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Bifurcation diagrams of one-dimensional Kirchhoff-type equations

  • Tetsutaro Shibata EMAIL logo
Published/Copyright: September 9, 2022
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

We study the one-dimensional Kirchhoff-type equation

− ( b + a ‖ u ′ ‖ 2 ) u ″ ( x ) = λ u ( x ) p , x ∈ I ≔ ( − 1 , 1 ) , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 ,

where ‖ u ′ ‖ = ∫ I u ′ ( x ) 2 d x 1 / 2 , a > 0 , b > 0 , p > 0 are given constants and λ > 0 is a bifurcation parameter. We establish the exact solution u λ ( x ) and complete shape of the bifurcation curves λ = λ ( ξ ) , where ξ ≔ ‖ u λ ‖ ∞ . We also study the nonlinear eigenvalue problem

− ‖ u ′ ‖ p − 1 u ″ ( x ) = μ u ( x ) p , x ∈ I , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 ,

where p > 1 is a given constant and μ > 0 is an eigenvalue parameter. We obtain the first eigenvalue and eigenfunction of this problem explicitly by using a simple time map method.

Keywords: Kirchhoff-type equations; bifurcation curves; exact value of the first eigenvalue
MSC 2010: 34C23; 37G99

1 Introduction

We consider the following nonlocal elliptic problem:

(1.1) − ( b + a ‖ u ′ ‖ 2 ) u ″ ( x ) = λ u ( x ) p , x ∈ I ≔ ( − 1 , 1 ) , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 ,

where ‖ u ′ ‖ = ∫ I u ′ ( x ) 2 d x 1 / 2 , a > 0 , b > 0 , p > 0 are given constants. If p = 3 , then problem (1.1) is known as the one-dimensional elliptic Kirchhoff-type equation. The purpose of this article is to establish the exact structure of the global solution curves of (1.1). The results give us the clear pictures of the bifurcation curves of (1.1) and better understanding of the structure of the solutions of nonlocal elliptic problems. To do this, we also obtain the first eigenvalue and eigenfunction associated with (1.1) explicitly. Such results as these seem to be novel and as far as the author knows, these are not obtained so far.

Nonlinear elliptic problems have a long history and have been studied intensively by many authors. We refer to [5] and references therein. In this field, nonlocal elliptic problems have also been studied by many investigators, since they are derived from several interesting physical and engineering phenomena. In particular, recently, fractional Kirchhoff problems and superlinear Schrödinger-Kirchhoff-type problems have been studied intensively. We refer to [1,12,14] and references therein.

Needless to say, the nonlinear bifurcation problems are one of the main topics in nonlinear elliptic problems. Therefore, there are also several studies which deal with bifurcation problems in nonlocal problems. We refer to [1,3,4,8,9,10] and references therein. As far as the author knows, however, there are few results which clarify the precise structures of bifurcation diagrams for nonlocal problems. Recently, the following typical problem was studied in [13] as the elliptic eigenvalue problems in bounded smooth domain Ω ⊂ R n ( n = 1 , 2 , 3 )

(1.2) − b + a ∫ Ω ∣ ∇ u ∣ 2 d x Δ u = λ u p in Ω , u > 0 in Ω , u = 0 on ∂ Ω ,

and using the degree argument and variational method, the following results have been obtained.

Theorem 1.1

[13]. Consider (1.2). Let 2 ∗ = ∞ for n = 1 , 2 and 2 ∗ = 6 for n = 3 .

  1. Assume that 0 < p < 1 . Then (1.2) has a branch of positive solutions bifurcating from zero at λ = 0 .

  2. Assume that 3 < p < 2 ∗ − 1 . Then (1.2) has a branch of positive solutions bifurcating from infinity at λ = 0 .

It should be mentioned that, in many cases, λ is parameterized by the maximum norm ξ ≔ ‖ u λ ‖ ∞ as λ = λ ( ξ ) , where u λ is a solution associated with λ . However, λ is not expressed explicitly by using ξ in [13]. Furthermore, in [13], the case 1 ≤ p ≤ 3 was not considered. Motivated by this, we focus on the case n = 1 and establish the explicit representation of λ ( ξ ) for (1.1), which is one of the most popular Kirchhoff-type bifurcation problems. The approach is the simple time map method. We put

(1.3) A p ≔ ∫ 0 1 1 1 − s p + 1 d s ,

(1.4) B p ≔ ∫ 0 1 1 − s p + 1 d s ,

(1.5) C p ≔ ∫ 0 1 s p + 1 1 − s p + 1 d s .

Let W p ( x ) ( 0 < p < 1 , p > 1 ) be a unique solution of

(1.6) − W ″ ( x ) = W ( x ) p , x ∈ I , W ( x ) > 0 , x ∈ I , W ( − 1 ) = W ( 1 ) = 0 .

The unique existence of W p ( x ) is known by [2,6].

Now we state the first result.

Theorem 1.2

Consider (1.1). Let a > 0 , b > 0 and p > 0 be given constants. Assume that 0 < p < 1 or 3 < p < ∞ . Then for any λ > 0 , there exists a unique solution pair ( λ , u λ ) ∈ R + × C 2 ( I ¯ ) . Put ξ ≔ ‖ u λ ‖ ∞ . Then λ is parameterized by ξ and the following formula holds for ξ > 0 .

(1.7) λ ( ξ ) = p + 1 2 A p 2 ( 2 A p B p a ξ 3 − p + b ξ 1 − p ) .

Furthermore, u λ ( x ) is given by

(1.8) u λ ( x ) = λ b + 2 A p B p a ξ ( λ ) 2 1 / ( 1 − p ) W p ( x ) ,

where ξ ( λ ) is the inverse function of λ ( ξ ) .

We note that if 0 < p < 1 (resp. 3 < p < ∞ ), then we see from (1.7) that λ ( ξ ) is strictly increasing (resp. decreasing) for ξ > 0 . Therefore, ξ ( λ ) exists.

By Theorem 1.2, we obtain the qualitative shapes of λ ( ξ ) for 0 < p < 1 and p > 3 (Figures 1 and 2).

Figure 1 The graph of λ ( ξ ) \lambda \left(\xi ) for 0 < p < 1 0\lt p\lt 1 .
Figure 1

The graph of λ ( ξ ) for 0 < p < 1 .

Figure 2 The graph of λ ( ξ ) \lambda \left(\xi ) for p > 3 p\gt 3 .
Figure 2

The graph of λ ( ξ ) for p > 3 .

Theorem 1.2 improves the results in Theorem 1.1 and gives the explicit formula for bifurcation curves of problem (1.1) for the case n = 1 .

Next, we consider the case where 1 < p < 3 . We apply ([1], Theorem 2) to (1.1) and obtain the following Theorem 1.3.

Theorem 1.3

Consider (1.1). Let a > 0 , b > 0 and 1 < p < 3 be given constants. For λ > 0 , put

(1.9) L ( λ ) ≔ p − 1 2 λ 2 / ( p − 1 ) ‖ W p ′ ‖ − 2 2 b 3 − p ( p − 3 ) / ( p − 1 ) .

  1. If L ( λ ) > a , then (1.1) has exactly two solutions.

  2. If L ( λ ) = a , then (1.1) has exactly one solution.

  3. If L ( λ ) < a , then (1.1) has no solutions.

Unfortunately, it is rather difficult to obtain the clear shape of λ ( ξ ) for general 1 < p < 3 . The reason is explained in Section 4. Therefore, we next concentrate on the special case p = 2 and establish the formulas for ξ = ξ ( λ ) . We put

(1.10) Q 1 ≔ 3 2 A 2 2 ,

(1.11) Q 2 ≔ 3 2 A 2 5 / 2 B 2 1 / 2 .

Theorem 1.4

Consider (1.1). Let p = 2 and a > 0 , b > 0 be given constants.

  1. Let

    (1.12) λ = 2 a b Q 2 .

    Then (1.1) has a unique solution u λ ( x ) = 2 b λ − 1 W 2 ( x ) . Moreover,

    (1.13) ξ = 2 b Q 1 λ − 1 = b a Q 1 Q 2 .

  2. Let λ > 2 a b Q 2 . Then there exist exactly two solutions u λ , 1 ( x ) and u λ , 2 ( x ) . Moreover, ξ are parameterized by λ such as ξ 1 = ξ 1 ( λ ) , ξ 2 = ξ 2 ( λ ) , which are represented as follows.

    (1.14) ξ 1 ( λ ) = λ Q 2 − 1 + λ 2 Q 2 − 2 − 4 a b 2 a Q 2 − 1 Q 1 ,

    (1.15) ξ 2 ( λ ) = λ Q 2 − 1 − λ 2 Q 2 − 2 − 4 a b 2 a Q 2 − 1 Q 1 .

  3. Let λ < 2 a b Q 2 . Then there are no solutions of (1.1).

The qualitative graph of ξ ( λ ) is as follows (Figure 3).

Figure 3 The graph of ξ ( λ ) \xi \left(\lambda ) for p = 2 p=2 .
Figure 3

The graph of ξ ( λ ) for p = 2 .

Before considering the case p = 1 and p = 3 for (1.1), we establish the first eigenvalue and eigenfunction of the following nonlocal eigenvalue problem:

(1.16) − ‖ u ′ ‖ p − 1 u ″ ( x ) = μ u ( x ) p , x ∈ I , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 ,

where p > 1 is a given constant, and μ > 0 is an eigenvalue parameter. Note that if p = 3 in (1.16), then it is also known as Kirchhoff-type eigenvalue problem. We refer to [3], which treated the case p = 3 precisely, and references therein. It is known that the first eigenvalue μ 1 of (1.16) is defined by

(1.17) μ 1 ≔ inf ‖ u ′ ‖ p + 1 : u ∈ H 0 1 ( I ) , ∫ I u ( x ) p + 1 d x = 1 .

Unfortunately, however, the explicit expression of μ 1 was not mentioned even in [3]. Since μ 1 also plays very important roles in the nonlocal elliptic bifurcation problems, we represent μ 1 explicitly by using simple time map method. We also obtain the explicit form of the first eigenfunction φ 1 ( x ) with ‖ φ 1 ‖ p + 1 = 1 associated with μ 1 . We note that μ 1 plays important roles in Theorems 1.6 and 1.7.

Theorem 1.5

Let p > 1 and consider (1.16). Then μ 1 is given by

(1.18) μ 1 = 2 ( p − 3 ) / 2 ( p + 1 ) A p ( p + 3 ) / 2 B p ( p − 1 ) / 2 .

Furthermore, let φ 1 be the first eigenfunction associated with μ 1 satisfying ‖ φ 1 ‖ p + 1 = 1 . Then

(1.19) φ 1 ( x ) = 2 2 / ( p 2 − 1 ) ( p + 1 ) − 1 / ( p − 1 ) A p − ( p + 3 ) / ( p 2 − 1 ) C p − 1 / ( p + 1 ) W p ( x ) .

Now we consider the case p = 1 and p = 3 for (1.1). For p = 3 , we know from ([11], Theorem 1.2) that the following result holds true.

Theorem 1.6

[11, Theorem 1.2] Assume that p = 3 . If a > 0 , b > 0 and λ > a μ 1 , then (1.1) has at least one positive solution.

We improve Theorem 1.6 for the case n = 1 .

Theorem 1.7

(i) Assume that p = 3 . Let λ > a μ 1 = 4 a A 3 3 B 3 . Then (1.1) has a unique solution

(1.20) u λ ( x ) = b ‖ W 3 ′ ‖ 2 λ − a ‖ W 3 ′ ‖ 2 ‖ W 3 ′ ‖ − 1 W 3 ( x ) ,

where ‖ W 3 ′ ‖ = 2 A 3 3 / 2 B 3 1 / 2 and

(1.21) λ ( ξ ) = 4 a A 3 3 B 3 + 2 A 3 2 b ξ − 2 ( ξ > 0 ) .

(ii) Assume that p = 1 . Let λ > π 2 4 b . Then the solution u λ ( x ) of (1.1) is given as follows:

(1.22) u λ ( x ) = 4 π 2 λ − ( π 2 b ) / 4 a cos π 2 x .

Furthermore,

(1.23) λ ( ξ ) = π 4 16 a ξ 2 + π 2 4 b ( ξ > 0 ) .

Figures 4 and 5 are rough shapes of λ ( ξ ) for p = 1 and p = 3 .

Figure 4 The graph of λ ( ξ ) \lambda \left(\xi ) for p = 3 p=3 .
Figure 4

The graph of λ ( ξ ) for p = 3 .

Figure 5 The graph of λ ( ξ ) \lambda \left(\xi ) for p = 1 p=1 .
Figure 5

The graph of λ ( ξ ) for p = 1 .

The remainder of this article is as follows. In Section 2, we prove Theorem 1.5 to introduce the time map method (cf. [7]). By using this argument, Theorems 1.2 and 1.4 are proved in Sections 2 and 3, respectively. Finally, we prove Theorem 1.7 in Section 5.

2 Proof of Theorem 1.5

We first prove (1.18). We have

(2.1) φ 1 ( x ) = φ 1 ( 1 − x ) ,

(2.2) ζ ≔ max x ∈ I φ 1 ( x ) = φ 1 ( 0 ) ,

(2.3) φ 1 ′ ( x ) > 0 , − 1 ≤ x < 0 .

We introduce the time map argument. From (1.16), we have

{ ‖ φ 1 ′ ‖ p − 1 φ 1 ″ ( x ) + μ 1 φ 1 ( x ) p } φ 1 ′ ( x ) = 0 .

This implies that

d d x 1 2 ‖ φ 1 ′ ‖ p − 1 φ 1 ′ ( x ) 2 + 1 p + 1 μ 1 φ 1 ( x ) p + 1 = 0 .

By putting x = 0 and (2.2), for x ∈ I , we have

1 2 ‖ φ 1 ′ ‖ p − 1 φ 1 ′ ( x ) 2 + 1 p + 1 μ 1 φ 1 ( x ) p + 1 = constant = 1 p + 1 μ 1 ζ p + 1 .

By this and (2.3), for − 1 ≤ x < 0 , we have

(2.4) φ 1 ′ ( x ) = 2 p + 1 ‖ φ 1 ′ ‖ − ( p − 1 ) / 2 μ 1 ( ζ p + 1 − φ 1 ( x ) p + 1 ) .

This along with (1.3) implies that

(2.5) μ 1 = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 ∫ − 1 0 φ 1 ′ ( x ) ζ p + 1 − φ 1 ( x ) p + 1 d x = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 ∫ 0 ζ 1 ζ p + 1 − θ p + 1 d θ = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 ζ ( 1 − p ) / 2 ∫ 0 1 1 1 − s p + 1 d s = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 ζ ( 1 − p ) / 2 A p .

Since ‖ φ 1 ‖ p + 1 = 1 , by using (1.5), (2.1) and (2.4), we have

(2.6) 1 2 = ∫ − 1 0 φ 1 ( x ) p + 1 d x = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 1 μ 1 ∫ − 1 0 φ 1 ( x ) p + 1 φ 1 ′ ( x ) ζ p + 1 − φ 1 ( x ) p + 1 d x = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 1 μ 1 ∫ 0 ζ θ p + 1 ζ p + 1 − θ p + 1 d θ = p + 1 2 ‖ φ 1 ′ ‖ ( p − 1 ) / 2 1 μ 1 ζ ( p + 3 ) / 2 C p .

By (2.5) and (2.6), we obtain

(2.7) ζ = A p 2 C p 1 / ( p + 1 ) .

By (1.4), (2.4) and (2.5), we have

(2.8) ‖ φ 1 ′ ‖ 2 = 2 ∫ − 1 0 φ 1 ′ ( x ) 2 d x = 2 2 p + 1 ‖ φ 1 ′ ‖ − ( p − 1 ) / 2 μ 1 ∫ − 1 0 ζ p + 1 − φ 1 ( x ) p + 1 φ 1 ′ ( x ) d x = 2 A p B p ζ 2 .

This along with (2.7) implies that

(2.9) ‖ φ 1 ′ ‖ = 2 A p B p A p 2 C p 1 / ( p + 1 ) .

By this, (2.5) and (2.7), we obtain

μ 1 = p + 1 2 2 ( p − 1 ) / 2 A p ( p + 3 ) / 2 B p ( p − 1 ) / 2 .

Thus, we obtain (1.18). We next prove (1.19). By (1.16), (1.18) and (2.9), we obtain

(2.10) − φ 1 ″ ( x ) = 2 − 2 / ( p + 1 ) ( p + 1 ) A p ( p + 3 ) / ( p + 1 ) C p ( p − 1 ) / ( p + 1 ) φ 1 ( x ) p .

We put

ν ≔ 2 2 / ( p 2 − 1 ) ( p + 1 ) − 1 / ( p − 1 ) A p − ( p + 3 ) / ( p 2 − 1 ) C p − 1 / ( p + 1 ) , φ 1 ( x ) = ν W p ( x ) .

Then it follows from this that φ 1 ( x ) satisfies (1.16) with μ 1 and ‖ φ 1 ‖ p + 1 = 1 . This implies (1.19). Thus, the proof of Theorem 1.5 is complete.

3 Proof of Theorem 1.2

Let p > 0 ( p ≠ 1 ). We consider the following equation of t ∈ R .

(3.1) a t + b = λ ‖ W p ′ ‖ 1 − p t ( p − 1 ) / 2 .

Assume that there exists a solution t λ > 0 of (3.1). Then we see from ([1], Theorem 2) that there exists a solution pair ( u λ , λ ) of (1.1) corresponding to t λ , and u λ ( x ) is given by

(3.2) u λ ( x ) = t λ 1 / 2 ‖ W p ′ ‖ − 1 W p ( x ) .

Indeed, let w λ be a unique solution of

(3.3) − w ″ ( x ) = λ w ( x ) p , x ∈ I , w ( x ) > 0 , x ∈ I , w ( − 1 ) = w ( 1 ) = 0 .

Then we see that w λ = λ 1 / ( 1 − p ) W p . We put γ = t λ 1 / 2 ‖ w λ ′ ‖ − 1 and u λ ≔ γ w λ . Then we have

b + a ‖ γ w λ ′ ‖ 2 = b + a t λ = t λ ( p − 1 ) / 2 ‖ w λ ′ ‖ 1 − p = γ p − 1 .

By this, we have

− ( b + a ‖ u λ ′ ‖ 2 ) u λ ″ = − γ p w λ ″ = λ γ p w λ p = λ u λ p .

We note that

u λ = γ w λ = t λ 1 / 2 λ 1 / ( p − 1 ) ‖ W p ′ ‖ − 1 λ 1 / ( 1 − p ) W p = t λ 1 / 2 ‖ W p ′ ‖ − 1 W p .

This implies (3.2). Therefore, if (3.1) has k positive solutions t λ , 1 , , t λ , 2 , … , t λ , k , then (1.1) also has k solutions corresponding to t λ , j ( j = 1 , 2 , … , k ). On the contrary, assume that u λ is a solution of (1.1). We put t λ ≔ ‖ u λ ′ ‖ 2 . Then we see that

u λ ( x ) = b + a t λ λ 1 / ( p − 1 ) W p ( x ) , t λ = ‖ u λ ′ ‖ 2 = b + a t λ λ 2 / ( p − 1 ) ‖ W p ′ ‖ 2 .

This implies that t λ satisfies (3.1). Therefore, the solutions of (1.1) correspond to those of (3.1). For t > 0 , we put

(3.4) g ( t ) ≔ a t + b − R t ( p − 1 ) / 2 ,

where R ≔ λ ‖ W p ′ ‖ 1 − p . Now we look for the solutions of g ( t ) = 0 . We have

(3.5) g ′ ( t ) = a − p − 1 2 R t ( p − 3 ) / 2 .

If p > 3 , then g ( t ) attains its maximum at

(3.6) t 0 ≔ 2 a ( p − 1 ) R 2 / ( p − 3 ) = 2 a ( p − 1 ) λ ‖ W p ′ ‖ 1 − p 2 / ( p − 3 ) .

Since g ( 0 ) = b and g ( t ) strictly increases in ( 0 , t 0 ) and attains its maximum g ( t 0 ) > 0 at t 0 and strictly decreases in ( t 0 , ∞ ) and g ( ∞ ) = − ∞ . So it is clear that there exists a unique t λ > t 0 , which satisfies g ( t λ ) = 0 . If 0 < p < 1 , then by (3.4) and (3.5), we have g ′ ( t ) > 0 for t > 0 and g ( 0 ) = − ∞ and g ( ∞ ) = ∞ . So it is clear that there exists a unique t λ > 0 , which satisfies g ( t λ ) = 0 . Certainly, if t λ is represented explicitly by λ , then it is natural to find the relationship between λ and ξ = ‖ u λ ‖ ∞ , and it seems possible to obtain the bifurcation curve λ ( ξ ) . Unfortunately, however, it is difficult to obtain t λ explicitly. To overcome this difficulty, we apply the time map method to obtain the bifurcation curves λ ( ξ ) of (1.1).

Let p > 3 or 0 < p < 1 be fixed. We apply the time map method to (1.1). Let an arbitrary λ > 0 be fixed and u λ ( x ) be a unique solution of (1.1). Then we have

(3.7) u λ ( x ) = u λ ( − x ) , x ∈ [ − 1 , 0 ] ,

(3.8) ξ ≔ ‖ u λ ‖ ∞ = max − 1 ≤ x ≤ 1 u λ ( x ) = u λ ( 0 ) ,

(3.9) u λ ′ ( x ) > 0 , x ∈ [ − 1 , 0 ) .

By (1.1), we have

{ ( b + a ‖ u λ ′ ‖ 2 ) u λ ″ ( x ) + λ u λ ( x ) p } u λ ′ ( x ) = 0 .

This implies that for − 1 ≤ x ≤ 0 ,

( b + a ‖ u λ ′ ‖ 2 ) u λ ′ ( x ) 2 + 2 p + 1 λ u λ ( x ) p + 1 = 2 p + 1 λ ξ p + 1 .

By this, for − 1 ≤ x ≤ 0 , we obtain

(3.10) u ′ λ ( x ) = 2 p + 1 1 b + a ‖ u λ ′ ‖ 2 λ ξ p + 1 − u λ ( x ) p + 1 .

By this and (1.3), we obtain

(3.11) λ = p + 1 2 b + a ‖ u λ ′ ‖ 2 ∫ − 1 0 u λ ′ ( x ) ξ p + 1 − u λ ( x ) p + 1 d x = p + 1 2 b + a ‖ u λ ′ ‖ 2 ∫ 0 ξ 1 ξ p + 1 − θ p + 1 d θ = p + 1 2 b + a ‖ u λ ′ ‖ 2 ξ ( 1 − p ) / 2 ∫ 0 1 1 1 − s p + 1 d s = p + 1 2 b + a ‖ u λ ′ ‖ 2 ξ ( 1 − p ) / 2 A p .

By (1.4) and (3.10), we have

(3.12) ‖ u λ ′ ‖ 2 = 2 ∫ − 1 0 u λ ′ ( x ) u λ ′ ( x ) d x = 2 ∫ − 1 0 2 p + 1 1 b + a ‖ u λ ′ ‖ 2 λ ξ p + 1 − u λ ( x ) p + 1 u λ ′ ( x ) d x = 2 2 p + 1 1 b + a ‖ u λ ′ ‖ 2 λ ∫ 0 ξ ξ p + 1 − θ p + 1 d θ = 2 2 p + 1 λ b + a ‖ u λ ′ ‖ 2 ξ ( p + 3 ) / 2 ∫ 0 1 1 − s p + 1 d s = 2 2 p + 1 λ b + a ‖ u λ ′ ‖ 2 ξ ( p + 3 ) / 2 B p .

Substitute (3.11) into (3.12). Then we have

(3.13) ‖ u λ ′ ‖ 2 = 2 A p B p ξ 2 .

By this and (3.11), we have

λ = p + 1 2 b + 2 A p B p a ξ 2 ξ ( 1 − p ) / 2 A p .

Namely,

λ = p + 1 2 A p 2 ( 2 A p B p a ξ 3 − p + b ξ 1 − p ) .

This implies (1.7). We next prove (1.8). By (1.1), (1.7) and (3.13), we have

− u λ ″ ( x ) = λ b + a ‖ u λ ′ ‖ 2 u λ ( x ) p = λ b + 2 A p B p a ξ ( λ ) 2 u λ ( x ) p .

This implies that

u λ ( x ) = λ b + 2 A p B p a ξ ( λ ) 2 1 / ( 1 − p ) W p ( x ) .

This implies (1.8). Thus, the proof of Theorem 2 is complete.

4 Proof of Theorems 1.3 and 1.4

We begin with the proof of Theorem 1.3.

Proof of Theorem 1.3

Let 1 < p < 3 . We fix b > 0 . Assume that y ( t ) = a t + b is the tangent line of g ( t ) = λ ‖ W p ′ ‖ 1 − p t ( p − 1 ) / 2 . Let t = t 0 be a point of tangency. Then we have

(4.1) a = p − 1 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 3 ) / 2 .

Since a t 0 + b = g ( t 0 ) , by (4.1), we have

(4.2) a t 0 + b = p − 1 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 1 ) / 2 + b = λ ‖ W p ′ ‖ 1 − p t 0 ( p − 1 ) / 2 .

By this, we have

b = 3 − p 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 1 ) / 2 .

This implies

(4.3) t 0 = 2 b ( 3 − p ) λ ‖ W p ′ ‖ 1 − p 2 / ( p − 1 ) .

By this and (4.1), we have

(4.4) a = p − 1 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 3 ) / 2 = p − 1 2 λ 2 / ( p − 1 ) ‖ W p ′ ‖ − 2 2 b 3 − p ( p − 3 ) / ( p − 1 ) .

Therefore, we see that the equation y ( t ) = g ( t ) has one solution t 0 > 0 if (4.4) holds. Then by (3.1) and (3.2), we know that u λ ( x ) ≔ t 0 1 / 2 ‖ W p ′ ‖ − 1 W p ( x ) is a unique solution to (1.1). Equally, if

a < p − 1 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 3 ) / 2 = p − 1 2 λ 2 / ( p − 1 ) ‖ W p ′ ‖ − 2 2 b 3 − p ( p − 3 ) / ( p − 1 ) ,

then y ( t ) = g ( t ) has exactly two solutions t 1 , t 2 , and (1.1) has exactly two solutions, and if

a > p − 1 2 λ ‖ W p ′ ‖ 1 − p t 0 ( p − 3 ) / 2 = p − 1 2 λ 2 / ( p − 1 ) ‖ W p ′ ‖ − 2 2 b 3 − p ( p − 3 ) / ( p − 1 ) ,

then the equation y ( t ) = g ( t ) has no solutions, and (1.1) has no solutions. Thus, the proof is complete.□

Proof of Theorem 1.4

In what follows, let p = 2 . We use (4.1) and (4.2) here. We calculate η = W 2 ( 0 ) and ‖ W 2 ′ ‖ . By (1.6) and the same argument as that to obtain (2.4), for − 1 ≤ x ≤ 0 , we have

1 2 W 2 ′ ( x ) 2 + 1 3 W 2 ( x ) 3 = 1 3 η 3 .

This implies that for − 1 ≤ x ≤ 0 ,

(4.5) W 2 ′ ( x ) = 2 3 η 3 − W 2 ( x ) 3 .

By this, we obtain

1 = ∫ − 1 0 3 2 W 2 ′ ( x ) η 3 − W 2 ( x ) 3 d x = 3 2 ∫ 0 η 1 η 3 − θ 3 d θ = 3 2 η − 1 / 2 ∫ 0 1 1 1 − s 3 d s .

By this, (1.3) and (1.10), we have

(4.6) η = W 2 ( 0 ) = Q 1 = 3 2 A 2 2 .

By (1.4), (4.5) and (4.6), we have

‖ W 2 ′ ‖ 2 = 2 ∫ − 1 0 2 3 η 3 − W 2 ( x ) 3 W 2 ′ ( x ) d x = 2 2 3 ∫ 0 η η 3 − θ 3 d θ = 2 2 3 η 5 / 2 ∫ 0 1 1 − s 3 d s = 9 2 A 2 5 B 2 .

This along with (1.11) implies

(4.7) ‖ W 2 ′ ‖ = 3 2 A 2 5 / 2 B 2 1 / 2 ≔ Q 2 .

Now we prove Theorem 1.4 (i). Put p = 2 in (4.4). Then if

(4.8) a = 1 4 b λ 2 ‖ W 2 ′ ‖ − 2 ,

namely, if λ = 2 a b ‖ W 2 ′ ‖ , then (1.1) has exactly one solution. By this, (4.3), (4.7) and (4.8), we have

(4.9) t 0 1 / 2 = b a , Q 2 − 1 = ‖ W 2 ′ ‖ − 1 = 2 λ − 1 a b .

By this and (3.2), we see that

u λ ( x ) = t 0 1 / 2 Q 2 − 1 W 2 ( x ) = 2 b λ − 1 W 2 ( x )

is the solution. By this, (4.6) and (4.9), we have

ξ = u λ ( 0 ) = 2 b λ − 1 Q 1 = b a Q 1 Q 2 .

This implies Theorem 1.4 (i). We next prove Theorem 1.4 (ii). By Theorem 1.3 (a), if λ > 2 a b Q 2 , then (1.1) has exactly two solutions. In this case, by (3.1) and (4.7), we have

(4.10) t 1 1 / 2 ( λ ) ≔ λ Q 2 − 1 + λ 2 Q 2 − 2 − 4 a b 2 a ,

(4.11) t 2 1 / 2 ( λ ) ≔ λ Q 2 − 1 − λ 2 Q 2 − 2 − 4 a b 2 a .

Then by (3.2) and (4.7), we have

(4.12) u λ , 1 ( x ) = λ Q 2 − 1 + λ 2 Q 2 − 2 − 4 a b 2 a Q 2 − 1 W 2 ( x ) ,

(4.13) u λ , 2 ( x ) = λ Q 2 − 1 − λ 2 Q 2 − 2 − 4 a b 2 a Q 2 − 1 W 2 ( x ) .

We put x = 0 in (4.12) and (4.13). Then by (4.6), we obtain (1.14) and (1.15). Thus, the proof is complete.□

5 Proof of Theorem 1.7

Proof of Theorem 1.7 (i)

Let p = 3 . If we have t λ in (3.1) for some λ > 0 , then we have a solution of (1.1) for p = 3 . We put p = 3 in (3.1). Then we have

(5.1) a t + b = λ ‖ W 3 ′ ‖ − 2 t .

Namely, if t λ = b ‖ W 3 ′ ‖ 2 λ − a ‖ W 3 ′ ‖ 2 exists, then we have a unique solution of (1.1) for p = 3 . Assume that

(5.2) λ > a ‖ W 3 ′ ‖ 2 .

Then by (3.2), we have the unique solution

(5.3) u λ ( x ) = b ‖ W 3 ′ ‖ 2 λ − a ‖ W 3 ′ ‖ 2 ‖ W 3 ′ ‖ − 1 W 3 ( x ) .

Now we calculate ‖ W 3 ′ ‖ . We put η ≔ ‖ W 3 ‖ ∞ . By (1.1) and the same argument as that in Section 4, we have

1 2 W 3 ′ ( x ) 2 + 1 4 W 3 ( x ) 4 = 1 4 η 4 .

By this, for − 1 ≤ x ≤ 0 , we have

(5.4) W 3 ′ ( x ) = 1 2 η 4 − W 3 ( x ) 4 .

By this and (1.4), we have

(5.5) ‖ W 3 ′ ‖ 2 = 2 ∫ − 1 0 1 2 η 4 − W 3 ( x ) 4 W 3 ′ ( x ) d x = 2 ∫ 0 η η 4 − θ 4 d θ = 2 η 3 ∫ 0 1 1 − s 4 d x = 2 η 3 B 3 .

By (1.3) and (5.4), we have

(5.6) 1 = ∫ − 1 0 2 W 3 ′ ( x ) η 4 − W 3 ( x ) 4 d x = 2 ∫ 0 η 1 η 4 − θ 4 d θ = 2 η − 1 ∫ 0 1 1 1 − s 4 d s = 2 η − 1 A 3 .

By (5.5) and (5.6), we have

(5.7) ‖ W 3 ′ ‖ 2 = 4 A 3 3 B 3 .

By this and (5.2), if λ > 4 a A 3 3 B 3 , then we have the unique solution. We know from (1.18) that μ 1 = 4 A 3 3 B 3 when p = 3 . Then the argument in the proof of Theorem 1.2 is also available for the case p = 3 , and we also obtain (1.7) for p = 3 . This implies (1.21). Thus, the proof of Theorem 1.7 (i) is complete.□

Proof of Theorem 1.7 (ii)

Let p = 1 . Then (1.1) is the linear eigenvalue problem with positive solution. Therefore, we have

(5.8) λ b + a ‖ u ′ ‖ 2 = λ 1 = π 2 4 .

We look for the solution u λ ( x ) = C φ 1 ( x ) , where C > 0 is a constant and φ 1 ( x ) ≔ cos π 2 x . We substitute C φ 1 ( x ) into (1.1) to obtain

(5.9) C = 4 π 2 a λ − π 2 4 b .

Therefore, if λ > b λ 1 , then we have a unique solution u λ ( x ) of (1.1) such as

(5.10) u λ ( x ) = 4 π 2 a λ − π 2 4 b cos π 2 x .

Thus, the proof of Theorem 1.7 (ii) is complete.□

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP21K03310.

  1. Conflict of interest: The author states no conflict of interest.

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Received: 2021-09-30
Revised: 2022-03-30
Accepted: 2022-07-14
Published Online: 2022-09-09

© 2023 Tetsutaro Shibata, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2022-0265
Keywords for this article
Kirchhoff-type equations; bifurcation curves; exact value of the first eigenvalue
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Articles in the same Issue

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
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