Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

Ziyatkhan S. Aliyev

and Leyla V. Nasirova

1Department of Mathematical Analysis, Baku State University, Z. Khalilov Str. 23, Baku, AZ1148, Azerbaijan

2Department of Differential Equations, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, B. Vahabzadeh Str. 9, Baku, AZ1141, Azerbaijan

3Department of Mathematical Analysis, Sumgait State University, Baku Str. 1, Sumgait, AZ5008, Azerbaijan

Received 27 January 2021, appeared 28 July 2021 Communicated by Alberto Cabada

Abstract. In this paper, we consider bifurcation from zero or infinity of nontrivial so- lutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontriv- ial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line R× {∞} (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

Keywords: nonlinear Sturm–Liouville problem, indefinite weight, population genetics, selection-migration model, bifurcation point, bifurcation interval, global continua.

2020 Mathematics Subject Classification: 34B15, 34B24, 34C10, 34C23, 34L15, 45C05, 47J10, 47J15.

1 Introduction

We consider the following nonlinear Sturm–Liouville eigenvalue problem

(`(u))(x)≡ −(p(x)u⁰(x))⁰+q(x)u(x) =λρ(x)u(x) +h(x,u(x),u⁰(x),λ), x∈ (0, 1), (1.1) α₀u(0)−β₀u⁰(0) =0, α₁u(1) +β₁u⁰(1) =0, (1.2) where λ ∈ _R is a spectral parameter, p ∈ C¹([0, 1];(0,+_∞)), q ∈ C([0, 1];[0,+_∞)), and ρ ∈ C([0, 1];R)such that there existς, ξ ∈[0, 1]for whichρ(ς)ρ(ξ)<0, andα_i,β_i,i=0, 1, are real

BCorresponding author. Email: z_aliyev@mail.ru

constants such that|α_i|+|β_i|>0 andα_iβ_i ≥0,i=0, 1. The functionhhas the representation h= f +g, where the functions f,g∈C([0, 1]×_R³;R)and satisfy the conditions

u f(x,u,s,λ)≤0, u g(x,u,s,λ)≤0; (1.3) there exists a constantM>0 such that

f(x,u,s,λ) u

≤ M, (x,u,s,λ)∈[0, 1]×_R²×_R, u6=0. (1.4) Moreover, at various points in the paper, we will impose one or the other or both of the following conditions on the functiong:

g(x,u,s,λ) =o(|u|+|s|), as|u|+|s| →_∞, (1.5) or

g(x,u,s,λ) =o(|u|+|s|), as|u|+|s| →0, (1.6) uniformly forx∈ [0, 1]andλ∈ Λ, for any bounded intervalΛ⊂_R.

Nonlinear eigenvalue problems of the type (1.1), (1.2) have been intensively studied re- cently, as they arise from selection-migration models in population genetics (see for exam- ple [2,3,9,10,14,16,18] and references therein). Note that population genetics is one of the important branches of biology, which studies the genetic structure and evolution of popu- lations. It has close ties to ecology, demography, epidemiology, phylogeny, genomics, and molecular evolution. Population genetics is mainly used in human genetics and medicine, as well as in animal and plant breeding. In the case of p(x) ≡ 1 and h(x,u(x),u⁰(x),λ) = λρ(x)[u(x)−m(u(x))], where m(u) = u(1−u)[h0(1−u) + (1−h0)u] and h0 ∈ (0, 1), Eq.

(1.1) is an one-dimensional reaction-diffusion equation, the interval [0,1] refers to the habi- tat of a species, the boundary conditions (1.2) for β₀ = β₁ = 0 means that no individuals cross the boundary of the habitat. Moreover, the weight function ρ(x) represents either the selective strength of the environment on genes, or the intrinsic growth rate of the species at locationx, and the real parameterλ corresponds to the reciprocal of the diffusion coefficient (see [10,14,18]).

If condition (1.6) is satisfied, then we consider bifurcation fromu=0, i.e. bifurcation from the line of trivial solutionsR₀ = _R× {0}. In the case whenρ(x)> 0 forx ∈ [0, 1]the global bifurcation of solutions of the nonlinear eigenvalue problem (1.1), (1.2) under conditions (1.4) and (1.6) (but without the conditionsα_iβ_i ≥0,i=0, 1) was considered in [8,11,12,22,23,25,26].

These papers it was shown the existence of two families of unbounded continua of nontrivial solutions in R×C¹[0, 1], possessing the usual nodal properties and bifurcating from points and intervals of the lineR₀ corresponding to the eigenvalues of the linear problem obtained from (1.1), (1.2) by settingh≡0. Similar results in nonlinear eigenvalue problems for ordinary differential equations of fourth order were established in the paper [1].

If condition (1.5) is satisfied, then problem (1.1), (1.2) for f ≡0 is asymptotically linear (see [17]), and, therefore, we must investigate the bifurcation from infinity, that is, the existence of non-trivial solutions to this problem with large norms. Note that in the case whenρ(x) > 0 forx ∈ [0, 1]the global bifurcation from infinity of nontrivial solutions of problem (1.1), (1.2) under conditions (1.4) and (1.5) (again without the conditionsα_iβ_i ≥ _0, i=0, 1) was studied in [11,12,22,24,25,27,28], where in particular, it was shown that there are two families of global continua of nontrivial solutions of this problem bifurcating from points and intervals of the setR_∞ = _R× {_∞}corresponding to the eigenvalues of the linear problem (1.1), (1.2)

with h ≡ 0 and having usual nodal properties in some neighborhoods these points and in- tervals. Moreover, it was also established that these continua either contain other asymptotic bifurcation points and intervals, or intersect the line R₀, or have an unbounded projection ontoR₀. Similar global results for fourth order nonlinear eigenvalue problems were obtained in the paper [6].

The problem (1.1), (1.2) in cases (i) f ≡ 0, and (ii) g ≡ 0 and f satisfies condition (1.4) for any (x,u,s,λ) ∈ [_{0, 1}]×_R²×_{R such that} u 6= _{0 and} |u|+|s| ≤ τ₀, where τ₀ > _{0 is} some constant, was considered in [7,21]. These papers prove the existence of four families of unbounded continua of solutions having the usual nodal properties and bifurcating from points and intervals of the line of trivial solutions corresponding to the positive and negative eigenvalues of linear problem (1.1), (1.2) withh ≡0.

The purpose of this paper is to study the location of bifurcation intervals in R₀ and R_∞, and the structure of global continua of nontrivial solutions of problem (1.1), (1.2) emanating from these bifurcation intervals.

In Section 2, we present the main properties of the eigenvalues and eigenfunctions of the linear problem (1.1), (1.2) with h ≡ 0. Here we introduce classes of functions inR×C¹[0, 1] with a fixed oscillation counter and also possessing other properties of the eigenfunctions of this linear problem. Here we consider problem (1.1), (1.2) under conditions (1.3), (1.4) and (1.6). Then we find the bifurcation intervals of the line of trivial solutions with respect to the above-mentioned oscillation classes and establish that the connected components of solutions emanating from bifurcation intervals are contained in the corresponding oscillation classes, and are unbounded in R×C¹[0, 1]. In Section 3 and 4 we consider problem (1.1), (1.2) under conditions (1.3)- (1.5). In Section 3 developing the approximation technique from [8], we prove the existence of nontrivial solutions of problem (1.1), (1.2) with large norms contained in the classes with a fixed oscillation count. Moreover, we find intervals containing asymptotically bifurcation points of problem (1.1), (1.2) with respect to these classes. Note that the approximation equation introduced here is more natural than those introduced in [24,25].

It is important to note that the solutions of problem (1.1), (1.2) from the global continuum emanating from the bifurcation interval of the set R_∞ with respect to a certain class of a fixed oscillation count and located outside a some neighborhood of this interval may not be included in this oscillation class. In Section 4, we present and prove the main result of this paper, namely, we show that there are four classes of unbounded continua of solutions of problem (1.1), (1.2) emanating from asymptotically bifurcation intervals which have usual nodal properties in a some neighborhoods of these intervals and for each of them one of the following statements holds: either contain other asymptotic bifurcation intervals, or intersect the lineR₀, or have an unbounded projection ontoR₀. Similar results in nonlinear eigenvalue problems for ordinary differential equations of fourth order and semi-linear elliptic partial differential equations with indefinite weight in the classes of positive and negative functions were obtained in recent papers [2–5]. In Section 5 we consider problem (1.1), (1.2) under both conditions (1.5) and (1.6). Here we manage to show that the continua emanating from asymptotically bifurcation intervals are contained in the corresponding oscillation classes and therefore they do not intersect other asymptotically bifurcation intervals. In Section 6 we consider problem (1.1), (1.2) in the case when the weight function ρ(x) ≥ 0 for x ∈ [0, 1]. Note that in this case linear problem obtained from (1.1), (1.2) by settingh ≡ 0 has only one sequence of positive simple eigenvalues, and consequently, in this case problem (1.1), (1.2) has two families of global connected components emanating from bifurcation intervals ofR₀ or R_∞, and having the properties of global continua from Sections 2 and 3–4, respectively.

Note that similar results was obtained in [11,12] in the case of a special form of the nonlinear term f.

2 Preliminary

By (b.c.) we denote the set of functions satisfying the boundary conditions (1.2).

It is known [15, Ch. 10, § 10·61] that the spectrum of the linear eigenvalue problem ((`(u))(x) =λρ(x)u(x), x∈ (0, 1),

u∈(b.c.), (2.1)

obtained from (1.1), (1.2) by settingh≡0 consists of two sequences of real and simple eigen- values

0<λ⁺₁ <λ₂⁺<· · ·< λ⁺_k 7→ +_{∞ and 0}>λ⁻₁ >λ⁻₂ > · · ·>λ⁻_k 7→ −_∞;

for each k ∈ _N the eigenfunctions u⁺_k andu⁻_k corresponding to the eigenvalues λ⁺_k and λ⁻_k , respectively, have exactlyk−1 simple nodal zeros in(0, 1)(by a nodal zero, we mean that the function changes sign at the zero, and at a simple nodal zero, the derivative of the function is nonzero). Moreover, according to [21, formula (2.10)] for eachk ∈_Nthe eigenfunctionsu⁺_k andu⁻_k satisfy the following relations

Z ₁

0

ρ(x)(u⁺_k(x))²dx>_{0 and}

Z ₁

0

ρ(x)(u⁻_k(x))²dx<_{0, (2.2)} respectively.

Let E = C¹[0, 1]∩(b.c.) be a Banach space with the usual norm kuk₁ = kuk_∞+ku⁰k_∞, wherekuk_∞ =max_x∈[0,1]|u(x)|.

From now onσ (νrespectively) will denote either +or−.

For eachk ∈_{N, each}σ and eachνwe denote byS_k,^ν_σ the set of functionsu∈ Esatisfying the following conditions:

(i) uhas exactlyk−1 simple nodal zeros in the interval(0, 1); (ii) σR1

0 ρ(x)u²(x)dx>_0;

(iii) νu(x)is positive in a deleted neighborhood of the pointx =0.

It follows from definition of the setS_k,σ^ν ,k ∈ N, that this set is open inEfor eachσand each ν. Note that for any(k,σ,ν)6= (k⁰,σ⁰,ν⁰)the following relation holds:

S_k,^ν_σ∩ S_k^ν0⁰,σ⁰ =_∅.

Moreover, ifu∈ ∂S_k,^ν_σ, then either (i) R1

0 ρ(x)u²(x)dx=0, or

(ii) there existsx₀ ∈[0, 1]that suchu(x₀) =u⁰(x₀) =0.

Remark 2.1. It follows from the above arguments that u^σ_k ∈ S_k,σ= S_k,⁺

σ∪ S_k,⁻

σ, k∈_N.

Remark 2.2. If(λ,u)∈_R×Ebe a nontrivial solution of problem (1.1), (1.2), then λ

Z ₁

0

ρ(x)u²(x)dx6=0.

Indeed, multiplying both sides of (1.1) byu(x), integrating the obtained equality in the range from 0 to 1, using the formula for integration by parts, and taking into account conditions (1.2), (1.3), we get

Z ₁

0

p(x)u⁰²(x) +q(x)u²(x) dx+N[u] =λ Z ₁

0

ρ(x)u²(x)dx +

Z ₁

0 f x,u(x),u⁰(x),λ

u(x)dx+

Z ₁

0 g x,u(x),u⁰(x),λ

u(x)dx, (2.3)

where N[u] =−(p(x)u⁰(x)u(x))|^x_x⁼₌¹₀. Since the conditions|α_i|+|β|>0 andα_iβ_i ≥0,i=0, 1, are satisfied it follows that N[u]≥0 for any function u∈ E. Consequently, the left hand side of (2.3) is positive, and if λ R1

0 ρ(x)u²(x)dx = 0, then by conditions (1.3) the right hand side of this relation is non-positive, a contradiction.

Let {λ⁺_k,M}^∞_k=1 and{λ⁻_k,M}^∞_k=1 be sequences of positive and negative eigenvalues, respec- tively, of the following spectral problem

((`(u))(x) +Mu(x) =λρ(x)u(x), x ∈(0, 1),

u∈ (b.c.), (2.4)

which are simple (see [15, Ch. 10, § 10·61]).

We introduce the notations:

I_k⁺= [λ⁺_k ,λ⁺_k,M], I_k⁻ = [λ⁻_k,M,λ⁻_k ],

R⁺= (0,+_∞), R⁻= (−_{∞, 0}), R⁺₀ =_R⁺× {0}, R⁻₀ =_R⁻× {0}, and

R⁺_∞=_R⁺× {_∞}_, R⁻_∞ =_R⁻× {_∞}_.

ByD we denote the set of nontrivial solutions of the nonlinear eigenvalue problem (1.1), (1.2). For any λ ∈ R, we say that a subset C ⊂ D meets (λ, 0) (respectively (λ,∞)) if there exists a sequence {(λ_n,u_n)}^∞_n=1 ⊂ C such that λ_n → λandku_nk₁ → 0 (respectivelyku_nk₁ → +_{∞) asn}→+∞. Furthermore, we will say thatC ⊂ Dmeets(_{λ, 0})(respectively(_λ,∞)_{) with} respect to the set R× S_k,^ν_σ, if the sequence{(λn,un)}^∞_n=1 can be chosen so that un ∈ S_k,^ν_σ for alln∈N. Moreover, we say(λ, 0)(respectively(λ,∞)) is a bifurcation point of problem (1.1), (1.2) with respect to the set R× S_k,^ν_σif there exists a sequence{(λ_n,un)}^∞_n=1⊂ D ∩(_R× S_k,^ν_σ) such that λ_n → λ and ku_nk₁ → 0 (respectively ku_nk₁ → +_{∞) as} n → +_{∞. If} I ⊂ _R is a bounded interval we say that C meets I× {0} (I× {_∞} respectively) if C meets (λ, 0) (respectively(λ,∞)) for someλ ∈ I. Furthermore, we will say thatC meets I× {0}(I× {_∞} respectively) with respect to the set R× S_k,^ν_σ, if C meets (λ, 0)(respectively (λ,∞)), λ ∈ I, with respect to the set R× S_k,^ν

σ (see [1,6,25]).

Now we consider problem (1.1), (1.2) under the conditions (1.3), (1.4) and (1.6). Following the corresponding reasoning given in [7,19,21] (see also [1,8]) and using Remark 2.2 we are convinced that the following results hold for this problem.

Lemma 2.3. Let (λ,u) ∈ _R×E be a solution of problem(1.1), (1.2) such that u ∈ ∂S_k,σ^ν , k ∈ _N, σ, ν∈ {+,−}. Then u ≡0.

Lemma 2.4. For each k∈_{N, each}σand eachνthe setB_k,^ν_σof bifurcation points (from zero) of problem (1.1),(1.2)with respect to the setR^σ× S_k,^ν

σis nonempty. Furthermore, B_k,^ν

σ⊂ I_k^σ× {0}.

For each k ∈_{N, each} σ and eachνbyD^ν,_k,σ^∗ we denote the union of all the components of D which meet I_k^σ× {0}with respect to the set R^σ× S_k,^ν

σ ([25, Theorem 3.2] and Lemma2.4 implies thatD_k,^ν,∗

σ 6= _∅). The setD_k,^ν,∗

σ may not be connected inR^σ×E, but joining the interval I_k^σ× {0}to this set gives a connected setD_k,^ν_σ=D^ν,_k,σ^∗∪(I_k^σ× {0}).

Remark 2.5. By Lemma2.4it follows from Remark2.2 thatD^ν_k,

σ ⊂_R^σ×E.

Theorem 2.6. For each k∈_N, eachσand eachνthe connected setD^ν_k,

σwhich contains I_k^σ× {₀}lies in(_R^σ× S_k,^ν_σ)∪(I_k^σ× {0})and is unbounded inR×E.

3 The existence of asymptotic bifurcation points of problem (1.1), (1.2) with respect to the set S

In the next two sections, we will consider problem (1.1), (1.2) under the conditions (1.3)–(1.5).

To study the structure of the set of asymptotic bifurcation points of problem (1.1), (1.2), we introduce the following modified nonlinear eigenvalue problem

(`(u) =λρ(x)u+ ₍^{f(x,|u|εu,u0,λ)}

1+|u|+|u⁰|)^2ε +g(x,u,u⁰,λ), x ∈(0, 1),

u∈(b.c.), (3.1)

whereε∈ (0, 1]. It is seen that this problem in a sense approximates problem (1.1), (1.2) for ε near 0. Note that approximations similar to this one were previously used in [6,25].

Since f ∈C([0, 1]×_R³;R)it follows that ^|₍^f₁⁽_+|^x,|_u^u_|+|^|εu,s,λ_s|)2ε^)| ∈ C([0, 1]×_R³;R)for anyε∈ (0, 1]. Moreover, by condition (1.4) we get

|f(x,|u|^εu,s,λ)|

(1+|u|+|s|)^2ε(|u|+|s|) ≤ ^M|u|^ε|u|

(|u|+|s|)^2ε(|u|+|s|) ≤ ^M (|u|+|s|)^ε, whence implies that for each bounded intervalΛ⊂_R

f(_x,|u|^εu,s,λ)

(1+|u|+|s|)^2ε =o(|u|+|s|) as|u|+|s| →_∞,

uniformly for(x,λ) ∈ [0, 1]×Λ. Then, by conditions (1.5), it follows from [24, Theorem 2.4]

that for each k ∈ _N, each σ and each ν there exist a neighborhood P_k,σ^ν of the point (λ^σ_k,∞) and a continuum D^ν_k,σ,

ε ⊂ _R^σ×E of the set of solutions of problem (3.1) bifurcating from (λ^σ_k,∞)such that

(i) (D^ν_k,σ,

ε∩ P_k,σ^ν )⊂_R× S_k,^ν

σ; (ii) either D^ν

k,σ,ε \ P_k,σ^ν is bounded in R×E, and in this case D^ν

k,σ,ε \ P_k,σ^ν meets R₀^σ, or D^ν_k,σ,

ε\ P_k,σ^ν is unbounded, and if in this caseD^ν_k,σ,

ε\ P_k,σ^ν has a bounded projection on R₀^σ, then this set meets (λ^σ_k0,∞)with respect toS_k^ν0⁰,σfor some(k⁰,ν⁰)6= (k,ν)_.

Lemma 3.1. For each k ∈ _{N, each}σ, eachνand any sufficiently large R > 0there exists a solution (λ^ν_k,σ,R,u^ν_k,σ,R)of problem(1.1),(1.2)such thatλ^ν_k,σ,R ∈_R^σ, u^ν_k,σ,R ∈ S_k,^ν_σ, andku^ν_k,σ,Rk₁ =R.

Proof. Let Rbe a sufficiently large positive number. Property (i) of the setD^ν

k,σ,ε implies that for any ε∈(0, 1)there exists a solution(λ^ν_k,σ,R,ε,u^ν_k,σ,R,ε)of problem (3.1) such that

λ^ν_k,σ,R,ε ∈_R^σ, u^ν_k,σ,R,ε ∈ S_k,^ν_σ, ku^ν_k,σ,R,εk₁= R.

Then it follows from (3.1) that(λ^ν_k,σ,R,ε,u^ν_k,σ,R,ε)solves the following problem ((`(u))(x) +ϕ^ν_k,σ,R,ε(x)u(x) =λρ(x)u(x) +g(x,u(x),u⁰(x),λ), x ∈(0, 1),

u ∈(b.c.), (3.2)

where

ϕ^ν_k,σ,R,ε(x) =







− ^{f(x,|uνk,σ,R,ε(x)|εuνk,σ,R,ε(x),(uνk,σ,R,ε)0(x),λνk,σ,R,ε)}

u^ν_k,σ,R,ε(x) (1+|u^ν_k,σ,R,ε(x)|+|(u^ν_k,σ,R,ε)⁰(x)|)^2ε ifu^ν_k,σ,R,ε(x)6=0,

0 ifu^ν_k,σ,R,ε(x) =0.

(3.3)

In view of conditions (1.3) and (1.4), by (3.3) we have ϕ^ν_k,σ,R,ε(x)≥0 and

|ϕ^ν_k,σ,R,ε(x)| ≤ ^M|u^ν_k,σ,R,ε(x)|^ε

(₁+|u^ν_k,σ,R,ε(x)|+|(u^ν_k,σ,R,ε)⁰(x)|)^2ε

≤ ^M

(1+|u^ν_k,σ,R,ε(x)|+|(u^ν_k,σ,R,ε)⁰(x)|)^ε ≤ M forx∈[0, 1].

(3.4)

Since C[_{0, 1}] is dense in L₁[_{0, 1}] and the function u^ν_k,σ,R,ε has a finite number of zeros in (0, 1), by relation (3.4), it follows from [15, Ch. 10, § 10·61] that the eigenvalues of problem

((`(u))(x) +ϕ^ν_k,σ,R,ε(x)u(x) =λρ(x)u(x), x∈(0, 1),

u∈(b.c.), (3.5)

are real, simple and form a positive infinitely increasing and negative infinitely decreasing sequences{λ^ν,_k,σ,⁺_R,ε}^∞_k=1and{λ^ν,_k,σ,⁻_R,ε}^∞_k=1respectively. In this case, for eachk∈_Nthe function u^ν,_k,⁺_σ,R,ε (u^ν,_k,⁻_σ,R,ε respectively) corresponding to the eigenvalue λ^ν,_k,σ,⁺_R,ε (λ^ν,_k,σ,⁻_R,ε respectively) hask−1 simple nodal zeros in the interval(0, 1). Moreover, by [21, Lemma 2.2], the following relations hold:

λ⁺_k ≤λ^ν,_k,⁺_+,R,ε ≤λ⁺_k,M and λ⁻_k,M ≤λ^ν,_k,−⁻_,R,ε ≤λ⁻_k , k∈_N. (3.6) Hence it follows from (3.6) that

λ^ν,_k,^σ_σ,R,ε ∈ I_k^σ, k∈_N. (3.7) Let

I_k⁺(δ) = [λ⁺_k −δ,λ⁺_k,M+δ], I_k⁻(δ) = [λ⁻_k,M−δ,λ⁻_k +δ], whereδ is a positive number.

By [17, Ch. 4, § 3, Theorem 3.1] for eachk ∈_{N, each}σand eachνthe point(λ^ν,_k,σ,^σ_R,ε,∞)is an unique asymptotic bifurcation point of the nonlinear eigenvalue problem (3.2) with respect to the setR^σ× S_k,^ν_σ. Then for any sufficiently largeR>0 there exists a sufficiently small τ_R,ε such that

λ^ν_k,σ,R,ε ∈[λ^ν,_k,σ,^σ_R,ε−τ_R,ε,λ^ν,_k,^σ_σ,R,ε+τ_R,ε]⊆ I_k^σ(τ_R,ε)⊂_R^σ. Letτ0=sup_R,ετR,ε. Hence it follows from last relation that

λ^ν_k,σ,R,ε ∈ I_k^σ(τ₀)⊂_R^σ. (3.8) Since ku^ν_k,σ,R,εk₁ = R and f, g ∈ C([0, 1]×_R³;R), by relation (3.8), it follows from (3.1) that

ku^ν_k,σ,R,εk₂ ≤ const,

wherekuk₂=kuk_∞+ku⁰k_∞+ku⁰⁰k_∞. Then by the Arzelà–Ascoli theorem the set{u^ν_k,σ,R,ε}_ε∈(0,1] is precompact in E. Hence we can choose the sequence {ε_n}^∞_n=1 ⊂ (0, 1) converging to 0 as n→_∞such that

(λ^ν_k,σ,R,εn,u^ν_k,σ,R,εn)→(λ^ν_k,σ,R,u^ν_k,σ,R) asn→_∞inR×E, and by (3.1) the sequence {(λ^ν_k,σ,R,ε

n,u^ν_k,σ,R,ε

n)}^∞_n=1 is convergent in R×C²[0, 1]. Putting (λ^ν_k,σ,R,ε

n,u^ν_k,σ,R,ε

n) _{instead of} (_λ,u) in (3.1) and passing to the limit (as n → ∞) in this re- lation we obtain that (λ^ν_k,σ,R,u^ν_k,σ,R) is a solution of problem (1.1), (1.2). It is obvious that (λ^ν_k,σ,R,u^ν_k,σ,R)has the following properties

λ^ν_k,σ,R ∈ I_k^σ(τ₀), ku^ν_k,σ,Rk₁ =R, and u^ν_k,σ,R ∈ S_k,^ν

σ =S_k,^ν_σ∪∂S_k,^ν_σ. Ifu^ν_k,σ,R ∈∂S_k,^ν

σ, then either (i) R1

0 ρ(x) (u^ν_k,σ,R(x))²dx=0, or

(ii) there existsx₀ ∈[0, 1]that suchu^ν_k,σ,R(x₀) = (u^ν_k,σ,R)⁰(x₀) =0.

Sinceku^ν_k,σ,Rk₁ =Rit follows from Remark2.2that R1

0 ρ(x) (u^ν_k,σ,R(x))²dx6=0.

Now let there exists x₀ ∈ [_{0, 1}]_{such that} u^ν_k,σ,R(x₀) = (u^ν_k,σ,R)⁰(x₀) =0. By (1.1), (1.2) we have

`(u^ν_k,σ,R) =λ^ν_k,σ,Rρ(x)u^ν_k,σ,R+ f(x,u^ν_k,σ,R(u^ν_k,σ,R)⁰,λ^ν_k,σ,R)

+g(x,u^ν_k,σ,R,(u^ν_k,σ,R)⁰_,λ^ν_k,σ,R)_, x∈ (_{0, 1})_, u^ν_k,σ,R∈(b.c.)_{. (3.9)} Dividing both sides of (3.9) byku^ν_k,σ,Rk₁ and settingv^ν_k,σ,R = _k^uνk,σ,R

u^ν_k,σ,Rk₁ we get

`(v^ν_k,σ,R) =λ^ν_k,σ,Rρ(x)v^ν_k,σ,R+ ^f(x,u^ν_k,σ,R (u^ν_k,σ,R)⁰_, λ^ν_k,σ,R) ku^ν_k,σ,Rk₁

+ ^g(x,u^ν_k,σ,R, (u^ν_k,σ,R)⁰, λ^ν_k,σ,R)

ku^ν_k,σ,Rk₁ ^{, x}∈(_{0, 1})_{. (3.10)} In view of (1.4) we have

f(_x,u^ν_k,σ,R (_u^ν_k,σ,R)⁰_, λ^ν_k,σ,R) ku^ν_k,σ,Rk₁

≤ M|v^ν_k,σ,R|. (3.11)

By virtue of condition (1.5) for any sufficiently small fixed e>0 there exists a sufficiently largeδ_e>0 such that

|g(x,u,s,λ)|<ε(|u|+|s|)2 for any(x,u,s,λ)∈[0, 1]×_R²×_Λ, |u|+|s|>δ_e, (3.12) where Λ∈_R^σis a bounded interval. In other hand, sinceg ∈C([0, 1]×_R³;R)it follows that there exists a positive numberKesuch that

|g(x,u,s,λ)| ≤Ke for any(x,u,s,λ)∈[0, 1]×_R²×_Λ, |u|+|s| ≤δ_e. (3.13) We chooseRlarge enough to satisfy the relations

R>δe and Ke <Re/2.

Then by (3.12) and (3.13) we have

g(_x,u^ν_k,σ,R(_x) (_u^ν_k,σ,R)⁰(_x)_, λ^ν_k,σ,R) ku^ν_k,σ,Rk₁

= ¹ R

g(x,u^ν_k,σ,R(x) (u^ν_k,σ,R)⁰(x), λ^ν_k,σ,R)

≤ ¹ R







n max

x∈[0,1]:|u^ν_k,σ,R(x)|⁺(u^ν_k,σ,R)⁰⁽x) ≤δe

o

g

x,u^ν_k,σ,R(x), u^ν_k,σ,R0

(x),λ^ν_k,σ,R

+_n max

x∈[0,1]:|u^ν_k,σ,R(x)|⁺(u^ν_k,σ,R)⁰⁽x) >δ_e

o

g

x,u^ν_k,σ,R(x), u^ν_k,σ,R0

(x),λ^ν_k,σ,R







≤ ¹

R{K_e+eR/2}= ^Ke

R +_e/2<_e/2+_e/2= e.

(3.14)

Taking into account (3.11) and (3.14), from (3.10) we obtain p₀|(v^ν_k,σ,R)⁰⁰(x)| ≤ |p(x)(v^ν_k,σ,R)⁰⁰(x)|

≤ |λ^ν_k,σ,R||ρ(x)|+|q(x)|+M

|v^ν_k,σ,R(x)|+|p⁰(x)||(v^ν_k,σ,R)⁰(x)|+e, which implies that

|(v^ν_k,σ,R)⁰⁰(x)| ≤c₀ |v^ν_k,σ,R(x)|+|(v^ν_k,σ,R)⁰(x)|+ ^e

p₀, (3.15)

where c₀ = ¹

p₀ max

x∈[0,1]

n maxn

λ⁺_k,M,|λ⁻_k,M|^o|ρ(x)|+|q(x)|+M,|p⁰(x)|^o, p₀ = min

x∈[0,1]p(x). Letw^ν_k,σ,R = ^v

νk,σ,R

(v^ν_k,σ,R)⁰

∈_R²with the norm that given by

|w^ν_k,σ,R|₂=|v^ν_k,σ,R|+|(v^ν_k,σ,R)⁰|. Then it follows from (3.15) that

|(w^ν_k,σ,R)⁰(x)|₂≤c₁|w_k,σ,R^ν (x)|₂+ ^e p₀,

wherec₁=c₀+1. Integrating the last relation in the range fromx₀to xwe have

Z _x

x0

|(w^ν_k,σ,R)⁰(t)|₂dt

≤c₁

Z _x

x0

|w^ν_k,σ,R(t)|₂dt

+ ^e

p₀. (3.16)

Using the relationv^ν_k,σ,R(x₀) = (v^ν_k,σ,R)⁰(x₀) =0 and inequality (3.16) we get

|w_k,σ,R^ν (x)|₂=

Z _x

x0

|(w_k,σ,R^ν )⁰(t)|₂dt

≤c₁

Z _x

x0

|w^ν_k,σ,R(t)|₂dt

+ ^e p₀, whence, with regard the Gronwall’s inequality, we get

|w^ν_k,σ,R(x)|₂ ≤ ^e p0

e^c1|x−x0| ≤ ^e p0

e^c1 <1, x∈ [0, 1], (3.17) (in advance we could choose e so small enough that the inequality e < _e^pc⁰1 holds). Then it follows from (3.17) thatkv^ν_k,σ,Rk₁<1 which contradicts the conditionkv^ν_k,σ,Rk₁ =1. Therefore, we haveu^ν_k,σ,R ∈ S_k,σ^ν .

Corollary 3.2. For each k ∈ _N, each σ and each ν there exists a sufficiently large positive number R^ν_k,σsuch that for any R ≥ R^ν_k,σproblem(1.1),(1.2)has a solution(λ,u)which satisfies the following properties:

λ∈ I_k^σ(τ₀)_, u∈ S_k,σ^ν _and kuk₁= R.

Recall that(λ,∞), λ ∈ _R^σ, is an asymptotic bifurcation point of problem (1.1), (1.2) with respect to the set R^σ× S_k,σ^ν , k ∈ N, if for any sufficiently smallr > 0 there exists a solution (λ^ν_k,σ,r,u^ν_k,σ,r)∈_R^σ×Esuch that

|λ^ν_k,σ,r−λ|<r, ku^ν_k,σ,rk₁ >r⁻¹ and u^ν_k,σ,r∈ S_k,σ^ν .

Remark 3.3. We add the points{(λ,∞):λ∈_R}to the spaceR×Eand define an appropriate topology on the resulting set.

For each k ∈ _{N, each} σ and each ν by B^ν_k,σ, k ∈ N, we denote the set of asymptotic bifurcation of (1.1), (1.2) with respect toR^σ× S_k,σ^ν _.

The following result immediately follows from Lemma3.1and Corollary3.2.

Corollary 3.4. For each k ∈ _{N, each} σ and each ν the setB^ν

k,σ is nonempty. Furthermore, B^ν

k,σ ⊂ I_k^σ× {_∞}.

4 Structures of global continua emanating from the asymptotic bi- furcation points of problem (1.1), (1.2)

Let conditions (1.3)–(1.5) hold.

For eachk∈ _{N, each}σand eachνwe define the setD^ν,_k,σ^∗as the union of all the components ofDbifurcating from I_k^σ× {_∞}with respect to the setR× S_k,σ^ν . It follows from Corollary3.4 that the set D^ν,_k,σ^∗ is nonempty. This set may not be connected in R×E, but the set D^ν_k,σ = D^ν,_k,σ^∗∪(I_k^σ× {_∞})will be connected in this space (see Remark3.3).

The main result of this paper is the following theorem.

Theorem 4.1. For each k∈_{N, each}σand eachνthe setD^ν_k,σis contained inR^σ×E and for this set at least one of the following statements holds: