Boundary Value Problems for a Super-Sublinear Asymmetric Oscillator: The Exact Number of Solutions

Abstract

Properties of asymmetric oscillator described by the equation 𝑥^′′=−𝜆(𝑥⁺)^𝑝+𝜇(𝑥⁻)^𝑞 (i), where 𝑝≥1 and 0<𝑞≤1, are studied. A set of (𝜆,𝜇) such that the problem (i), 𝑥(0)=0=𝑥(1) (ii), and |𝑥^′(0)|=𝛼 (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.

1. Introduction

Asymmetric oscillators were studied intensively starting from the works by Kufner and Fučík; see [1] and references therein. Simple equations like (2) given with the boundary conditions allow for complete investigation of spectra. It is known that the spectrum of the problem (2), (4) is a set of hyperbola looking curves in the (𝜆,𝜇)-plane. On the other hand, there is a plenty of works devoted to one-parameter case of equations 𝑥^′′+𝜆𝑓(𝑥)=0 given together with the two-point boundary conditions. Due to nonlinearity of 𝑓 one should consider solutions 𝑥(𝑡;𝛼) with different 𝛼=𝑥^′(0). Bifurcation diagrams in terms of 𝛼 and 𝜆, or ‖𝑥‖ and 𝜆, can serve then to evaluate the number of solutions [2–4].

In this paper we consider differential equations of the form 𝑥^′′=−𝜆𝑓𝑥⁺+𝜇𝑔(𝑥⁻), (1) where 𝑓=𝑥^𝑝,𝑝≥1, and 𝑔=𝑥^𝑞,0<𝑞≤1. Here 𝜆 and 𝜇 are nonnegative parameters, 𝑥⁺=max{𝑥,0}, 𝑥⁻=max{−𝑥,0}. This equation describes asymmetric oscillator with different nonlinear restoring forces on both sides of 𝑥=0. If 𝑓=𝑔=𝑥, then equation becomes famous Fučík equation 𝑥^′′=−𝜆𝑥⁺+𝜇𝑥⁻. (2) Properties of the Fučík spectrum are well known (the Fučík spectrum is a set of all pairs (𝜆,𝜇) where 𝜆,𝜇≥0, such that the Dirichlet problem—(2) with boundary conditions 𝑥(0)=0=𝑥(1)—has a non-trivial solution).

The aim of our study in this paper is to describe properties of the spectrum of the problem 𝑥^′′=−𝜆𝑥^+𝑝+𝜇(𝑥⁻)^𝑞𝑥(0)(1),0<𝑞≤1,1≤𝑝,=0,𝑥=0. (3)(4)

For this we study first the time maps for the related functions (Section 2), then we give the analytical description of the spectrum (Section 3), and formulate the properties of the spectrum (Section 4), including the asymptotics. In Section 5 we consider the solution sets and solution surfaces which bear information on multiplicity of solutions to the problem. Analysis of properties of solution surfaces (Section 6) can give us estimations of the number of solutions to the problem (Section 7). These estimations contain also information of properties of solutions such as the number of zeros and evaluations of 𝑥^′(0).

This paper continues series of publications by the authors devoted to nonlinear asymmetric oscillations [5–8].

2. Time Maps

Consider the Cauchy problem 𝑥^′′=−𝜆𝑥^𝑟,𝑥(0)=0,𝑥^′(0)=𝛼, (5) where 𝑟>0. Solutions of this problem for 𝛼 and 𝜆 positive have a zero. This zero will be denoted 𝑇_𝑟(𝛼,𝜆) and called time map function; more on time maps can be found in [9, 10].

Proposition 1. Suppose 𝑟,𝛼,𝜆>0, 𝑝>1 and 0<𝑞<1.  (1) For any 𝛼,𝜆>0 the formula is valid: 𝑇_𝑟()=1𝛼,𝜆√𝜆𝑡_𝑟𝛼√𝜆, (6) where 𝑡_𝑟(𝛾)=𝑇_𝑟(𝛾,1). (2) The function 𝑇_𝑟(𝛼,𝜆) for the problem (5) is 𝑇_𝑟(𝛼,𝜆)=2^{𝑟/(𝑟+1)}(𝑟+1)^1/(𝑟+1)𝐴(𝑟)𝛼^{−(𝑟−1)/(𝑟+1)}𝜆^{−1/(𝑟+1)}, (7) where 𝐴(𝑟)=∫¹₀(𝑑𝑠/√1−𝑠^𝑟+1). (3) The function 𝑇_𝑟(𝛼,𝜆) for fixed 𝑟 and 𝛼 is strictly decreasing function of 𝜆 and possesses the properties lim_𝜆→0+𝑇_𝑟(𝛼,𝜆)=+∞,lim_𝜆→+∞𝑇_𝑟(𝛼,𝜆)=0. (8) (4) The function 𝑇_𝑝(𝛼,𝜆) for fixed 𝑝 and 𝜆 is decreasing function of 𝛼. (5) The function 𝑇_𝑞(𝛼,𝜆) for fixed 𝑞 and 𝜆 is increasing function of 𝛼.

Proof. By standard computations.

Remark 2. 𝑇_𝑟(𝛼,𝜆)=𝜋/√𝜆 for 𝑟=1, irrespective of 𝛼.

3. Spectrum

Suppose that the problems (3) and (4) are considered with the additional condition 𝑥^′(0)=𝛼>0. (9)

Definition 3. For given 𝛼>0 a set of all nonnegative (𝜆,𝜇) for which the problems (3), (4), and (9) have a nontrivial solution is called 𝛼-spectrum.

Remark 4. A solution of (3) is a 𝐶²-function. Therefore 𝑥^′(𝑡) is continuous. If 𝑧₁ and 𝑧₂ are two consecutive zeros of 𝑥(𝑡), then |𝑥^′(𝑧₁)|=|𝑥^′(𝑧₂)| since a solution in the interval (𝑧₁,𝑧₂) is symmetric with respect to the middle point. Therefore |𝑥^′(𝑧)|=𝛼 for any zero point 𝑧 and signs of 𝑥^′(𝑡) alternate.

Theorem 5. The 𝛼-spectrum for the problem 𝑥^′′=−𝜆𝑥^+𝑝+𝜇(𝑥⁻)^𝑞(0)𝑥(1)𝑥,𝑥=0,=0,^′(0)=𝛼>0 (10) consists of the following 𝛼-branches: 𝐹⁺₀(𝛼)=𝜆,𝜇∶𝑇_𝑝(),𝐹𝛼,𝜆=1,𝜇≥0₋₀(𝛼)=𝜆,𝜇∶𝜆≥0,𝑇_𝑞,𝐹𝛼,𝜇=1_+2𝑖−1(𝛼)=𝜆;𝜇∶𝑖𝑇_𝑝(𝛼,𝜆)+𝑖𝑇_𝑞,𝐹𝛼,𝜇=1_−2𝑖−1(𝛼)=𝜆;𝜇∶𝑖𝑇_𝑞(𝛼,𝜆)+𝑖𝑇_𝑝,𝐹𝛼,𝜇=1_+2𝑖(𝛼)=𝜆;𝜇∶(𝑖+1)𝑇_𝑝(𝛼,𝜆)+𝑖𝑇_𝑞,𝐹𝛼,𝜇=1_−2𝑖(𝛼)=𝜆;𝜇∶(𝑖+1)𝑇_𝑞𝛼,𝜇+𝑖𝑇_𝑝(𝛼,𝜆)=1. (11) The notation 𝐹⁺_𝑖(𝛼), respectively: 𝐹⁻_𝑖(𝛼), refers to solutions 𝑥 which satisfy the initial conditions 𝑥(0)=0,𝑥^′(0)=𝛼>0 (resp.: 𝑥^′(0)=−𝛼<0) and have exactly 𝑖 zeros in the interval (0,1).

Proof. We prove the theorem only for solutions which have exactly one zero in (0,1) and satisfy the initial condition |𝑥^′(0)|=𝛼>0.
Consider the case 𝑥^′(0)=𝛼>0. The first zero of 𝑥(𝑡) appears at 𝑡=𝑇_𝑝 and 𝑥^′(𝑇_𝑝)=−𝛼. The second zero is at 𝑡=𝑇_𝑝+𝑇_𝑞. One has that for solutions with exactly one zero in (0,1) the relation 𝑇_𝑝+𝑇_𝑞=1 holds, which defines the branch 𝐹⁺₁(𝛼).
Suppose 𝑥^′(0)=−𝛼. The first zero now is at 𝑡=𝑇_𝑞. The second one is at 𝑡=𝑇_𝑞+𝑇_𝑝 and therefore 𝑇_𝑞+𝑇_𝑝=1.
The branches 𝐹⁺₁(𝛼) and 𝐹⁻₁(𝛼) are given by the equivalent relations 𝑇_𝑝+𝑇_𝑞=1 and 𝑇_𝑞+𝑇_𝑝=1, respectively, and therefore coincide.
Proof for solutions with different nodal structure is similar.

Remark 6. If 𝑝=𝑞=1, then for any 𝛼>0 the 𝛼-spectrum is 𝐹⁺₀(𝛼)=∶𝜋𝜆,𝜇√𝜆,𝐹=1,𝜇≥0₋₀(𝛼)=𝜋𝜆,𝜇∶𝜆≥0,√𝜇,𝐹=1_+2𝑖−1(𝛼)=𝜋𝜆;𝜇∶𝑖√𝜆𝜋+𝑖√𝜇,𝐹=1_−2𝑖−1(𝛼)=𝜋𝜆;𝜇∶𝑖√𝜇𝜋+𝑖√𝜆,𝐹=1_+2𝑖(𝛼)=∶()𝜋𝜆;𝜇𝑖+1√𝜆𝜋+𝑖√𝜇,𝐹=1_−2𝑖(𝛼)=∶()𝜋𝜆;𝜇𝑖+1√𝜇𝜋+𝑖√𝜆=1. (12) It is classical Fučík spectrum for the problems (2) and (4), see [1].

4. Properties of the α-Spectrum

Proposition 7. (1)The branches 𝐹⁺_2𝑖−1(𝛼) and 𝐹⁻_2𝑖−1(𝛼) coincide.(2)The branches 𝐹^±_2𝑖−1(𝛼) and 𝐹^±_2𝑖−1(𝛼) do not intersect unless 𝑖≠𝑗.(3)The branches 𝐹⁺_2𝑖(𝛼) and 𝐹⁺_2𝑗(𝛼) do not intersect unless 𝑖≠𝑗.(4)The branches 𝐹⁻_2𝑖(𝛼) and 𝐹⁻_2𝑗(𝛼) do not intersect unless 𝑖≠𝑗.(5)Any branch 𝐹^𝑠_𝑖(𝛼) where 𝑖≥1 and 𝑠 is either “+” or “−” is a graph of monotonically decreasing function 𝜇=𝜇(𝜆).(6)The branches 𝐹⁺_2𝑖(𝛼) and 𝐹⁻_2𝑖(𝛼) intersect once.

Proof. (1) It follows from (11) that 𝐹⁺_2𝑖−1(𝛼) coincides with 𝐹⁻_2𝑖−1(𝛼), since both sets (branches) are defined by symmetric relations: 𝑖𝑇_𝑝(𝛼,𝜆)+𝑖𝑇_𝑞𝛼,𝜇=1,𝑖𝑇_𝑞𝛼,𝜇+𝑖𝑇_𝑝(𝛼,𝜆)=1. (13)(2), (3) and (4) follows from (11), but (5) follows from the relations (11) and (7). (6) Indeed, any point of intersection satisfies the system (𝑖+1)𝑇_𝑝(𝛼,𝜆)+𝑖𝑇_𝑞()𝑇𝛼,𝜇=1,𝑖+1_𝑞𝛼,𝜇+𝑖𝑇_𝑝(𝛼,𝜆)=1. (14) It follows that 𝑇_𝑝(𝛼,𝜆)=𝑇_𝑞𝛼,𝜇 (15) or 2^{𝑝/(𝑝+1)}𝑝+1^1/(𝑝+1)𝐴𝑝𝛼^{−(𝑝−1)/(𝑝+1)}𝜆^{−1/(𝑝+1)}=2^{𝑞/(𝑞+1)}𝑞+1^1/(𝑞+1)𝐴𝑞𝛼^{−(𝑞−1)/(𝑞+1)}𝜇^{−1/(𝑞+1)}. (16) The above relation defines a curve which is a graph of monotonically increasing function 𝜇=𝐶(𝑝,𝑞,𝛼)𝜆^{(𝑞+1)/(𝑝+1)}, where 𝐶(𝑝,𝑞,𝛼) is a constant computable from (16). This curve emanates from the origin and intersects the graph of monotonically decreasing from +∞ to 0 function 𝜇=𝜇(𝜆) from (5) only once.

Let 𝜆_𝑖(𝛼,𝑝) be a unique solution of the equation 𝑇_𝑝(𝛼,𝜆)=1/𝑖; similarly, let 𝜇_𝑖(𝛼,𝑞) be a unique solution of the equation 𝑇_𝑞(𝛼,𝜇)=1/𝑖.

Proposition 8. Suppose 𝛼,𝜆,𝜇>0.  (1) The branch 𝐹^±_2𝑖−1(𝛼)(𝑖∈ℕ) is located in the sector 𝜆,𝜇∶𝜆>𝜆_𝑖𝛼,𝑝,𝜇>𝜇_𝑖𝛼,𝑞 (17) and is a hyperbola looking curve in (𝜆,𝜇) plane with vertical asymptote 𝜆=𝜆_𝑖(𝛼,𝑝) and horizontal asymptote 𝜇=𝜇_𝑖(𝛼,𝑞). (2) The branch 𝐹⁺_2𝑖(𝛼)(𝑖∈ℕ) is located in the sector 𝜆,𝜇∶𝜆>𝜆_𝑖−1𝛼,𝑝,𝜇>𝜇_𝑖𝛼,𝑞 (18) and is a hyperbola looking curve in (𝜆,𝜇) plane with vertical asymptote 𝜆=𝜆_𝑖−1(𝛼,𝑝) and horizontal asymptote 𝜇=𝜇_𝑖(𝛼,𝑞). (3) The branch 𝐹⁻_2𝑖(𝛼)(𝑖∈ℕ) is located in the sector 𝜆,𝜇∶𝜆>𝜆_𝑖𝛼,𝑝,𝜇>𝜇_𝑖−1𝛼,𝑞 (19) and is a hyperbola looking curve in (𝜆,𝜇) plane with vertical asymptote 𝜆=𝜆_𝑖(𝛼,𝑝) and horizontal asymptote 𝜇=𝜇_𝑖−1(𝛼,𝑞).

Proof. Follows from the relations (11) and Proposition 7.

Remark 9. So the positive part of the 𝛼-spectrum 𝐹⁺(𝛼)=^+∞_𝑖=0𝐹⁺_𝑖(𝛼) (20) in the extended (𝜆,𝜇)-plane may be schematically described by the chain 𝜆₁,0−−−−→^𝐹+0(𝛼)𝜆₁,+∞−−−−→^𝐹+1(𝛼)+∞,𝜇−−−−→₁^𝐹+2(𝛼)𝜆₂,+∞_𝐹+3(𝛼)−−−−→+∞,𝜇−−−−→₃^𝐹+4(𝛼)𝜆₃,+∞⋯. (21) Similarly, the negative part of the 𝛼-spectrum 𝐹⁻(𝛼)=^+∞_𝑖=0𝐹⁻_𝑖(𝛼) (22) in the extended (𝜆,𝜇)-plane may be described as follows: 0,𝜇−−−−→₁^{𝐹−0(𝛼)}+∞,𝜇−−−−→₁^{𝐹−1(𝛼)}𝜆₁,+∞−−−−→^{𝐹−2(𝛼)}+∞,𝜇_{2𝐹−3(𝛼)}−−−−→𝜆₂,+∞−−−−→^{𝐹−4(𝛼)}+∞,𝜇₃⋯. (23)

Proposition 10. Let 𝑝>1 and 0<𝑞<1 be fixed. It is true that lim_𝛼→+0𝜆₁𝛼,𝑝=+∞,lim_𝛼→+∞𝜆₁𝛼,𝑝=+0,lim_𝛼→+0𝜇₁𝛼,𝑝=+0,lim_𝛼→+∞𝜇₁𝛼,𝑝=+∞. (24)

Proof. Follows from Propositions 1 and 10.

5. Solution Sets and Solution Surfaces

Definition 11. A solution set of the problems (3) and (4) is a set 𝐹 of all triples (𝜆,𝜇,𝛼)  (𝜆≥0, 𝜇≥0,𝛼>0) such that there exists a nontrivial solution of the problem.

Let us distinguish between solutions of the problems (3) and (4) with different number of zeros in the interval (0,1). Let 𝐹⁺_𝑖 be a set of all triples (𝜆,𝜇,𝛼) such that there exists a nontrivial solution of the respective problems (3) and (4), 𝑥^′(0)=𝛼>0 which has exactly 𝑖 zeros in (0,1),𝑖=0,1,…, but 𝐹⁻_𝑖 be a set of all triples (𝜆,𝜇,𝛼) such that there exists a nontrivial solution of the respective problems (3) and (4), 𝑥^′(0)=−𝛼<0 which has exactly 𝑖 zeros in (0,1),𝑖=0,1,…

Definition 12. 𝐹⁺_𝑖 will be called a positive 𝑖-solution surface, but 𝐹⁻_𝑖-a negative 𝑖-solution surface.

5.1. Description and Properties of a Solution Set

We will identify the cross section of a solution surface 𝐹^𝑠_𝑖 with the plane 𝛼=𝛼₀>0 (𝛼₀ is fixed) with its projection to the (𝜆,𝜇)-plane. This projection is in fact the respective branch 𝐹^𝑠_𝑖(𝛼₀) of the spectrum of the problem (3), (4), and (9).

Theorem 13.  (1) Solution surfaces are unions of respective 𝛼-branches: 𝐹⁺_𝑖=_𝛼>0𝐹⁺_𝑖(𝛼),𝐹⁻_𝑖=_𝛼>0𝐹⁻_𝑖(𝛼). (25) (2) A solution set 𝐹 of the problems (3) and (4) is a union of all 𝑖-solution surfaces and is a union of all 𝛼-branches: 𝐹=^∞_𝑖=1𝐹^±_𝑖=_𝛼>0𝐹^±_𝑖(𝛼). (26) (3) Solution surfaces 𝐹⁺_2𝑖−1 and 𝐹⁻_2𝑖−1 coincide. (4) Solution surfaces 𝐹^±_2𝑖−1 and 𝐹^±_2𝑗−1 do not intersect unless 𝑖=𝑗. (5) For given 𝑖≠𝑗 the solution surfaces 𝐹^±_2𝑖−1 and 𝐹^±_2𝑗−1 are centroaffine equivalent under the mapping Φ_𝑖,𝑗∶ℝ³→ℝ³: 𝜆,𝜇,𝛼↦^Φ𝑖,𝑗𝜆,𝜇,𝛼, (27) where 𝜆=(𝑗/𝑖)²𝜆, 𝜇=(𝑗/𝑖)²𝜇, 𝛼=(𝑗/𝑖)𝛼.

Proof. (1), (2), (3), and (4) follow from definitions of 𝐹, 𝐹⁺_𝑖, 𝐹⁻_𝑖, 𝐹⁺_𝑖(𝛼), 𝐹_𝑖(𝛼)⁻ and Proposition 7.
(5) First observe that for 𝛼,𝛽,𝜆>0 we have by making use of the formula (6) that 𝑇=1𝛽,𝜆√𝜆𝑇𝛽√𝜆=1,1√𝜆𝑇𝛼√𝜆𝛽𝛼=1,1√𝜆𝑇𝛼𝜆𝛼²/𝛽²=1,1𝜆𝛼²/𝛽²𝛼𝛽𝑇𝛼𝜆𝛼²/𝛽²=𝛼,1𝛽𝑇𝛼𝛼,𝜆²𝛽². (28) The above formula is applicable to 𝑇_𝑝 and to 𝑇_𝑞.
For given 𝑖≠𝑗 and 𝛼>0 set 𝛼=(𝑗/𝑖)𝛼. Suppose (𝜆,𝜇,𝛼)∈𝐹^±_2𝑖−1: 𝑖𝑇_𝑝(𝛼,𝜆)+𝑖𝑇_𝑞𝛼,𝜇=1. (29) Applying the rescaling formula (28), where 𝛼 replaces 𝛽, to the previous equation one has 𝑖𝛼𝛼𝑇_𝑝𝛼,𝜆𝛼²𝛼²+𝑖𝛼𝛼𝑇_𝑞𝛼,𝜇𝛼²𝛼²=1,𝑗𝑇_𝑝𝑗𝛼,𝑖²𝜆+𝑗𝑇_𝑞𝑗𝛼,𝑖²𝜇=1,𝑗𝑇_𝑝𝛼,𝜆+𝑗𝑇_𝑞𝛼,𝜇=1, (30) therefore (𝜆,𝜇,𝛼)∈𝐹⁺_2𝑗−1. Since Φ⁻¹_𝑖,𝑗=Φ_𝑗,𝑖, one has Φ_𝑖,𝑗(𝐹^±_2𝑖−1)=𝐹^±_2𝑗−1, and the surfaces 𝐹^±_2𝑖−1 and 𝐹^±_2𝑗−1 are centroaffine equivalent under the mapping Φ_𝑖,𝑗.

Remark 14. Since solution surfaces 𝐹^±_2𝑖−1 and 𝐹^±_2𝑗−1(𝑖≠𝑗) are centro-affine equivalent, they have similar shape. Therefore it is enough to study properties of the solution surface 𝐹^±₁, in order to know properties of other odd-numbered solution surfaces. The same is true for 𝐹⁻_2𝑖−1.

5.2. Cross-Sections of Solution Surfaces with the Planes α = Const

A cross-section of any solution surface 𝐹⁺_𝑖 or 𝐹⁻_𝑖 by the plane 𝛼=const>0 locates in the sector 𝑄₁(𝛼)={(𝜆,𝜇)∶𝜆>𝜆₁(𝛼),𝜇>𝜇₁(𝛼)}.

Irrespective of the choice of 𝛼 no oscillatory (with at least one zero in (0,1)) solution of the problems (3), (4), and (9) exists for (𝜆,𝜇) in the “dead zone” below the envelope (see Figure 1).

Figure 1 

“Dead zone” beneath the envelope (in red) of branches 𝐹±1(𝛼) of the spectrum of the problems (3) and (4) for 𝑝=3, 𝑞=1/3 and 𝛼=7,9,12.

The analytical description of envelopes, corresponding to branches of spectra, follows.

6. Envelopes of Solution Surfaces

Any solution surface for (3) is defined by one of the following relations: 𝐹^±_2𝑖−1=𝜆,𝜇,𝛼∶𝑖𝑇_𝑓(𝛼,𝜆)+𝑖𝑇_𝑔(),𝐹𝛼,𝜇=1𝑖=1,2,…_+2𝑖=𝜆,𝜇,𝛼∶(𝑖+1)𝑇_𝑓(𝛼,𝜆)+𝑖𝑇_𝑔(),𝐹𝛼,𝜇=1𝑖=0,1,…_−2𝑖=𝜆,𝜇,𝛼∶𝑖𝑇_𝑓(𝛼,𝜆)+(𝑖+1)𝑇_𝑔().𝛼,𝜇=1𝑖=0,1,… (31) We use the unifying formula ℱ∶𝑘𝑇_𝑓(𝛼,𝜆)+𝑚𝑇_𝑔𝛼,𝜇=1 (32) and consider 𝐻𝜆,𝜇,𝛼∶=𝑘𝑇_𝑓(𝛼,𝜆)+𝑚𝑇_𝑔𝛼,𝜇−1=0. (33)

Treat 𝛼 as a parameter. The family of envelopes for 𝐻(𝜆,𝜇,𝛼)=0 can be determined from the system 𝐻=0,𝜕𝐻𝜕𝛼=0. (34)

For the case of (3) one gets 𝐻𝜆,𝜇,𝛼=𝑘ℎ𝑝𝛼^{−(𝑝−1)/(𝑝+1)}𝜆^{−1/(𝑝+1)}+𝑚ℎ𝑞𝛼^{−(𝑞−1)/(𝑞+1)}𝜇^{−1/(𝑞+1)}−1, (35) where ℎ𝑝=2^{𝑝/(𝑝+1)}𝑝+1^1/(𝑝+1)𝐴𝑝,𝐴𝑝=¹₀𝑑𝑠√1−𝑠^𝑝+1,𝜕𝐻𝜕𝛼=−𝑘𝑝−1ℎ𝑝𝛼𝑝+1^{−2𝑝/(𝑝+1)}𝜆^{−1/(𝑝+1)}−𝑚𝑞−1ℎ𝑞𝛼𝑞+1^{−2𝑞/(𝑞+1)}𝜇^{−1/(𝑞+1)}. (36)

Consider the system 𝑘ℎ𝑝𝛼^{−(𝑝−1)/(𝑝+1)}𝜆^{−1/(𝑝+1)}+𝑚ℎ𝑞𝛼^{−(𝑞−1)/(𝑞+1)}𝜇^{−1/(𝑞+1)}𝑘=1,𝑝−1ℎ𝑝𝛼𝑝+1^{−2𝑝/(𝑝+1)}𝜆^{−1/(𝑝+1)}+𝑚𝑞−1ℎ𝑞𝛼𝑞+1^{−2𝑞/(𝑞+1)}𝜇^{−1/(𝑞+1)}=0 (37) and exclude the parameter 𝛼. One obtains that 𝜇=𝜓(𝑝,𝑞)𝑚^{(𝑞+1)(𝑝−1)}𝜓(𝑞,𝑝)𝑘^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)}, (38) where 𝜓𝑝,𝑞=𝑞−1𝑝+12𝑞−𝑝ℎ𝑝^{(𝑝+1)(𝑞−1)}. (39) (1) For 𝐹^±_2𝑖−1∶𝑖𝑇_𝑓(𝛼,𝜆)+𝑖𝑇_𝑔𝛼,𝜇=1, (40) 𝑘=𝑖,𝑚=𝑖, and the equation of the envelope is ℰ^±_2𝑖−1∶𝜇=𝜔^±_2𝑖−1(𝜆)𝜓𝑖∶=𝑝,𝑞^{(𝑞+1)(𝑝−1)}𝜓𝑞,𝑝𝑖^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)}. (41) (2) For 𝐹⁺_2𝑖∶(𝑖+1)𝑇_𝑓(𝛼,𝜆)+𝑖𝑇_𝑔𝛼,𝜇=1, (42) 𝑘=𝑖+1, 𝑚=𝑖, the equation of the envelope is ℰ⁺_2𝑖∶𝜇=𝜔⁺_2𝑖(𝜆)𝜓𝑖∶=𝑝,𝑞^{(𝑞+1)(𝑝−1)}𝜓𝑞,𝑝(𝑖+1)^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)}. (43) (3) For 𝐹⁻_2𝑖∶𝑖𝑇_𝑓(𝛼,𝜆)+(𝑖+1)𝑇_𝑔𝛼,𝜇=1, (44) 𝑘=𝑖, 𝑚=𝑖+1, the equation of the envelope is ℰ⁻_2𝑖∶𝜇=𝜔⁻_2𝑖(𝜆)𝜓)∶=𝑝,𝑞(𝑖+1^{(𝑞+1)(𝑝−1)}𝜓𝑞,𝑝𝑖^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)}. (45)

Proposition 15. For given 𝑝>1 and 0<𝑞<1 the location of the envelopes is as follows:  (1)ℰ^±_2𝑖−1≺ℰ⁺_2𝑖,  (2)ℰ^±_2𝑖−1≺ℰ⁻_2𝑖,  (3)   (a) if 𝑝𝑞−1<0, then ℰ⁻_2𝑖≺ℰ⁺_2𝑖,   (b) if 𝑝𝑞−1=0, then ℰ⁻_2𝑖=ℰ⁺_2𝑖,   (c) if 𝑝𝑞−1>0, then ℰ⁺_2𝑖≺ℰ⁻_2𝑖,   (4) ℰ⁺_2𝑖≺ℰ^±_{2(𝑖+1)−1},  (5) ℰ⁻_2𝑖≺ℰ^±_{2(𝑖+1)−1},where ℰ^𝑠_𝑛≺ℰ^𝑡_𝑚 means that 𝜔^𝑠_𝑛(𝜆)<𝜔^𝑡_𝑚(𝜆) for any 𝜆>0.

Proof. (1) First consider ℰ^±_2𝑖−1∶𝜇=𝜔^±_2𝑖−1(𝜆)𝜓𝑖∶=𝑝,𝑞^{(𝑞+1)(𝑝−1)}𝜓𝑞,𝑝𝑖^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)},ℰ⁺_2𝑖∶𝜇=𝜔⁺_2𝑖(𝜆)𝜓𝑖∶=𝑝,𝑞^{(𝑞+1)(𝑝−1)}𝜓𝑞,𝑝(𝑖+1)^{(𝑝+1)(𝑞−1)1/(𝑝−1)}𝜆^{(𝑞−1)/(𝑝−1)}. (46) For any 𝜆>0, 𝜔⁺_2𝑖(𝜆)𝜔^±_2𝑖−1(𝜆)=𝑖^{(𝑞+1)(𝑝−1)}𝑖^{(𝑝+1)(𝑞−1)}(𝑖+1)^{(𝑝+1)(𝑞−1)}𝑖^{(𝑞+1)(𝑝−1)}=𝑖𝑖+1^{(𝑝+1)(𝑞−1)}>1, (47) then ℰ^±_2𝑖−1≺ℰ⁺_2𝑖.

Remark 16. Layout of envelopes depends on sign(𝑝𝑞−1) (see Figure 2).

(a) the case 𝑝=5 and 𝑞=1/3. Since 𝑝𝑞−1>0, the envelopes are ordered as ℰ±1≺ℰ+2≺ℰ−2≺ℰ±3 < ⋯

(b) the case 𝑝=3 and 𝑞=1/3. Since 𝑝𝑞−1=0, the envelopes are ordered as ℰ±1≺ℰ±2≺ℰ±3 < ⋯

(c) the case 𝑝=2 and 𝑞=1/3. Since 𝑝𝑞−1<0, the envelopes are ordered as ℰ±1≺ℰ−2≺ℰ+2≺ℰ±3 < ⋯

Figure 2 

Layout of envelopes depends on sign(𝑝𝑞−1).

7. The Number of Solutions by Geometrical Analysis of Solution Surfaces

We can detect the precise number of solutions to the problem for given positive (𝜆,𝜇). We can evaluate the initial values 𝑥^′(0) for solutions on a basis of geometrical analysis of solution surfaces and the respective envelopes. The nodal structure of solutions can be described also.

Let 𝜇=𝑐²𝜆 be the family of rays, covering the first quadrant of the (𝜆,𝜇)-plane. Consider the cross-section of a solution surface (32) by the plane 𝜇=𝑐²𝜆 in the (𝜆,𝜇,𝛼)-space, for example, the curve ℱ|_{𝜇=𝑐2𝜆}, which is defined by the relations 𝜇=𝑐²𝜆 and 𝑘𝑇_𝑝(𝛼,𝜆)+𝑚𝑇_𝑞𝛼,𝑐²𝜆=1. (48) This 3D curve ℱ|_{𝜇=𝑐2𝜆} can be regularly parameterized as 𝜆=𝜆(𝑟)∶=𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟)𝑐²,𝜇=𝜇(𝑟)∶=𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟)²,𝛼=𝛼(𝑟)∶=𝑟𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟), (49) where 𝑟>0. These formulas define homeomorphism ℝ₊→ℱ|_{𝜇=𝑐2𝜆}, where ℝ₊=(0,+∞).

Let 𝒢_𝑐 be projection of ℱ|_{𝜇=𝑐2𝜆} to the (𝜆,𝛼) plane, that is: 𝒢_𝑐=(𝜆,𝛼)∈ℝ²₊∶𝑘𝑇_𝑓(𝛼,𝜆)+𝑚𝑇_𝑔𝛼,𝑐²𝜆=1. (50) The curve 𝒢_𝑐 can be regularly [11] parameterized as 𝜆=𝜆(𝑟)∶=𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟)𝑐²,𝛼=𝛼(𝑟)∶=𝑟𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟), (51) where 𝑟>0. These formulas define homeomorphism ℝ₊→𝒢_𝑐.

Proposition 17. There exists a unique parameter 𝑟₀>0 such that the line 𝜆=𝜆_∗>0(a)does not intersect the curve 𝒢_𝑐 if 0<𝜆_∗<𝜆(𝑟₀),(b)intersects the curve 𝒢_𝑐 only once if 𝜆_∗=𝜆(𝑟₀),(c)intersects the curve 𝒢_𝑐 exactly at two points if 𝜆_∗>𝜆(𝑟₀).

Proof. (a) Consider equation 𝜆^′(𝑟)=𝑘𝑐²𝑡^′_𝑝(𝑐𝑟)+𝑚𝑡^′_𝑞(𝑟)=0, which turns to 𝑘𝑐²𝑝−1ℎ𝑝𝑐𝑝+1^{−2𝑝/(𝑝+1)}𝑟^{−2𝑝/(𝑝+1)}+𝑚𝑞−1ℎ𝑞𝑟𝑞+1^{−2𝑞/(𝑞+1)}=0. (52)
It can be found from the above that 𝑟₀=−𝑚𝑞−1𝑝+1ℎ𝑞𝑘𝑝−1𝑞+1ℎ𝑝𝑐^{2/(𝑝+1)−(𝑝+1)(𝑞+1)/2(𝑝−𝑞)} (53) is the only root of the equation 𝜆^′(𝑟)=0. Hence at the point (𝜆₀,𝛼₀)=(𝜆(𝑟₀),𝛼(𝑟₀)) the curve 𝒢_𝑐 has a unique tangent line parallel to the 𝑂𝛼 axis.
(b) Since the given parametrization of the curve 𝒢_𝑐 is regular, then 𝛼^′(𝑟₀)≠0 and in some neighborhood of the point (𝜆₀,𝛼₀) the curve 𝒢_𝑐 can be represented as the graph of the function 𝜆=𝜑(𝛼). Now find 𝜑^′′𝑟₀=𝜆^′′𝑟₀𝛼^′𝑟₀²=2𝑐𝑘𝑡_𝑓(𝑐𝑟)+𝑚𝑡_𝑔(𝑟)𝑘𝑐³𝑡_𝑓^′′𝑐𝑟₀+𝑚𝑡_𝑔^′′𝑟₀𝑐²𝛼^′(𝑟₀)². (54) The next step is to detect the sign of the expression 𝑘𝑐³𝑡_𝑝^′′(𝑐𝑟)+𝑚𝑡_𝑞^′′(𝑟)=𝑘𝑐³2𝑝𝑝−1𝑝+1²ℎ𝑝𝑐^{−(3𝑝+1)/(𝑝+1)}𝑟^{−(3𝑝+1)/(𝑝+1)}+𝑚2𝑞𝑞−1𝑞+1²ℎ𝑞𝑟^{−(3𝑞+1)/(𝑞+1)}=𝑘2𝑝𝑝−1𝑝+1²ℎ𝑝𝑐^{3−(3𝑝+1)/(𝑝+1)}𝑟^{−(3𝑝+1)/(𝑝+1)}+𝑚2𝑞𝑞−1𝑞+1²ℎ𝑞𝑟^{−(3𝑞+1)/(𝑞+1)}=𝑟^{−(3𝑞+1)/(𝑞+1)}×𝑘2𝑝𝑝−1𝑝+1²ℎ𝑝𝑐^{3−(3𝑝+1)/(𝑝+1)}×𝑟^{−(3(𝑝+1)/(𝑝+1))+((3𝑞+1)/(𝑞+1))}+𝑚2𝑞𝑞−1𝑞+1²ℎ𝑞. (55) The routine calculations show that the expression in parentheses is equal to −𝑚𝑞−1ℎ𝑞𝑞+12𝑝−𝑝+12𝑞𝑞+1=−𝑚𝑞−1ℎ𝑞2𝑞+1𝑝−𝑞𝑝+1𝑞+1>0. (56) Hence the function 𝜆=𝜑(𝛼) has the strict local minimum at the point 𝛼=𝛼₀.
(c) It follows from the relations lim_𝑟→0+𝑡_𝑝(𝑟)=+∞,lim_𝑟→+∞𝑡_𝑝(𝑟)=0+,lim_𝑟→0+𝑟𝑡_𝑝(𝑟)=0+,lim_𝑟→+∞𝑟𝑡_𝑝(𝑟)=+∞,lim_𝑟→0+𝑡_𝑞(𝑟)=0+,lim_𝑟→+∞𝑡_𝑞(𝑟)=+∞,lim_𝑟→0+𝑟𝑡_𝑞(𝑟)=0+,lim_𝑟→+∞𝑟𝑡_𝑞(𝑟)=+∞ (57) that if a parameter 𝑟>0 goes from 0 to +∞, then a point (𝜆,𝛼)∈𝒢_𝑐 goes from the point (+∞,0+) to the point (+∞,+∞).
(d) It follows from the above argument that if the parameter 𝑟>0 goes from 0 to +∞ then a point (𝜆,𝛼)∈𝒢_𝑐 goes from (+∞,0+) to (𝜆₀,𝛼₀), then turns to the right, and goes to (+∞,+∞). Since the parametrization of the curve 𝒢_𝑐 is without self-intersection points, one can deduce that the curve 𝒢_𝑐 is a union of two branches (i.e., graphs of the functions 𝛼=𝜓₁(𝜆) and 𝛼=𝜓₂(𝜆), where (𝜆₀<𝜆<+∞) which do not intersect and are continuously “glued” at the point (𝜆₀,𝛼₀).

Before presenting “the exact number of solutions” result we make the following conventions: (1) solution means a nontrivial solution of the problems (3) and (4),(2)ℰ^𝑠_𝑛 means the envelope, where 𝑛∈ℕ and 𝑠 is either + or − or ±,(3) we mean by 𝑠-solution a solution of the problem (3), (4): (a) a solution with 𝑥^′(0)>0 if 𝑠=+; (b) a solution with 𝑥^′(0)<0 if 𝑠=−; (c) two solutions with 𝑥^′(0)=𝛼>0 and 𝑥^′(0)=−𝛼<0 if 𝑠=±.

Proposition 18. Consider an envelope ℰ^𝑠_𝑛=𝜆,𝜇∈ℝ²₊∶𝜇=𝜔^𝑠_𝑛(𝜆),𝑛>0. (58) The problems (3) and (4) have (a)no 𝑠-solutions with 𝑛 zeroes in (0,1) if 𝜇<𝜔^𝑠_𝑛(𝜆),(b)exactly one 𝑠-solution with even number 𝑛 zeroes in (0,1) if 𝑠=+ or 𝑠=− and 𝜇=𝜔^𝑠_𝑛(𝜆),(c)exactly two 𝑠-solutions with 𝑛 zeroes in (0,1) if  𝑛 is odd, 𝜇=𝜔^𝑠_𝑛(𝜆) and 𝑠=± or 𝑛 is even, 𝜇>𝜔^𝑠_𝑛(𝜆) and 𝑠=+ or 𝑠=−,(d)exactly four 𝑠-solutions with odd number 𝑛 of zeroes in (0,1) if 𝑠=± and 𝜇>𝜔^𝑠_𝑛(𝜆).

Proof. It can be verified analytically that (𝜆₀,𝜇₀)∈ℰ^𝑠_𝑛, where 𝜆₀=𝜆𝑟₀=1𝑐²𝑐𝑘ℎ𝑝𝑐𝑟₀^{−(𝑝−1)/(𝑝+1)}+𝑚ℎ𝑞𝑟₀^{−(𝑞−1)/(𝑞+1)2}. (59) and 𝜇₀=𝑐²𝜆(𝑟₀).
Recall that the curve 𝒢_𝑐 is the projection of the 3D curve ℱ|_{𝜇=𝑐2𝜆} to the (𝜆,𝛼)-plane. Taking in mind Proposition 17 we can assert that there exists a unique parameter 𝑟₀>0, see (53), such that the line parallel to the 𝑂𝛼 axis in the (𝜆,𝜇,𝛼)-space going through the point (𝜆,𝜇), where 𝜆>0 and 𝜇=𝑐²𝜆, and the curve ℱ|_{𝜇=𝑐2𝜆} (hence the solution surface ℱ also) (i)does not intersect if 𝜇<𝜔^𝑠_𝑛(𝜆),(ii)intersects only once if 𝜇=𝜔^𝑠_𝑛(𝜆),(iii)intersects exactly at two points if 𝜇>𝜔^𝑠_𝑛(𝜆), (see Figure 3).
Depending on the value of 𝑠 one can detect the number of 𝑠-solutions as the theorem states. For example, in the case (d): if 𝑛 is odd, 𝜇>𝜔^𝑠_𝑛(𝜆) and 𝑠=±, then the mentioned above line intersects the solution surface 𝐹⁺_𝑛 exactly twice. Since 𝐹⁻_𝑛=𝐹⁺_𝑛 the mentioned above line intersects the solution surface 𝐹⁻_𝑛exactly twice also. Hence the problem (3), (4) has exactly four solutions with 𝑛 zeros in (0;1).

Figure 3 

The case 𝑝=3, 𝑞=1/2, 𝑘=2, 𝑚=1 and 𝑐=1.2.

Now we are able to prove the main theorem. In this theorem we suppose that 𝜔⁻₀(𝜆)=0 and 𝜔⁺₀(𝜆)=0 for all 𝜆>0 and will use auxiliary envelopes ℰ⁻₀∶𝜇=𝜔⁻₀(𝜆) and ℰ⁺₀∶𝜇=𝜔⁺₀(𝜆).

Theorem 19. Suppose 0<𝑞<1<𝑝. (1)If 𝑝𝑞−1>0 and 𝑖∈ℕ, then the exact number of nontrivial solutions to the problems (3) and (4) is(a)8𝑖−6 if 𝜔⁻_2𝑖−2(𝜆)<𝜇<𝜔^±_2𝑖−1(𝜆),(b)8𝑖−4 if 𝜇=𝜔^±_2𝑖−1(𝜆), (c)8𝑖−2 if 𝜔^±_2𝑖−1(𝜆)<𝜇<𝜔⁺_2𝑖(𝜆),(d)8𝑖−1 if 𝜇=𝜔⁺_2𝑖(𝜆), (e)8𝑖 if 𝜔⁺_2𝑖(𝜆)<𝜇<𝜔⁻_2𝑖(𝜆),(f)8𝑖+1 if 𝜇=𝜔⁻_2𝑖(𝜆). (2)If 𝑝𝑞−1=0 and 𝑖∈ℕ, then the exact number of nontrivial solutions to the problems (3) and (4) is (a)8𝑖−6 if 𝜔⁺_2𝑖−2(𝜆)<𝜇<𝜔^±_2𝑖−1(𝜆), (b)8𝑖−4 if 𝜇=𝜔^±_2𝑖−1(𝜆),(c)8𝑖−2 if 𝜔^±_2𝑖−1(𝜆)<𝜇<𝜔⁺_2𝑖(𝜆), (d)8𝑖if 𝜇=𝜔⁺_2𝑖(𝜆). (3)If 𝑝𝑞−1<0 and 𝑖∈ℕ then the exact number of nontrivial solutions to the problems (3) and (4) is(a)8𝑖−6 if 𝜔⁺_2𝑖−2(𝜆)<𝜇<𝜔^±_2𝑖−1(𝜆),(b)8𝑖−4 if 𝜇=𝜔^±_2𝑖−1(𝜆),(c)8𝑖−2 if 𝜔^±_2𝑖−1(𝜆)<𝜇<𝜔⁻_2𝑖(𝜆), (d)8𝑖−1 if 𝜇=𝜔⁻_2𝑖(𝜆), (e)8𝑖 if 𝜔⁻_2𝑖(𝜆)<𝜇<𝜔⁺_2𝑖(𝜆),(f)8𝑖+1 if 𝜇=𝜔⁺_2𝑖(𝜆).

Proof. First notice that if 𝜆>0 and 𝜇>0 then there exists exactly one ±-solution without zeros in (0,1), in fact, accordingly to conventions, two solutions without zeros in (0,1), and of opposite sign values of 𝑥^′(0).
We consider only the case 𝑝𝑞−1>0. Similarly two other possible cases 𝑝𝑞−1<0 and 𝑝𝑞−1=0 can be treated.
It follows from Proposition 15 that the envelopes are ordered as ℰ⁻₀≺ℰ^±₁≺ℰ⁺₂≺ℰ⁻₂≺ℰ^±₃≺⋯≺ℰ^±_2𝑖−1≺ℰ⁺_2𝑖≺ℰ⁻_2𝑖≺ℰ^±_2𝑖+1≺⋯. (60)
Let us consider the case (1), when 𝑝𝑞−1>0, and the subcase (a): 𝜔⁻_2𝑖−2(𝜆)<𝜇<𝜔^±_2𝑖−1(𝜆) for 𝑖=1,2,….(i)if 𝑖=1 then there are 2=8⋅1−6 solutions without zeros;(ii)if 𝑖=2, then there are two solutions without zeros, two ±-solutions with 1 zero (that is, four solutions with 1 zero), two “+”-solution with 2 zeros and two “−”-solutions with 2 zeros; totally there are 10=8⋅2−6 (nontrivial) solutions;(iii)if 𝑖=3, then in addition to 10 solutions mentioned in the previous step, there are also two ±-solutions with 3 zeros (that is, four solutions with 3 zeros), two “+”-solutions with 4 zeros and two “−”-solution with 4 zeros; totally there are 18=8⋅3−6 (nontrivial) solutions and so on;(iv)if 𝑖∈ℕ, then totally there are 8𝑖−6 (nontrivial) solutions.
The other cases can be considered analogously.