(u(r)):

Assume now for contradiction that h(r)¿ 0 (r ∈ [0; r2]). Then A′(r)¿ 0 (r ∈ (0; r2)),hence

A(r2)¿ 0:

However, this contradicts

A(r2) = (p− 1)|u′(r2)|p−2u′(r2)h(r2)6 0:


Proposition 11. Let f∈C1[0;∞) be strictly p-convex. If c∈D(T ) then h= @cu hasat most one root in [0; T (c)].

Proof. Let

A(r) := f(u(r))h(r)− (p− 1)|u′(r)|p−2u′(r)h′(r) (r ∈ [0; T (c)]):

If u′(r) �= 0 then v= u′ is C1 at r. Hence from (5) we obtain

A(r) = (p− 1)|u′(r)|p−2(v(r)h′(r)− v′(r)h(r));

therefore,

A′(r) = (p− 1)((|u′|p−2h′)′v− (|u′|p−2v′)′h)

= (p− 1)(f′(u)hv− f′(u)hv) = 0:

That is, A(r) ≡ const: on intervals where u′ �= 0. Since the roots of u′ are isolated andA is continuous on [0; T (c)], therefore A is constant on [0; T (c)], i.e.

A(r) ≡ A(0) = f(c) (r ∈ [0; T (c)]):

Let r1 ∈ [0; T (c)]) be an arbitrary root of h. Then

f(c) = A(r1) = (p− 1)|u′(r1)|p−2u′(r1)h′(r1): (29)

Here, owing to c∈D(T ), we have f(c)¿ 0. We show that consequently at most oner1 may satisfy (29).If f(c)¿ 0 then (29) yields h′(r1)¡ 0, i.e. h is strictly decreasing at each of its

roots, which implies obviously that it can have at most one root.In the case f(c)=0 there holds A(r1)=0. If h′(r1) �= 0 then u′(r1)=0. If h′(r1)=0,

then in (23) uniqueness is violated at r1 (since h ≡ 0 would also solve it), whichyields again u′(r1) = 0. Hence Proposition 3 yields

F(u(r1)) = F(c): (30)

By Proposition 6 f is convex, hence Lemma 3 and Remark 2 implies that F mayassume F(c) only once in [0; c). Therefore F ◦ u may also assume F(c) only once in(0; T (c)], i.e. (30) may hold at most for one r1.

Lemma 7. Let f∈C2[0;∞) be a strictly p-convex function having a root, and �¿ 0be its minimizer. Then T is strictly decreasing on D(T ) ∩ (�;∞).

Proof. Propositions 10 and 11 yield that h has exactly one root r1 in [0; T (c)], andthis satis6es r1¡T (c). Here c∈D(T ) implies f(c)¿ 0, since f(c) = 0 with c¿�would imply that f changes its sign at � from negative to positive, which, by statement(ii) of Corollary 1, would contradict c∈D(T ). Consequently, (29) yields h′(r1)¡ 0.Hence h(T (c))¡ 0, and from (26) T ′(c)¡ 0.

Lemma 8. Let %¿ 0, f∈C[0; %] be strictly decreasing on [0; %]. Then T is strictlyincreasing on [0; %].


Proof. Let c1¿c2¿ 0, u1(r) := u(r; c1) and u2(r) := u(r; c2). We prove that u1¿u2on [0; T (c2)], which implies that T (c1)¿T (c2).

Assume for contradiction that there exists R∈ [0; T (c2)] such that u1(R)=u2(R). Fori = 1; 2 we have

(’(u′i))′ + f(ui) = 0:

Multiplying by uj and integrating by parts in [0; R], we obtain

’(u′i(R))uj(R)−∫ R

0’(u′i)u

′j +

∫ R

0f(ui)uj = 0

for j = 1; 2. Suitable arrangement yields

(’(u′1(R))− ’(u′2(R)))(u1(R)− u2(R))−∫ R

0(’(u′1)− ’(u′2))(u

′1 − u′2)

+∫ R

0(f(u1)− f(u2))(u1 − u2) = 0:

This is impossible, since u1(R) − u2(R) = 0, and the monotonicity properties imply(’(u′1)− ’(u′2))(u

′1 − u′2)¿ 0 and (f(u1)− f(u2))(u1 − u2)¡ 0 on [0; R].

Remark 8. Lemma 8 has been proved by a standard monotonicity argument. Therefore,it is proved in the same way for an arbitrary domain , ⊂ RN that if f∈C[0; %] isstrictly decreasing on [0; %], then the problem

div(|∇u|p−2∇u) + f(u) = 0; (31)

u|@, = 0 (32)

has at most one solution that satis6es u¡% on ,.

5. Classi�cation of the number of positive solutions

This section contains the main result of this paper: classi6cation is given for strictlyp-convex functions f∈C2 having a non-negative root according to the number ofpositive solutions of (1) and (2).The two cases f(0)6 0 or f(0)¿ 0 will prove to be diNerent concerning the appli-

cation of the shooting method, hence they serve as the 6rst level of our classi6cationand are considered in separate subsections. In the 6rst case the shooting method canbe applied directly to count the solutions, owing to the global uniqueness of the IVPs.The resulting possibilities are 0 or 1 solutions. In the second case global uniquenessof the IVP may be violated, hence the solutions of the BVP are obtained as suitablejunctions of minimal solutions of the IVP. In the case of two positive roots of f thisfact leads to the existence of continuums of solutions. The solutions may contain deadcores, i.e. subintervals where u is constant.We note that by Proposition 6 strict p-convexity implies strict convexity. Hence,

the starting assumption on (1) and (2), namely, that the roots of f are isolated, isautomatically satis6ed with at most two roots of f.


Since f has a non-negative root, therefore its minimum satis6es f(�)6 0. Hence,Proposition 8 implies that for u large enough the function f(u)=up−1 is increasing,therefore the limit limu→+∞ f(u)=up−1 exists. Moreover, this limit is positive sincef is convex and eventually increasing. The two possible cases, i.e. super-p-linear(limu→+∞ f(u)=up−1 = +∞) and asymptotically p-linear (limu→+∞(f(u)=up−1)∈(0;+∞)) functions f are the second level of our classi6cation, and are considered inseparate theorems within both subsections.

5.1. Classi6cation for strictly p-convex functions with f(0)6 0

Owing to Corollaries 2 and 3(ii), the solutions of the boundary value problem (1)and (2) now coincide with those of (19) and (20) in the sense of Lemma 2, and thustheir number is given by the shape of the time-map. As has been mentioned in thepreceding paragraph, we consider two cases depending whether limu→+∞ f(u)=up−1 is6nite. We note that in the case f(0)=0 p-convexity implies f′(0)6 0, since otherwisethe minimizer would be �=0, and we would have contradiction with the de6nition ofp-convexity. In the case f′(0) = 0 we also have limu→0 f(u)=up−1¡+ ∞ owing toProposition 8. Hence the case f(0)=0 can be divided into three parts, as is consideredin the following theorems.

Theorem 1. Let f∈C2[0;∞) be strictly p-convex, f(0)6 0, and limu→+∞f(u)=up−1 = +∞.

(i) If f(0)= 0 and limu→0 f(u)=up−1 = 0, then for all R¿ 0 (1)–(2) has a uniquesolution.

(ii) Let f(0)=0 and limu→0 f(u)=up−1 =m∈ (0;∞). Let R0 := ((p−1)=m)1=p&p=2.Then (1) and (2) has a unique solution for R¡R0 and no solution for R¿R0.

(iii) Let f(0) = 0 and f′(0)¡ 0, or let f(0)¡ 0. Then there exists R1¿ 0 suchthat (1) and (2) has a unique solution for R6R1 and no solution for R¿R1.

Proof. Lemma 3 and the convexity of f imply that +∞∈ @D(T ), hence according toLemma 5, limu→+∞ f(u)=up−1 = +∞ implies that lim+∞ T = 0.

(i)–(ii) Since now f′(0) = 0, the p-convexity of f implies f(u)¿ 0 (u¿ 0), henceD(T ) = (0;∞). Lemma 7 yields that T is strictly decreasing. In case (i)Lemma 4 yields lim0 T=∞, hence T assumes each value R¿ 0 once. In case(ii) Lemma 4 yields lim0 T = R0, hence T assumes each value 0¡R¡R0

once.(iii) Now the integral function F is 6rst decreasing, then it strictly increases to

∞. Hence by Lemma 3 D(T ) is a half-line with left endpoint % satisfyingF(%)¿ 0; moreover, %∈D(T ). Clearly %¿� if � minimizes f, hence Lemma7 yields that T is strictly decreasing on [%;∞). Therefore, our result followsusing notation R1 := T (%).


Theorem 2. Let f∈C2[0;∞) be strictly p-convex, f(0)6 0, and limu→+∞ f(u)=up−1 = L∈ (0;+∞). Let R∞ := ((p− 1)=L)1=p&p=2.

(i) If f(0) = 0 and limu→0 f(u)=up−1 = 0, then (1) and (2) has no solution forR6R∞ and one solution for R¿R∞.

(ii) Let f(0)=0 and limu→0 f(u)=up−1 =m∈ (0;∞). Let R0 := ((p−1)=m)1=p&p=2.Then (1) and (2) has no solution for R6R∞ or R¿R0 and one solution forR∞¡R¡R0.

(iii) Let f(0) = 0 and f′(0)¡ 0, or let f(0)¡ 0. Then there exists R1¿ 0 suchthat (1) and (2) has no solution for R6R∞ or R¿R1 and one solution forR∞¡R6R1.

Proof. The proof proceeds just in the same way as in Theorem 1, now using lim+∞ T=R∞ from Lemma 5. We note that in (ii) f(u)=up−1 is increasing by Proposition 8,hence m¡L.

5.2. Classi6cation for strictly p-convex functions with f(0)¿ 0 and positive root

As mentioned earlier, now global uniqueness may be violated, hence the solutionsof the BVP are obtained as suitable junctions of minimal solutions of the IVP (seeRemark 7). When f has two positive roots, it turns out that most times there areuncountably many solutions, but the structure of these can be described. The solutionsmay contain subintervals where u is constant. Since we consider positive solutions, itis reasonable to consider these subintervals as dead cores (instead of the more usualnotion when u ≡ 0 on the subinterval).

Theorem 3. Let f∈C2[0;∞) be strictly p-convex, f(0)¿ 0, and limu→+∞f(u)=up−1 = +∞.

(i) Assume that f has one root �. If � �∈ Dlow then (1) and (2) has two solutionsfor all R¿ 0. In case �∈Dlow let

R� := �p

∫ �

0

1p√F(�)− F(s)

ds;

where �p = ((p − 1)=p)1=p as in (11). Then (1) and (2) has two solutions forR¡R� and one solution for R¿R�.

(ii) Assume that f has two roots �1¡�2. Then there exists a unique %¿�1 suchthat F(�1) = F(%). Let R∗ := T (%). (Then R∗¿T (�1).)If R6R∗ then (1) and (2) has two solutions. If R¿R∗ then (1) and (2) hasuncountably many solutions. The latter can be characterized as follows. Let

R1 := T (�1); u1(r) := u(r; �1) (−R16 r6R1);

R2 := R∗ − R1; u2(r) := u(r; %) (−R26 r6R2):

Then for any solution u the following holds: u|[R−R1 ;R] is a translate of u1|[0;R1]

(i.e. u(r)=u1(r−R+R1), R−R16 r6R), u|[−R;−R+R1] is a translate of u1|[−R1 ;0]


(in similar sense), and u|[−R+R1 ;R−R1] is the junction of at most n non-overlappingtranslates of u2 and constants between these (where n∈N+ is determined byR1 + nR2¡R6R1 + (n+ 1)R2.)

Proof. (i) If � �∈ Dlow, then from Lemma 3 we have D(T ) = (0; �)∪ (�;∞), and fromLemma 2 all solutions of the BVP are minimal solutions of the IVP, obtained directlyfrom the time-map. Lemmas 4–6 imply that lim0 T=lim+∞ T=0, lim� T=∞. Finally,Lemmas 7 and 8 imply that T assumes each value R¿ 0 twice (once in (0; �) andonce in (�;∞)).In the case �∈Dlow Lemma 3 yields D(T ) = (0;∞). Here, T (�) = R� from (27).

The limits at 0 and ∞ and the monotonicity of T are the same as above, hence Tassumes each value R¡R� twice and R= R� once. However, now the uniqueness ofthe IVP is violated at �, since it has at least two solutions from u(0) = �, namely, theconstant � and the minimal solution u� attaining 0 in T (�) = R�. Owing to Lemma1 there are no more solutions locally. Now we can rely on Remark 7, hence all theother solutions u of (1) and (2) with some 6xed R¿R� are obtained as u|[0;R−R�] ≡ �,u(r)=u�(r−R+R�) (r ∈ [R−R�; R]) for r¿ 0, and symmetrically for r6 0. If c �= �then it follows from Proposition 3 that all solutions of the IVP are minimal (since Fis strictly increasing, therefore u′¡ 0), hence the only solutions of the BVPs that arenot given by the time-map are the ones above. This shows that for all R¿R� the BVPhas one solution, the one with u(0) = �.(ii) Now the integral function F is increasing on [0; �1], decreasing on [�1; �2],

then it strictly increases to ∞. Hence by Lemma 3 D(T ) = [0; �1] ∪ [%;∞) with theunique %¿�1 satisfying F(�1) = F(%). The strict p-convexity of f implies that fis strictly decreasing on [0; �1]. Lemmas 8 and 7 yield that T is strictly increasing(decreasing) on [0; �1] ([%;∞)), respectively. From Proposition 3 the function u(:; %)satis6es u′(r; %) = 0 when u(r; %) = �1, hence (from the uniqueness of right decreasingsolutions) afterwards it coincides with the translate of u(:; �1), therefore T (%)¿T (�1).Summing up, using the notations R∗=T (%) and R1=T (�1), we obtain for the BVP twominimal solutions for R6R1, one minimal solution for R1¡R6R∗ and no minimalsolution for R¿R∗.To count the non-minimal solutions, we 6rst note that similarly to part (i), for c �= �1

or c �= % Proposition 3 and the shape of F imply that all solutions of the IVP fromu(0) = c are minimal. Further, according to the above, any non-minimal solution usatis6es u′(r)= 0 when u(r)= �1. This implies that non-minimal solutions exist if andonly if R¿R1, and a non-minimal solution u coincides on [R−R1; R] ([−R;−R+R1])with the translate of u(:; �1)|[0;R1] (u(:; �1)|[−R1 ;0]), respectively. Here u may even be notsymmetric, but we know from Proposition 3 that u′(r0)=0 implies u(r0)=�1 or u(r0)=%.Since we also have uniqueness of the IVP at %, therefore in the case u′(r0) = 0 andu(r0) = % we obtain that u is symmetric with respect to r0 and u|[r0−R2 ; r0+R2] coincideswith the translate of u(:; %)|[−R2 ; R2], where u(R2; %) = �1 (since u′ �= 0 until u attains�1). If −R + R1¡r0¡R − R1, u′(r0) = 0 and u(r0) = �1, then on both sides of r0u is either locally constant or coincides with an above-mentioned translate of u(:; %).Altogether, either u|[−R+R1 ;R−R1] ≡ �1, or u is the junction of humps (non-overlappingtranslates of u2 := u(:; %)|[−R2 ;R2]) and constants between these. The number of the


humps is limited by the length of the interval [ − R + R1; R − R1], and a hump mayhave room if R− R1¿R2. (The latter means R∗ := T (%)¡R since T (%) = R1 + R2).Summing up, for R1¡R6R∗ we thus obtain a non-minimal solution which is constanton [− R+ R1; R− R1], and this just provides a second solution to BVP in addition tothe minimal one obtained in the 6rst paragraph. For R¿R∗ there is room for humps,and the preceding considerations verify the required description of the solutions.

Remark 9. If f is uniformly p-convex, i.e.(f′(u)

|u− �|p−2

)′¿m¿ 0;

then in (i) � �∈ Dlow, hence (1) and (2) has two solutions for all R¿ 0.

Theorem 4. Let f∈C2[0;∞) be strictly p-convex, f(0)¿ 0, and limu→+∞ f(u)=up−1 = L∈ (0;+∞). Let R∞ := ((p− 1)=L)1=p&p=2.

(i) Assume that f has one root �. If � �∈ Dlow then (1) and (2) has one solution forR6R∞ and two solutions for R¿R∞. In case �∈Dlow let R� be as in Theorem3. Then (1) and (2) has one solution for R6R∞ or R¿R� and two solutionsfor R∞¡R¡R�.

(ii) Assume that f has two roots �1¡�2. Then there exists a unique %¿�1 suchthat F(�1) = F(%). Let R∗ := T (%). (Then R∗¿R1.)Then (1) and (2) has one solution if R6R∞ and two solutions if R∞¡R6R∗.If R¿R∗ then (1) and (2) has uncountably many solutions, characterized in thesame way as in Theorem 3.

Proof. The proof is the same as for Theorem 3. The diNerent behaviour of f atin6nity has only inSuence on the number of solutions corresponding to small valuesof R, i.e. R¡R� in case (i) and R6R∗ in case (ii). Namely, Lemma 5 now implieslim+∞ T = R∞. In case (i) we have R∞¡R�, since from Lemma 7 T is strictlydecreasing for c¿�; in case (ii) we have R∞¡R∗, since again from Lemma 7 T isstrictly decreasing for c¿%.

Remark 10. Note that in both theorems, denoting by � the 6rst root of f, a dead coreoccurs if and only if∫ �

0

1p√F(�)− F(s)

ds¡∞;

since this is automatically satis6ed in case (ii) at �1. This phenomenon is in coherencewith the general results in [7].

Remark 11. The structure of the continuum of solutions is similar to the one aris-ing in [8] for the convex-concave nonlinearity f(u) = uq − �um with �, ¿ 0 and0¡m¡q6p− 1, or in [14] for a generalized eigenvalue problem.


Acknowledgements

The authors are grateful to Professors J.I. D)Taz, )A. Elbert, J. Hern)andez and L.A.Peletier for the useful discussions.

References

[1] A. Ambrosetti, J. Garcia Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations,J. Funct. Anal. 137 (1996) 219–242.

[2] H. Berestycki, P.L. Lions, L.A. Peletier, An ODE approach to the existence of positive solutions forsemilinear problems in Rn, Indiana Univ. Math. J. 30 (1981) 141–157.

[3] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64.[4] A. Castro, S. Gadam, R. Shivaji, Positive solution curves of semipositone problems with concave

nonlinearities, Proc. Roy. Soc. Edinburgh 127A (1997) 921–934.[5] A. Castro, R. Shivaji, Positive solutions for a concave semipositone Dirichlet problem, Nonlinear Anal.

31 (1998) 91–98.[6] M. Del Pino, R. Man)asevich, M. Elgueta, A homotopic deformation along p of a Leray Schauder degree

result and existence for (|u′|p−2u′)′ + f(t; u) = 0; u(0) = u(T ) = 0, p¿ 1, J. DiNerential Equations80 (1989) 1–13.

[7] J.I. D)Taz, Nonlinear Partial DiNerential Equations and Free Boundaries, Vol. I: Elliptic Equations, Pitman,London, Res. No. 106, 1985.

[8] J.I. D)Taz, J. Hern)andez, Global bifurcation and continua of nonnegative solutions for a quasilinear ellipticproblem, C. R. Acad. Sci. Paris Ser. I Math. 329 (7) (1999) 587–592.

[9] J.I. D)Taz, J.E. Saa, Existence et unicit)e de solutions positives pour certaines )equations elliptiquesquasilin)eaires, C. R. Acad. Sci. Paris Ser. I. Math. 305 (12) (1987) 521–524.

[10] )A. Elbert, A half-linear second order diNerential equation, Coll. Math. Soc. J. Bolyai 30 (1981)153–180.

[11] )A. Elbert, On the half-linear second order diNerential equations, Acta Math. Hungar. 49 (1987)487–508.

[12] J. Garcia Azorero, I. Peral Alonso, Some results about the existence of a second positive solution in aquasilinear critical problem, Indiana Univ. Math. J. 43 (1994) 941–957.

[13] B. Gidas, W.N. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun.Math. Phys. 68 (1979) 209–243.

[14] M. Guedda, L. Veron, Bifurcation phenomena associated to the p-Laplace operator, Trans. AMS 310(1988) 419–431.

[15] D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. RationalMech. Anal. 49 (1973) 241–269.

[16] J. Kar)atson, P.L. Simon, Bifurcations of semilinear elliptic equations with convex nonlinearity, Electron.J. DiNerential Equations 1999 (43) (1999) 1–16.

[17] J. Kar)atson, P.L. Simon, On the stability properties of nonnegative solutions of semilinear problemswith convex or concave nonlinearity, J. Comput. Appl. Math 131 (2001) 497–501.

[18] J. Kar)atson, P.L. Simon, On the linearized stability of positive solutions of quasilinear problems withp-convex or p-concave nonlinearity, Nonlinear Anal. 47 (2001) 4513–4520.

[19] Ph. Korman, T. Ouyang, Exact multiplicity results for two classes of boundary value problems,DiNerential Integral Equations 6 (1993) 1507–1517.

[20] M.K. Kwong, Uniqueness of positive radial solutions of Vu− u+ up = 0 in Rn, Arch. Rational Mech.Anal. 105 (1989) 243–266.

[21] T. Laetsch, The theory of forced, convex, autonomous, two point boundary value problems, RockyMountain J. Math. 15 (1985) 133–154.

[22] R. Man)asevich, K. Schmitt, Boundary value problems for quasilinear second order diNerential equations,in: F. Zanolin (Ed.), Nonlinear Analysis and Boundary Value Problems for Ordinary DiNerentialEquations, Courses and Lectures, Vol. 371, Springer, Berlin, 1996, pp. 79–119.


[23] R. Man)asevich, F. Zanolin, Time-mappings and multiplicity of solutions for the one dimensionalp-Laplacian, Nonlinear Anal. 21 (1993) 269–291.

[24] C. Maya, R. Shivaji, Instability of nonnegative solutions for a class of semilinear elliptic boundaryvalue problems, J. Comput. Appl. Math. 88 (1998) 125–128.

[25] K. McLeod, Uniqueness of positive radial solutions of Vu + f(u) = 0 in Rn II, Trans. Amer. Math.Soc. 339 (1993) 495–505.

[26] F. Merle, L.A. Peletier, Positive solutions of elliptic equations involving supercritical growth, Proc. Roy.Soc. Edinburgh 118A (1991) 49–62.

[27] M. Otani, On certain second order ordinary diNerential equations associated with Sobolev–Poincar)e-typeinequalities, Nonlinear Anal. 8 (1984) 1255–1270.

[28] R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, in: Springer LectureNotes in Mathematics, Vol. 1458, Springer, Berlin, 1990.

[29] J.A. Smoller, A.G. Wassermann, Global bifurcation of steady state solutions, J. DiNerential Equations39 (1981) 269–290.

[30] J.A. Smoller, A.G. Wassermann, On the monotonicity of the time-map, J. DiNerential Equations 77(1989) 287–303.

[31] P.N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, DiNerential Integral Equations6 (3) (1993) 663–670.

Exact multiplicity for degenerate two-point boundary value problems with p-convex nonlinearity

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