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Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function
Applied Mathematics and Computation 215 (2009) 1077–1083
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function q Ruyun Ma *, Xiaoling Han Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, PR China
a r t i c l e
i n f o
a b s t r a c t This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem
Keywords: Indefinite weight problem Bifurcation Positive solutions
u00 þ kaðtÞf ðuÞ ¼ 0;
0 < t < 1;
uð0Þ ¼ uð1Þ ¼ 0;
where a 2 C½0; 1 changes sign. The proof of our main result is based upon bifurcation techniques. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and main result In this paper we consider the existence and multiplicity of positive solutions of the boundary value problem
u00 þ kaðtÞf ðuÞ ¼ 0;
0 < t < 1;
uð0Þ ¼ uð1Þ ¼ 0;
ð1:1Þ ð1:2Þ
where a 2 C½0; 1 may change sign, k is a parameter. Problem (1.1) and (1.2) and its higher dimensional analogue
Du þ kgðxÞf ðuÞ ¼ 0; u ¼ 0; x 2 @ X;
x 2 X;
ð1:3Þ ð1:4Þ
(where X is a bounded domain with smooth boundary) arises from many branches of applied mathematics. For example, the generalized Emden–Fowler equation, where f ¼ up ; p > 0 and a > 0 in (1.1) arises in the fields of gas dynamics, nuclear physics, and chemically reacting system [18]; the Thomas–Fermi equation, where f ðuÞ ¼ u3=2 and aðtÞ ¼ t 1=2 was developed in studies of atomic structures [18]; and the selection–migration model in population genetics where g in (1.3) changes sign in X [12]. Existence and multiplicity of positive solutions of (1.1)–(1.4) with a nonnegative weight function has been extensively studied, see, for example, Erbe and Wang [9], Fink et al. [10,11], Lions [13], Henderson and Wang [15], Lan and Webb [18], Wang [24] and the references therein. The main tools used in these paper are fixed point theorem in cones and the fixed point index theory in cones.
q Supported by the NSFC (No.10671158), the NSF of Gansu Province (No. 3ZS051-A25-016), NWNUKJCXGC- 03-17, the Spring-sun program (No. Z2004-162033), SRFDP (No. 20060736001), the SRF for ROCS, SEM (2006[311]), NWNU-KJCXGC-3-18, NWNU-KJCXGC-3-17, NWNU-KJCXGC-03-40, and NSF of Gansu Province (No. 3ZS061-A25-016). * Corresponding author. E-mail address: [email protected] (R. Ma).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.042
1078
R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
The existence of positive solutions of (1.1)–(1.4) with a indefinite weight function has also been studied by several authors, see Afrouzi and Brown [1], Các et al. [7], Các et al. [8] and Hai [14]. However, the existence results of positive solutions in these papers were established only when the parameter k > 0 is small enough. It is the purpose of this paper to study the indefinite weight boundary value problem (1.1) and (1.2) under the assumptions: (H1) f 2 C 2 ðR; RÞ with f ð0Þ ¼ f ðs1 Þ ¼ f ðs2 Þ ¼ 0; 0 < s1 6 s2 ; f ðsÞ > 0 for s 2 ð0; s1 Þ [ ðs2 ; þ1Þ; f ðsÞ < 0 for s 2 ðs1 ; s2 Þ; (H2) f 00 ðsÞ < 0 for s 2 ½0; s1 Þ; (H3) there exist f0 ; f1 2 ð0; 1Þ such that
f0 ¼ lim s!0
f ðsÞ ; s
f 1 ¼ lim
s!1
f ðsÞ ; s
(H4) a 2 C½0; 1 changes sign. We will show the existence, multiplicity of positive solutions of (1.1) and (1.2) (in which a changes sign) via global bifurcation techniques. For earlier work related to the nodal solutions with the case aðtÞ > 0 on ½0; 1, see Ma [19]. It is well known (cf. Binding [2], Bôcher [3], Brown and Lin [5] and Ince [16]) that the indefinite weight linear eigenvalue problem
u00 þ kaðtÞu ¼ 0;
0 < t < 1;
ð1:5Þ
uð0Þ ¼ uð1Þ ¼ 0; have two infinite sequences of simple eigenvalues
0 < kþ < kþ1 < kþ2 < < kþk < ;
lim kþk ¼ þ1
k!1
and
0 > k > k1 > k2 > > kk > ;
lim kk ¼ 1
k!1
such that the eigenfunction corresponding to kmk ; m 2 fþ; g, has exactly k simple zeros in (0, 1), and km ; m 2 fþ; g is principle eigenvalue(i.e., an eigenvalue having a positive eigenfunction). Let uþ ; u be the positive eigenfunctions corresponding to kþ and k , respectively. Similar spectrum results were established for discrete analogue of (1.5) by Binding [2]. Remark 1.1. It is worth remarking that if u is a nontrivial solution of (1.1) and (1.2) in which f satisfy (H1), then any initial value problem (IVP) associated with (1.1) has a unique solution, so u have n simple zeros in (0, 1) for some n 2 N. In fact, if a nontrivial solution uðtÞ has a double zero t 0 2 ð0; 1Þ, then uðt 0 Þ ¼ u0 ðt 0 Þ ¼ 0. Thus, by the uniqueness of solutions of the associated IVP, we have uðtÞ 0 on [0, 1], contradicting the assumption that uðtÞ is nontrivial. Suppose ðk; uÞ is a solution of (1.1) and (1.2), u is said to be stable if all the eigenvalues of the linearized operator associated with (1.1) and (1.2) at ðk; uÞ are strictly positive. Our main result is the following: Theorem 1.1. Let (H1)–(H4) hold. (a) Assume that f0 < f1 . Then (1.1) and (1.2) have at least two positive solutions for k 2 1; kf0 [ kf0þ ; þ1 , and one of them is stable; there exist at least one positive solution for k 2 ½kf0 ; kf1 Þ [ ðkf1þ ; kf0þ . (b) Assume that f1 < f0 . Then (1.1) and (1.2) have at least two positive solutions for k 2 1; kf1 [ kf1þ ; þ1 , and one of them is stable; there exist at least one positive solution for k 2 ½kf1 ; kf0 Þ [ ðkf0þ ; kf1þ and it is stable. If the condition (H3) is replaced by
ðH3Þ
f 0 2 ð0; 1Þ; f1 ¼ 1;
then we obtain the following result: Theorem 1.2. Let (H1),(H2), ðH3Þ and (H4) hold. Then (1.1) and (1.2) has a positive solution if and only if k–0. Moreover, for k 2 1; kf0 [ kf0þ ; þ1 there exist at least two positive solutions, one of them is stable. The proof of above theorems are based on the following two theorems. Theorem A (Crandall–Rabinowitz Theorem [17, Theorem I.5.1]). Consider mappings F : V U ! Z with open sets U X; V Y, where X and Z are Banach spaces, and Y ¼ R. Suppose: (i) Fðk; 0Þ ¼ 0; 8 k 2 R, and dim NðDx Fðk0 ; 0ÞÞ ¼ codim RðDx Fðk0 ; 0ÞÞ ¼ 1; (ii) F 2 C 2 ðV U; ZÞ, where 0 2 U X; k0 2 V R;
R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
1079
(iii) NðDx Fðk0 ; 0ÞÞ ¼ span½v 0 ; v 0 2 X; kv 0 k ¼ 1; D2xk Fðk0 ; 0Þv 0 R RðDx Fðk0 ; 0ÞÞ. Then there is a nontrivial continuously differentiable curve through ðk0 ; 0Þ,
fðkðsÞ; xðsÞÞjs 2 ðd; dÞ; ðkð0Þ; xð0ÞÞ ¼ ðk0 ; 0Þg;
ð1:6Þ
such that
FðkðsÞ; xðsÞÞ ¼ 0;
s 2 ðd; dÞ
and all solutions of Fðk; xÞ ¼ 0 in a neighborhood of ðk0 ; 0Þ are on the trivial solution line or on the nontrivial curve (1.6). Consider the global solution behavior of the bifurcation branches of the equation
x ¼ lðLx þ NxÞ;
l 2 K; x 2 X:
ð1:7Þ
The closure of the set of nontrivial solutions of (1.7) will be denoted by S. Suppose (A1) The operators L; N : X ! X are compact on the B-space X over K where K ¼ R. Furthermore, L is linear and kNxk n kxk ! 0 as kxk ! 0. (A2) The real number l0 is a characteristic number of L of odd algebraic multiplicity.
Theorem B (Rabinowitz Theorem [22, Theorem 1.3]). Suppose (A1) and (A2) are given. Then S possesses a maximal subcontinuum Cl such that ðl0 ; 0Þ 2 Cl and Cl either (i) Cl is unbounded, or (ii) Cl is compact and in addition to ðl0 ; 0Þ also contains a further point of the trivial solution branch T.
Remark 1.2. In Brown [6] and Ma and Han [20], the Rabinowitz’s global bifurcation theorem were used to prove the existence of nodal/positive solutions of some indefinite weight problems.
2. Proof of main theorem To prove our main result, we need some preliminary results. Let X ¼ fu 2 C 2 ½0; 1 : uð0Þ ¼ uð1Þ ¼ 0g; E ¼ fu 2 C 1 ½0; 1 : uð0Þ ¼ uð1Þ ¼ 0g; Y ¼ C½0; 1. Define L : X ! Y by setting
Lu :¼ u00 ;
u 2 X:
Then (1.1) and (1.2) is equivalent to
Lu ¼ kaðtÞf ðuÞ;
u 2 X:
Lemma 2.1. For all p P 1, we have
Z
1
k 0
aðtÞupþ1 dt > 0:
ð2:1Þ
Proof. Multiplying Lu ¼ k aðtÞu by up we obtain Lu up ¼ k aðtÞupþ1 on (0, 1), integrating from 0 to 1, it gives rise to the relation
Z k 0
1
aðtÞupþ1 dt ¼ p
Z
1
0
up1 ðu0 Þ2 dt:
Hence, (2.1) hold. h Consider F : R X ! Y defined by
Fðk; uÞ ¼ Lu kaðtÞf ðuÞ; then F is a smooth map with Fréchet derivative
F u ðk; 0Þw ¼ Lw kf0 aðtÞw:
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R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
R1 Thus, N F u kf0 ; 0 ¼ ½u and R F u kf0 ; 0 ¼ ½u ? ¼ fu 2 C½0; 1 : 0 uu dt ¼ 0g. Moreover F ku kf0 ; 0 u ¼ aðtÞu , by Lemma 2.1,
Z k 0
1
aðtÞu2 dt > 0;
it follows that F ku kf0 ; 0 u R R F u kf0 ; 0 . Thus, by Theorem A, there exist two nontrivial continuously differentiable curves passing through kf0 ; 0 , respectively, and with the form
fðk ðsÞ; u ðsÞÞjs 2 ðd; dÞg such that
ðk ðsÞ; u ðsÞÞ ¼ for s 2 ðd; dÞ with
k þ l ðsÞ; sðu þ w ðsÞÞ f0
ð2:2Þ
l ð0Þ ¼ 0; w ð0Þ ¼ 0; w ðsÞ 2 X \ ½u ? , and
Fðk ðsÞ; u ðsÞÞ ¼ 0 for s 2 ðd; dÞ and all nontrivial solutions of Fðk; uÞ ¼ 0 in a neighborhood of kf0 ; 0 are on the curve (2.2). By Theorem B and Remark 1.1, these two continua Cþ and C of solutions bifurcating from kf0þ ; 0 and kf0 ; 0 is unbounded, respectively. Lemma 2.2. Suppose (H2)–(H4) hold. Then Cþ bifurcates to the right at
kþ f0
; 0 ; C bifurcates to the left at kf0 ; 0 .
Proof. Note that (1.1) and (1.2) is equivalent to
Lu ¼ kf0 aðtÞu þ kaðtÞðf ðuÞ f0 uÞ;
ð2:3Þ
substituting expression (2.2) in (2.3) and using that Lu ¼ k aðtÞu , we deduce that
ðL k aÞw ðsÞ ¼ f0 l ðsÞaðu þ w ðsÞÞ þ
k f ðu ðsÞÞ f0 u ðsÞ þ l ðsÞ a ; s f0
ð2:4Þ
dividing (2.4) by s and take limit as s ! 0, we have
ðL k aÞw0 ð0Þ ¼ f0 l0 ð0Þau þ
k f 00 ð0Þ 2 au ; 2f 0
ð2:5Þ
00 this means f0 l0 ð0Þau þ k2ff 0ð0Þ au2 2 RðL k aÞ ¼ R F u kf0 ; 0 ¼ ½u ? , which is equivalent to
Z
1
0
k f 00 ð0Þ 2 f0 l0 ð0Þau þ au u dt ¼ 0; 2f 0
so we get
l0 ð0Þ ¼
R1 k f 00 ð0Þ 0 aðtÞu3 dt : R 1 2f02 aðtÞu2 dt 0
from condition (H2) and Lemma 2.1 we have l0þ ð0Þ > 0 and l0 ð0Þ < 0. So Cþ bifurcates to the right at cates to the left at kf0 ; 0 . h We now analyze the global behavior of Cþ and C .
kþ f0
; 0 and C bifur-
Lemma 2.3. Suppose (H1), (H3) and (H4) hold. Then for ðk; uÞ 2 Cþ [ C ,
0 6 uðtÞ < s1 ;
t 2 ½0; 1:
ð2:6Þ
Proof. Suppose on the contrary that there exists ðk; uÞ 2 Cþ such that
max uðtÞ ¼ s1 :
ð2:7Þ
t2½0;1
By (H1), (H3) and (H4), there exists m P 0 such that aðtÞf ðsÞ þ ms is strictly increasing in s for s 2 ½0; s1 . Then
Lu þ kmu ¼ kðaðtÞf ðuÞ þ muÞ;
t 2 ð0; 1Þ
ð2:8Þ
and, since Ls1 ¼ 0 ¼ f ðs1 Þ,
ðL þ kmÞs1 ¼ kðaðtÞf ðs1 Þ þ ms1 Þ:
ð2:9Þ
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R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
Subtracting, we get
ðL þ kmÞðs1 uðtÞÞ P 0;
t 2 ð0; 1Þ
ð2:10Þ
and
s1 uð0Þ > 0;
s1 uð1Þ > 0:
ð2:11Þ
The maximum principle [21] implies that s1 > uðtÞ in ½0; 1, it is a contradiction to (2.7). Therefore
uðtÞ < s1 ;
t 2 ½0; 1:
If there exists ðk; uÞ 2 C such that
max uðtÞ ¼ s1 : t2½0;1
Note that in this case k < 0, so we can choose m P 0 such that aðtÞf ðsÞ ms is strictly decreasing in s for s 2 ½0; s1 . Then
Lu kmu ¼ kðaðtÞf ðuÞ muÞ;
t 2 ð0; 1Þ
and, since Ls1 ¼ 0 ¼ f ðs1 Þ,
ðL kmÞs1 ¼ kðaðtÞf ðs1 Þ ms1 Þ: Subtracting, we get
ðL kmÞðs1 uðtÞÞ P 0;
t 2 ð0; 1Þ
and
s1 uð0Þ > 0;
s1 uð1Þ > 0:
The maximum principle implies that s1 > uðtÞ in ½0; 1, this is a contradiction too. h Define P : R C½0; 1 ! R, the projection map onto R, i.e., Pðk; uÞ ¼ k. Lemma 2.4. Suppose (H1)–(H4) hold. Then for ðk0 ; u0 Þ 2 Cþ [ C ; u0 is stable. So PðCþ Þ ¼
kþ f0
; þ1 ; PðC Þ ¼ 1; kf0 .
Proof. The proof is motivated by Brown and Hess [4]. We need to prove all the eigenvalues of the operator Lw k0 aðtÞf 0 ðu0 Þw are strictly positive. To do this, we only need to prove that all eigenvalues l of the linearized problem associated with (1.1) and (1.2), viz
Lw k0 aðtÞf 0 ðu0 Þw ¼ lw
ð2:12Þ
are positive. Suppose that lðk0 Þ and / are the principle eigenvalue and eigenfunction of the linearized operator L k0 aðtÞf 0 ðu0 Þ. Then we have
Lu0 k0 aðtÞf ðu0 Þ ¼ 0;
t 2 ð0; 1Þ
ð2:13Þ
and
L/ k0 aðtÞf 0 ðu0 Þ/ ¼ lðk0 Þ/;
t 2 ð0; 1Þ:
ð2:14Þ
0
Multiplying (2.13) by f ðu0 Þ/ and (2.14) by f ðu0 Þ, subtracting and integrating gives
2
Z 0
1
ðu00 Þ2 f 00 ðu0 Þ/dt ¼ lðk0 Þ
Z
1
f ðu0 Þ/dt:
ð2:15Þ
0
By Lemma 2.3, 0 6 u0 < s1 for t 2 ½0; 1, from (H1),(H2) we have f ðu0 Þ > 0; f 00 ðu0 Þ < 0, it follows that lðk0 Þ > 0, so u0 is stable. Further more we know that F u ðk0 ; u0 Þw ¼ Lw k0 aðtÞf 0 ðu0 Þw is an isomorphism, from implicit function theorem we know that all solutions of (1.1) and (1.2) in R X near to ðk0 ; u0 Þ lie on a C 1 curve passing through ðk0 ; u0 Þ and parametrised by k. This curve of nontrivial solutions interval of definition over the k axis. This together with can be continuedto a maximal Lemma 2.2 we get that PðCþ Þ ¼ kf0þ ; þ1 ; PðC Þ ¼ 1; kf0 . h In the following we will investigate the other positive solutions of problem (1.1) and (1.2). Let n 2 CðR; RÞ be such that
f ðuÞ ¼ f1 u þ nðuÞ:
ð2:16Þ
Clearly
lim
u!1
nðuÞ ¼ 0: u
ð2:17Þ
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R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
Let us consider
Lu kaðtÞf1 u ¼ kaðtÞnðuÞ
ð2:18Þ
as a bifurcation problem from infinity. We note that (2.18) is equivalent to (1.1) and (1.2). The results of Rabinowitz can [23]for (2.18) be stated as follows: there exists a continuum Dþ and D of positive solutions of (2.18) meeting kf1þ ; 1 and kf1 ; 1 , respectively. Lemma 2.5. Suppose (H1), (H3) and (H4) hold. Then for ðk; uÞ 2 Dþ [ D , we have
max uðtÞ > s2 : t2½0;1
Proof. It is similar to the proof of Lemma 2.3, so we omit it. h Lemma 2.6. Suppose (H1), (H3) and (H4) hold. Then
kþ f1
; þ1 PðDþ Þ and 1; kf1 PðD Þ.
kþ Proof. Firstly, we proof kf1þ ; þ1 PðDþ Þ.Take K R is an interval such that K \ fkf1þ ; f1j jj 2 N n f0gg ¼ fkf1þ g and M is a neighborhood of kf1þ ; 1 whose projection on R lies in K and whose projection on E is bounded away from 0. Then by [21, Theorem 1.6 and Corollary 1.8], we have that, either (1) Dþ n M is bounded in R E in which case Dþ n M meetsfðk; 0Þjk 2 Rg. (2) Dþ n M is unbounded. n þo ^ ; 1Þ where l ^ 2 kf k . Moreover if (2) occurs and Dþ n M has a bounded projection on R, then Dþ n M meets ðl 1 Obviously Lemma 2.5 implies that (1) do not occur. So Dþ n M is unbounded. kþ Remark 1.1 guarantees that Dþ is a component of positive solutions of (2.18) which meets f1 ; 1 . Therefore there is no þ k k 2 N n f0g such that Dþ also meets f1k ; 1 . Otherwise, there will exist ðg; yÞ 2 Dþ such that y has a multiple zero point in (0,1). However this contradicts with Remark 1.1, and consequently PðDþ n MÞ is unbounded. Thus
kþ ; þ1 PðDþ Þ f1 and similarly we have
k 1; PðD Þ: f1
Proof of Theorems 1.1 and 1.2. From Lemma 2.1–2.6 we already complete the proof of Theorem 1.1. We note that if f1 ¼ 1; then kf1 ¼ 0, then (1.7) has a positive solution if and only if k – 0. Acknowledgement The authors are very grateful to the anonymous referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
G.A. Afrouzi, K.J. Brown, Positive solutions for a semilinear elliptic problem with sign-changing nonlinearity, Nonlinear Anal. 36 (4) (1999) 507–510. P. Binding, Variational principles for indefinite eigenvalue problems, Linear Algebra Appl. 218 (1995) 251–262. M. Bôcher, The smallest characteristic numbers in a certain exceptional case, Bull. Amer. Math. Soc. 21 (1914) 6–9. K.J. Brown, P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Diff. Integral Equat. 3 (2) (1990) 201–207. K.J. Brown, S.S. Lin, On the existence of positive eigenfunction for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl. 75 (1980) 112–120. K.J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Diff. Equat. 239 (2007) 296–310. N.P. Các, A.M. Fink, J.A. Gatica, Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign, Proc. Amer. Math. Soc. 123 (5) (1995) 1393–1398. N.P. Các, J.A. Gatica, Y. Li, Positive solutions to semilinear problems with coefficient that changes sign, Nonlinear Anal. 37 (4) (1999) 501–510. L.H. Erbe, H.Y. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (3) (1994) 743–748. A.M. Fink, J.A. Gatica, Positive solutions of second order systems of boundary value problems, J. Math. Anal. Appl. 180 (1) (1993) 93–108. A.M. Fink, J.A. Gatica, G.E. Hernández, Eigenvalues of generalized Gelfand models, Nonlinear Anal. 20 (12) (1993) 1453–1468. W.H. Fleming, A selection–migration model in population genetics, J. Math. Biol. 2 (3) (1975) 219–233. P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (4) (1982) 441–467. D.D. Hai, Positive solutions to a class of elliptic boundary value problems, J. Math. Anal. Appl. 227 (1998) 195–199. J. Henderson, H.Y. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl. 208 (1997) 252–259. E.L. Ince, Ordinary Differential Equations, Dover, New York, 1926. H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer-Verlag, New York, 2004. K.Q. Lan, J.R.L. Webb, Positive solutions of semilinear differential equations with singularities, J. Diff. Equat. 148 (2) (1998) 407–421.
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[19] R.Y. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett. 21 (7) (2008) 754–760. [20] R.Y. Ma, X.L. Han, Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function, Nonlinear Anal. 71 (2009) 2119– 2125. [21] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. [22] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513. [23] P.H. Rabinowitz, On bifurcation from infinity, J. Diff. Equat. 14 (1973) 462–475. [24] H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Diff. Equat. 109 (1) (1994) 1–7.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function q Ruyun Ma *, Xiaoling Han Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, PR China
a r t i c l e
i n f o
a b s t r a c t This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem
Keywords: Indefinite weight problem Bifurcation Positive solutions
u00 þ kaðtÞf ðuÞ ¼ 0;
0 < t < 1;
uð0Þ ¼ uð1Þ ¼ 0;
where a 2 C½0; 1 changes sign. The proof of our main result is based upon bifurcation techniques. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and main result In this paper we consider the existence and multiplicity of positive solutions of the boundary value problem
u00 þ kaðtÞf ðuÞ ¼ 0;
0 < t < 1;
uð0Þ ¼ uð1Þ ¼ 0;
ð1:1Þ ð1:2Þ
where a 2 C½0; 1 may change sign, k is a parameter. Problem (1.1) and (1.2) and its higher dimensional analogue
Du þ kgðxÞf ðuÞ ¼ 0; u ¼ 0; x 2 @ X;
x 2 X;
ð1:3Þ ð1:4Þ
(where X is a bounded domain with smooth boundary) arises from many branches of applied mathematics. For example, the generalized Emden–Fowler equation, where f ¼ up ; p > 0 and a > 0 in (1.1) arises in the fields of gas dynamics, nuclear physics, and chemically reacting system [18]; the Thomas–Fermi equation, where f ðuÞ ¼ u3=2 and aðtÞ ¼ t 1=2 was developed in studies of atomic structures [18]; and the selection–migration model in population genetics where g in (1.3) changes sign in X [12]. Existence and multiplicity of positive solutions of (1.1)–(1.4) with a nonnegative weight function has been extensively studied, see, for example, Erbe and Wang [9], Fink et al. [10,11], Lions [13], Henderson and Wang [15], Lan and Webb [18], Wang [24] and the references therein. The main tools used in these paper are fixed point theorem in cones and the fixed point index theory in cones.
q Supported by the NSFC (No.10671158), the NSF of Gansu Province (No. 3ZS051-A25-016), NWNUKJCXGC- 03-17, the Spring-sun program (No. Z2004-162033), SRFDP (No. 20060736001), the SRF for ROCS, SEM (2006[311]), NWNU-KJCXGC-3-18, NWNU-KJCXGC-3-17, NWNU-KJCXGC-03-40, and NSF of Gansu Province (No. 3ZS061-A25-016). * Corresponding author. E-mail address: [email protected] (R. Ma).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.042
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R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
The existence of positive solutions of (1.1)–(1.4) with a indefinite weight function has also been studied by several authors, see Afrouzi and Brown [1], Các et al. [7], Các et al. [8] and Hai [14]. However, the existence results of positive solutions in these papers were established only when the parameter k > 0 is small enough. It is the purpose of this paper to study the indefinite weight boundary value problem (1.1) and (1.2) under the assumptions: (H1) f 2 C 2 ðR; RÞ with f ð0Þ ¼ f ðs1 Þ ¼ f ðs2 Þ ¼ 0; 0 < s1 6 s2 ; f ðsÞ > 0 for s 2 ð0; s1 Þ [ ðs2 ; þ1Þ; f ðsÞ < 0 for s 2 ðs1 ; s2 Þ; (H2) f 00 ðsÞ < 0 for s 2 ½0; s1 Þ; (H3) there exist f0 ; f1 2 ð0; 1Þ such that
f0 ¼ lim s!0
f ðsÞ ; s
f 1 ¼ lim
s!1
f ðsÞ ; s
(H4) a 2 C½0; 1 changes sign. We will show the existence, multiplicity of positive solutions of (1.1) and (1.2) (in which a changes sign) via global bifurcation techniques. For earlier work related to the nodal solutions with the case aðtÞ > 0 on ½0; 1, see Ma [19]. It is well known (cf. Binding [2], Bôcher [3], Brown and Lin [5] and Ince [16]) that the indefinite weight linear eigenvalue problem
u00 þ kaðtÞu ¼ 0;
0 < t < 1;
ð1:5Þ
uð0Þ ¼ uð1Þ ¼ 0; have two infinite sequences of simple eigenvalues
0 < kþ < kþ1 < kþ2 < < kþk < ;
lim kþk ¼ þ1
k!1
and
0 > k > k1 > k2 > > kk > ;
lim kk ¼ 1
k!1
such that the eigenfunction corresponding to kmk ; m 2 fþ; g, has exactly k simple zeros in (0, 1), and km ; m 2 fþ; g is principle eigenvalue(i.e., an eigenvalue having a positive eigenfunction). Let uþ ; u be the positive eigenfunctions corresponding to kþ and k , respectively. Similar spectrum results were established for discrete analogue of (1.5) by Binding [2]. Remark 1.1. It is worth remarking that if u is a nontrivial solution of (1.1) and (1.2) in which f satisfy (H1), then any initial value problem (IVP) associated with (1.1) has a unique solution, so u have n simple zeros in (0, 1) for some n 2 N. In fact, if a nontrivial solution uðtÞ has a double zero t 0 2 ð0; 1Þ, then uðt 0 Þ ¼ u0 ðt 0 Þ ¼ 0. Thus, by the uniqueness of solutions of the associated IVP, we have uðtÞ 0 on [0, 1], contradicting the assumption that uðtÞ is nontrivial. Suppose ðk; uÞ is a solution of (1.1) and (1.2), u is said to be stable if all the eigenvalues of the linearized operator associated with (1.1) and (1.2) at ðk; uÞ are strictly positive. Our main result is the following: Theorem 1.1. Let (H1)–(H4) hold. (a) Assume that f0 < f1 . Then (1.1) and (1.2) have at least two positive solutions for k 2 1; kf0 [ kf0þ ; þ1 , and one of them is stable; there exist at least one positive solution for k 2 ½kf0 ; kf1 Þ [ ðkf1þ ; kf0þ . (b) Assume that f1 < f0 . Then (1.1) and (1.2) have at least two positive solutions for k 2 1; kf1 [ kf1þ ; þ1 , and one of them is stable; there exist at least one positive solution for k 2 ½kf1 ; kf0 Þ [ ðkf0þ ; kf1þ and it is stable. If the condition (H3) is replaced by
ðH3Þ
f 0 2 ð0; 1Þ; f1 ¼ 1;
then we obtain the following result: Theorem 1.2. Let (H1),(H2), ðH3Þ and (H4) hold. Then (1.1) and (1.2) has a positive solution if and only if k–0. Moreover, for k 2 1; kf0 [ kf0þ ; þ1 there exist at least two positive solutions, one of them is stable. The proof of above theorems are based on the following two theorems. Theorem A (Crandall–Rabinowitz Theorem [17, Theorem I.5.1]). Consider mappings F : V U ! Z with open sets U X; V Y, where X and Z are Banach spaces, and Y ¼ R. Suppose: (i) Fðk; 0Þ ¼ 0; 8 k 2 R, and dim NðDx Fðk0 ; 0ÞÞ ¼ codim RðDx Fðk0 ; 0ÞÞ ¼ 1; (ii) F 2 C 2 ðV U; ZÞ, where 0 2 U X; k0 2 V R;
R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
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(iii) NðDx Fðk0 ; 0ÞÞ ¼ span½v 0 ; v 0 2 X; kv 0 k ¼ 1; D2xk Fðk0 ; 0Þv 0 R RðDx Fðk0 ; 0ÞÞ. Then there is a nontrivial continuously differentiable curve through ðk0 ; 0Þ,
fðkðsÞ; xðsÞÞjs 2 ðd; dÞ; ðkð0Þ; xð0ÞÞ ¼ ðk0 ; 0Þg;
ð1:6Þ
such that
FðkðsÞ; xðsÞÞ ¼ 0;
s 2 ðd; dÞ
and all solutions of Fðk; xÞ ¼ 0 in a neighborhood of ðk0 ; 0Þ are on the trivial solution line or on the nontrivial curve (1.6). Consider the global solution behavior of the bifurcation branches of the equation
x ¼ lðLx þ NxÞ;
l 2 K; x 2 X:
ð1:7Þ
The closure of the set of nontrivial solutions of (1.7) will be denoted by S. Suppose (A1) The operators L; N : X ! X are compact on the B-space X over K where K ¼ R. Furthermore, L is linear and kNxk n kxk ! 0 as kxk ! 0. (A2) The real number l0 is a characteristic number of L of odd algebraic multiplicity.
Theorem B (Rabinowitz Theorem [22, Theorem 1.3]). Suppose (A1) and (A2) are given. Then S possesses a maximal subcontinuum Cl such that ðl0 ; 0Þ 2 Cl and Cl either (i) Cl is unbounded, or (ii) Cl is compact and in addition to ðl0 ; 0Þ also contains a further point of the trivial solution branch T.
Remark 1.2. In Brown [6] and Ma and Han [20], the Rabinowitz’s global bifurcation theorem were used to prove the existence of nodal/positive solutions of some indefinite weight problems.
2. Proof of main theorem To prove our main result, we need some preliminary results. Let X ¼ fu 2 C 2 ½0; 1 : uð0Þ ¼ uð1Þ ¼ 0g; E ¼ fu 2 C 1 ½0; 1 : uð0Þ ¼ uð1Þ ¼ 0g; Y ¼ C½0; 1. Define L : X ! Y by setting
Lu :¼ u00 ;
u 2 X:
Then (1.1) and (1.2) is equivalent to
Lu ¼ kaðtÞf ðuÞ;
u 2 X:
Lemma 2.1. For all p P 1, we have
Z
1
k 0
aðtÞupþ1 dt > 0:
ð2:1Þ
Proof. Multiplying Lu ¼ k aðtÞu by up we obtain Lu up ¼ k aðtÞupþ1 on (0, 1), integrating from 0 to 1, it gives rise to the relation
Z k 0
1
aðtÞupþ1 dt ¼ p
Z
1
0
up1 ðu0 Þ2 dt:
Hence, (2.1) hold. h Consider F : R X ! Y defined by
Fðk; uÞ ¼ Lu kaðtÞf ðuÞ; then F is a smooth map with Fréchet derivative
F u ðk; 0Þw ¼ Lw kf0 aðtÞw:
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R. Ma, X. Han / Applied Mathematics and Computation 215 (2009) 1077–1083
R1 Thus, N F u kf0 ; 0 ¼ ½u and R F u kf0 ; 0 ¼ ½u ? ¼ fu 2 C½0; 1 : 0 uu dt ¼ 0g. Moreover F ku kf0 ; 0 u ¼ aðtÞu , by Lemma 2.1,
Z k 0
1
aðtÞu2 dt > 0;
it follows that F ku kf0 ; 0 u R R F u kf0 ; 0 . Thus, by Theorem A, there exist two nontrivial continuously differentiable curves passing through kf0 ; 0 , respectively, and with the form
fðk ðsÞ; u ðsÞÞjs 2 ðd; dÞg such that
ðk ðsÞ; u ðsÞÞ ¼ for s 2 ðd; dÞ with
k þ l ðsÞ; sðu þ w ðsÞÞ f0
ð2:2Þ
l ð0Þ ¼ 0; w ð0Þ ¼ 0; w ðsÞ 2 X \ ½u ? , and
Fðk ðsÞ; u ðsÞÞ ¼ 0 for s 2 ðd; dÞ and all nontrivial solutions of Fðk; uÞ ¼ 0 in a neighborhood of kf0 ; 0 are on the curve (2.2). By Theorem B and Remark 1.1, these two continua Cþ and C of solutions bifurcating from kf0þ ; 0 and kf0 ; 0 is unbounded, respectively. Lemma 2.2. Suppose (H2)–(H4) hold. Then Cþ bifurcates to the right at
kþ f0
; 0 ; C bifurcates to the left at kf0 ; 0 .
Proof. Note that (1.1) and (1.2) is equivalent to
Lu ¼ kf0 aðtÞu þ kaðtÞðf ðuÞ f0 uÞ;
ð2:3Þ
substituting expression (2.2) in (2.3) and using that Lu ¼ k aðtÞu , we deduce that
ðL k aÞw ðsÞ ¼ f0 l ðsÞaðu þ w ðsÞÞ þ
k f ðu ðsÞÞ f0 u ðsÞ þ l ðsÞ a ; s f0
ð2:4Þ
dividing (2.4) by s and take limit as s ! 0, we have
ðL k aÞw0 ð0Þ ¼ f0 l0 ð0Þau þ
k f 00 ð0Þ 2 au ; 2f 0
ð2:5Þ
00 this means f0 l0 ð0Þau þ k2ff 0ð0Þ au2 2 RðL k aÞ ¼ R F u kf0 ; 0 ¼ ½u ? , which is equivalent to
Z
1
0
k f 00 ð0Þ 2 f0 l0 ð0Þau þ au u dt ¼ 0; 2f 0
so we get
l0 ð0Þ ¼
R1 k f 00 ð0Þ 0 aðtÞu3 dt : R 1 2f02 aðtÞu2 dt 0
from condition (H2) and Lemma 2.1 we have l0þ ð0Þ > 0 and l0 ð0Þ < 0. So Cþ bifurcates to the right at cates to the left at kf0 ; 0 . h We now analyze the global behavior of Cþ and C .
kþ f0
; 0 and C bifur-
Lemma 2.3. Suppose (H1), (H3) and (H4) hold. Then for ðk; uÞ 2 Cþ [ C ,
0 6 uðtÞ < s1 ;
t 2 ½0; 1:
ð2:6Þ
Proof. Suppose on the contrary that there exists ðk; uÞ 2 Cþ such that
max uðtÞ ¼ s1 :
ð2:7Þ
t2½0;1
By (H1), (H3) and (H4), there exists m P 0 such that aðtÞf ðsÞ þ ms is strictly increasing in s for s 2 ½0; s1 . Then
Lu þ kmu ¼ kðaðtÞf ðuÞ þ muÞ;
t 2 ð0; 1Þ
ð2:8Þ
and, since Ls1 ¼ 0 ¼ f ðs1 Þ,
ðL þ kmÞs1 ¼ kðaðtÞf ðs1 Þ þ ms1 Þ:
ð2:9Þ
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Subtracting, we get
ðL þ kmÞðs1 uðtÞÞ P 0;
t 2 ð0; 1Þ
ð2:10Þ
and
s1 uð0Þ > 0;
s1 uð1Þ > 0:
ð2:11Þ
The maximum principle [21] implies that s1 > uðtÞ in ½0; 1, it is a contradiction to (2.7). Therefore
uðtÞ < s1 ;
t 2 ½0; 1:
If there exists ðk; uÞ 2 C such that
max uðtÞ ¼ s1 : t2½0;1
Note that in this case k < 0, so we can choose m P 0 such that aðtÞf ðsÞ ms is strictly decreasing in s for s 2 ½0; s1 . Then
Lu kmu ¼ kðaðtÞf ðuÞ muÞ;
t 2 ð0; 1Þ
and, since Ls1 ¼ 0 ¼ f ðs1 Þ,
ðL kmÞs1 ¼ kðaðtÞf ðs1 Þ ms1 Þ: Subtracting, we get
ðL kmÞðs1 uðtÞÞ P 0;
t 2 ð0; 1Þ
and
s1 uð0Þ > 0;
s1 uð1Þ > 0:
The maximum principle implies that s1 > uðtÞ in ½0; 1, this is a contradiction too. h Define P : R C½0; 1 ! R, the projection map onto R, i.e., Pðk; uÞ ¼ k. Lemma 2.4. Suppose (H1)–(H4) hold. Then for ðk0 ; u0 Þ 2 Cþ [ C ; u0 is stable. So PðCþ Þ ¼
kþ f0
; þ1 ; PðC Þ ¼ 1; kf0 .
Proof. The proof is motivated by Brown and Hess [4]. We need to prove all the eigenvalues of the operator Lw k0 aðtÞf 0 ðu0 Þw are strictly positive. To do this, we only need to prove that all eigenvalues l of the linearized problem associated with (1.1) and (1.2), viz
Lw k0 aðtÞf 0 ðu0 Þw ¼ lw
ð2:12Þ
are positive. Suppose that lðk0 Þ and / are the principle eigenvalue and eigenfunction of the linearized operator L k0 aðtÞf 0 ðu0 Þ. Then we have
Lu0 k0 aðtÞf ðu0 Þ ¼ 0;
t 2 ð0; 1Þ
ð2:13Þ
and
L/ k0 aðtÞf 0 ðu0 Þ/ ¼ lðk0 Þ/;
t 2 ð0; 1Þ:
ð2:14Þ
0
Multiplying (2.13) by f ðu0 Þ/ and (2.14) by f ðu0 Þ, subtracting and integrating gives
2
Z 0
1
ðu00 Þ2 f 00 ðu0 Þ/dt ¼ lðk0 Þ
Z
1
f ðu0 Þ/dt:
ð2:15Þ
0
By Lemma 2.3, 0 6 u0 < s1 for t 2 ½0; 1, from (H1),(H2) we have f ðu0 Þ > 0; f 00 ðu0 Þ < 0, it follows that lðk0 Þ > 0, so u0 is stable. Further more we know that F u ðk0 ; u0 Þw ¼ Lw k0 aðtÞf 0 ðu0 Þw is an isomorphism, from implicit function theorem we know that all solutions of (1.1) and (1.2) in R X near to ðk0 ; u0 Þ lie on a C 1 curve passing through ðk0 ; u0 Þ and parametrised by k. This curve of nontrivial solutions interval of definition over the k axis. This together with can be continuedto a maximal Lemma 2.2 we get that PðCþ Þ ¼ kf0þ ; þ1 ; PðC Þ ¼ 1; kf0 . h In the following we will investigate the other positive solutions of problem (1.1) and (1.2). Let n 2 CðR; RÞ be such that
f ðuÞ ¼ f1 u þ nðuÞ:
ð2:16Þ
Clearly
lim
u!1
nðuÞ ¼ 0: u
ð2:17Þ
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Let us consider
Lu kaðtÞf1 u ¼ kaðtÞnðuÞ
ð2:18Þ
as a bifurcation problem from infinity. We note that (2.18) is equivalent to (1.1) and (1.2). The results of Rabinowitz can [23]for (2.18) be stated as follows: there exists a continuum Dþ and D of positive solutions of (2.18) meeting kf1þ ; 1 and kf1 ; 1 , respectively. Lemma 2.5. Suppose (H1), (H3) and (H4) hold. Then for ðk; uÞ 2 Dþ [ D , we have
max uðtÞ > s2 : t2½0;1
Proof. It is similar to the proof of Lemma 2.3, so we omit it. h Lemma 2.6. Suppose (H1), (H3) and (H4) hold. Then
kþ f1
; þ1 PðDþ Þ and 1; kf1 PðD Þ.
kþ Proof. Firstly, we proof kf1þ ; þ1 PðDþ Þ.Take K R is an interval such that K \ fkf1þ ; f1j jj 2 N n f0gg ¼ fkf1þ g and M is a neighborhood of kf1þ ; 1 whose projection on R lies in K and whose projection on E is bounded away from 0. Then by [21, Theorem 1.6 and Corollary 1.8], we have that, either (1) Dþ n M is bounded in R E in which case Dþ n M meetsfðk; 0Þjk 2 Rg. (2) Dþ n M is unbounded. n þo ^ ; 1Þ where l ^ 2 kf k . Moreover if (2) occurs and Dþ n M has a bounded projection on R, then Dþ n M meets ðl 1 Obviously Lemma 2.5 implies that (1) do not occur. So Dþ n M is unbounded. kþ Remark 1.1 guarantees that Dþ is a component of positive solutions of (2.18) which meets f1 ; 1 . Therefore there is no þ k k 2 N n f0g such that Dþ also meets f1k ; 1 . Otherwise, there will exist ðg; yÞ 2 Dþ such that y has a multiple zero point in (0,1). However this contradicts with Remark 1.1, and consequently PðDþ n MÞ is unbounded. Thus
kþ ; þ1 PðDþ Þ f1 and similarly we have
k 1; PðD Þ: f1
Proof of Theorems 1.1 and 1.2. From Lemma 2.1–2.6 we already complete the proof of Theorem 1.1. We note that if f1 ¼ 1; then kf1 ¼ 0, then (1.7) has a positive solution if and only if k – 0. Acknowledgement The authors are very grateful to the anonymous referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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