Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight

Liping Wei; Shunchang Su

doi:10.1515/math-2023-0122

Artikel Open Access

Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight

Liping Wei und Shunchang Su

Veröffentlicht/Copyright: 26. September 2023

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Abstract

In this article, we prove the existence of eigenvalues for the problem

( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) ϕ p ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , A u ( 0 ) − A ′ u ′ ( 0 ) = 0 , B u ( 1 ) + B ′ u ′ ( 1 ) = 0

under hypotheses that ϕ p ( s ) = ∣ s ∣ p − 2 s , p > 1 , and h is a nonnegative measurable function on ( 0 , 1 ) , which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem:

( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 ,

where λ is a real parameter, a ≥ 0 , f ∈ C ( ( 0 , ∞ ) , ( 0 , ∞ ) ) satisfies inf s ∈ ( 0 , ∞ ) f ( s ) > 0 and limsup s → 0 s α f ( s ) < ∞ for some α > 0 .

Keywords: p-Laplacian; spectrum; nonlinear boundary conditions; bifurcation

MSC 2010: 34B18; 34C10; 34C23

1 Introduction

In this article, we study the existence and properties of eigenvalues for the following one-dimensional p -Laplacian problem under the Sturm-Liouville boundary condition:

(1.1) ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) ϕ p ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , A u ( 0 ) − A ′ u ′ ( 0 ) = 0 , B u ( 1 ) + B ′ u ′ ( 1 ) = 0 ,

where ϕ p ( s ) = ∣ s ∣ p − 2 s , p > 1 , λ is a real parameter, and the weight h is a nonnegative measurable function on ( 0 , 1 ) , which may be singular at 0 and/or 1. In boundary conditions, A , A ′ , B , and B ′ are given nonnegative real numbers such that A 2 + A ′ 2 ≠ 0 , B 2 + B ′ 2 ≠ 0 , and A 2 + B 2 ≠ 0 .

Since our concern is focused on the singular weight, let us summarize a history about (1.1) with several kinds of weights briefly. In the case of h ∈ C [ 0 , 1 ] , Kusano and Naito’s results [1] refer to a sign changing weight, they proved that the totality of eigenvalues consists two sequences { λ n + } n = 0 ∞ and { λ n − } n = 0 ∞ such that ⋯ < λ n − < ⋯ < λ 1 − < λ 0 − < 0 < λ 0 + < λ 1 + < ⋯ < λ n + < ⋯ lim n → ∞ λ n + = + ∞ , lim n → ∞ λ n − = − ∞ . The eigenfunctions associated with λ = λ n + and λ n − have exactly n zeros in ( 0 , 1 ) . In the case of h ∈ L 1 ( 0 , 1 ) with h ≥ 0 and A ′ = B ′ = 0 , Zhang [2] and Lee and Sim [3] proved the following properties by Sturm type comparison and Picone identity:

the set of all eigenvalues of (1.1) is a countable set { λ k ∣ k ∈ N } satisfying 0 < λ 1 < λ 2 < ⋯ < λ k < ⋯ → ∞ ;
for each k , the solution space is a subspace of C 1 [ 0 , 1 ] and its dimension is one;
let μ k be a corresponding eigenfunction to λ k , then the number of interior zeros of μ k is k − 1 .

We will show the similar properties (i), (ii), and (iii) for problem (1.1). For the proof, a variant of the generalized Prüfer transformation for the quasilinear problem (1.1) plays a crucial role. This transformation involves the generalized sine function and the generalized cosine function. To overcome the difficulty that Sturmian comparison is not applicable directly, we use the generalized Picone identity to quasilinear operator.

On the basis of the existence and properties of eigenvalues for (1.1), we investigate the existence of connected set of positive solutions for the nonlinear problems of the form

(1.2) ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 ,

where a ≥ 0 . Then h , f , and c satisfy the following conditions:

f ∈ C ( ( 0 , ∞ ) , ( 0 , ∞ ) ) with inf s ∈ ( 0 , ∞ ) f ( s ) > 0 and limsup s → 0 s α f ( s ) < ∞ for some α > 0 ;
h ∈ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) ∩ L 1 ( ( 0 , 1 ) , ( 0 , ∞ ) ) with ∫ 0 1 h ( t ) d ( t ) α d t < ∞ , where d ( t ) ≔ min { t , 1 − t } ;
c ∈ C ( ( 0 , ∞ ) × [ 0 , ∞ ) , [ 0 , ∞ ) ) is such that c ( r , 0 ) = 0 for all r ∈ ( 0 , + ∞ ) and c ( r , s ) is nondecreasing in s ∈ [ 0 , ∞ ) . There exists a constant c * ∈ ( 0 , ∞ ) , such that
lim s → + ∞ c ( r , s ) s = c *
uniformly for r in any compact interval of ( 0 , ∞ ) .

For the case a = 0 (i.e., (1.2) has the Dirichlet boundary condition at t = 1 ), Ko et al. [4] showed the multiplicity of positive solutions for a certain range of λ by a method of sub-super solutions; Dai [5], Dai and Ma [6] investigated the existence and multiplicity of one-sign solutions and nodal solutions of the p -Laplacian involving a linear/superlinear nonlinearity by using bifurcation method. For the case a > 0 (i.e., (1.2) has a nonlinear boundary condition at t = 1 ), in the study by Hai and Wang, [7], this type of problem is considered for a nonlinearity, f that is, superlinear or sublinear at infinity, where the existence and multiplicity results were shown by Krasnoselskii type fixed point theorem. Further, Sim and Son [8] considered singular nonlinear elliptic problems involving nonhomogeneous operators on annular domains, which can be reduced to (1.2). Once again, they analyzed the existence and multiplicity of positive radial solutions according to the behavior of nonlinearity near infinity via a Krasnoselskii type fixed point theorem.

The shape of the bifurcation branches provides many important clues to the qualitative properties of (1.2). For this reason and motivated by the aforementioned studies, we will show the existence and shape of the connected set of positive solutions of (1.2) by bifurcation and topological methods. However, the bifurcation results of [9–13] cannot be applied directly to quasilinear problems, which are very important tools in dealing with semilinear boundary value problems. So we will use a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator established by Dai [5]. The following theorem is the first result on the multiplicity of positive solutions of problem (1.2).

Theorem 1.1

Assume (A1)–(A3) and f ∞ ≔ lim s → + ∞ f ( s ) ϕ p ( s ) = ∞ . Then there exists λ * > 0 such that for all λ ∈ ( 0 , λ * ) , (1.2) has at least two positive solutions.

Our result involving the sublinear growth at infinity is the following theorem:

Theorem 1.2

Assume (A1)–(A3) and f ∞ = 0 . Then (1.2) has at least one positive solution for all λ > 0 . Moreover, if

f ( s ) = g ( s ) s α , where g is continuous and nondecreasing;
there exist σ and τ such that σ < 4 τ and f ( τ ) ϕ p ( τ ) f ( σ ) ϕ p ( σ ) > 4 α h * ϕ p ( 16 ) h * are satisfied, where h * ≔ ∫ 0 1 h ( t ) d ( t ) α d t , h * ≔ ∫ 1 ⁄ 4 1 ⁄ 2 h ( t ) d t .

Then there exist λ * , λ * > 0 such that for λ ∈ ( λ * , λ * ) , (1.2) has at least three positive solutions.

We state the concept related to the shapes of bifurcation branch on the ( λ , ‖ u ‖ ∞ ) -plane.

Definition 1.3

We say that, a connected component C of positive solutions of (1.2) is of S * -shaped, if C contains ( 0 , 0 ) , and there exist positive constants r 1 , r 2 , Λ 1 , Λ 2 with

r 1 < r 2 , Λ 1 > Λ 2 ,

such that

( λ , u ) ∈ C with ‖ u ‖ ∞ = r 1 ⇒ λ ≥ Λ 1 , ( λ , u ) ∈ C with ‖ u ‖ ∞ = r 2 ⇒ λ ≤ Λ 2 ,

and eventually continues to ( + ∞ , + ∞ ) .

This article is organized as follows. In Section 2, we state a Whyburn type limit theorem and define a completely continuous operator. Section 3 is devoted to establish a sequence { λ k } of eigenvalues of (1.1) under the assumption on h . In the last Section, we study the bifurcation phenomenon from infinity of problem (1.2) with superlinear/sublinear growth nonlinearity at infinity, and then give the proofs of Theorems 1.1 and 1.2.

2 Preliminaries

In this section, we introduce a topological result involving superior limit and lower bounded estimates.

2.1 Whyburn type limit theorem

The Whyburn limit theorem [14, Theorem 9.1] is an important tool in the study of differential equations theory, see, e.g., [15–18], and the references cited therein. However, if the collection of the infinite sequence of sets is unbounded, Whyburn’s limit theorem cannot be used directly because the collection may not be relatively compact. To deal with the case of superlinear/sublinear growth at infinity, we introduce a topological lemma established by Dai [5] as follows.

Lemma 2.1

[5] Let X be a normal space with norm ‖ . ‖ X , and let { C n ∣ n = 1 , 2 , … } be a sequence of unbounded connected subsets of X. Assume that

there exists z * ∈ liminf n → + ∞ C n with ‖ z * ‖ X < + ∞ ;
for every R > 0 , ( ∪ n = 1 + ∞ C n ) ∩ B ¯ R is a relative compact set of X, where B R = { x ∈ X : ‖ x ‖ X < R } .

Then C ≔ limsup n → ∞ C n is unbounded closed connected.

Remark 2.2

By the proof of [5, Lemma 2.5], it is easy to see that z * ∈ C and C is a compact set of X .

2.2 Lower bound estimates

Lemma 2.3

Assume (A1) and (A2). Let v λ be the solution of the boundary value problem

− ( ϕ p ( v ′ ( t ) ) ) ′ = λ h ( t ) f * , t ∈ ( 0 , 1 ) , v ( 0 ) = v ( 1 ) = 0 ,

where f * = inf s ∈ ( 0 , ∞ ) f ( s ) > 0 . Then v λ ( t ) ≥ ‖ v λ ‖ ∞ d ( t ) .

Proof

From ( ϕ p ( v λ ′ ( t ) ) ) ′ = − λ h ( t ) f * , it follows ϕ p ( v λ ′ ) is strictly decreasing because f * > 0 . Since ϕ p is an odd increasing homeomorphism, v λ ′ is strictly decreasing on [ 0 , 1 ] . So v λ has only one critical point, say at t m ∈ ( 0 , 1 ) by boundary conditions. It is sufficient to show that v λ ′ > 0 on [ 0 , t m ) and v λ ′ < 0 on ( t m , 1 ] , which ensures that v λ is strictly concave and v λ > 0 in ( 0 , 1 ) . Thus, we have

v λ ( t ) ≥ ‖ v λ ‖ ∞ t m t , t < t m , ‖ v λ ‖ ∞ 1 − t m ( 1 − t ) , t > t m .

Therefore, we obtain v λ ( t ) ≥ ‖ v λ ‖ ∞ d ( t ) .□

Define c λ : [ 0 , ∞ ) → ( 0 , ∞ ) by

c λ ( s ) = s , a = 0 , c ( λ , s ) a , a > 0 .

Let 1 p + 1 q = 1 and m u ∈ ( 0 , ∞ ) be the constant such that

m u sgn ( a ) = ∫ 0 1 ϕ q − ϕ p ( c λ ( m u ) ) + λ ∫ s 1 h ( τ ) f ( max { u , v λ } ) d τ d s .

Then m u is uniquely determined according to (A3). Let us define T λ : C [ 0 , 1 ] → C [ 0 , 1 ] by

T λ u ( t ) ≔ ∫ 0 t ϕ q − ϕ p ( c λ ( m u ) ) + λ ∫ s 1 h ( τ ) f ( max { u , v λ } ) d τ d s .

By the arguments in [8], we have that T λ is well defined and completely continuous.

It is remarkable that T λ u is the solution of

− ( ϕ p ( u ′ ( t ) ) ) ′ = λ h ( t ) f ( max { u , v λ } ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = a u ′ ( 1 ) + c ( u ( 1 ) ) ,

and T λ u ≥ v λ ( t ) . Further, T λ u ( t ) can be rewritten as follows:

T λ u ( t ) = ∫ 0 t ϕ q λ ∫ s t m h ( τ ) f ( max { u , v λ } ) d τ d s , t ∈ ( 0 , t m ) , T λ u ( 1 ) + ∫ t 1 ϕ q λ ∫ t m s h ( τ ) f ( max { u , v λ } ) d τ d s , t ∈ ( t m , 1 ) ,

where t m ∈ ( 0 , 1 ) is the only critical point of T λ u .

3 Spectrum of singular nonlinear eigenvalue problems

We start with the eigenvalue problem. To obtain the spectrum of problem (1.1), we need to introduce the Sturm type comparison theorem for p -Laplacian problem.

Lemma 3.1

Let b 1 , b 2 ∈ L 1 ( 0 , 1 ) satisfy b 2 ≥ b 1 for t ∈ ( 0 , 1 ) . Also let y , z be solutions of the following differential equations

( ϕ p ( y ′ ) ) ′ + b 1 ( t ) ϕ p ( y ) = 0 , ( ϕ p ( z ′ ) ) ′ + b 2 ( t ) ϕ p ( z ) = 0 ,

respectively. If ( c , d ) ⊂ ( 0 , 1 ) and y ( c ) = y ( d ) = 0 but without any zeros on ( c , d ) , then either there exists τ ∈ ( c , d ) such that τ is a zero of z or b 2 = b 1 and z ( t ) = ν y ( t ) for some constant ν ≠ 0 .

Proof

If z has a zero in ( c , d ) , the conclusion is done. If there are no zeros of z on ( c , d ) , then we can assume without loss of generality that y ( t ) > 0 , z ( t ) > 0 in ( c , d ) . Integrating the generalized Picone type identity [19], we have

(3.1) ∫ c d ∣ y ∣ p ϕ p ( z ′ ) ϕ p ( z ) − y ϕ p ( y ′ ) ′ d t = ∫ c d ( b 1 − b 2 ) ∣ y ∣ p − ∣ y ′ ∣ p + ( p − 1 ) y z ′ z p − p ϕ p ( y ) y ′ ϕ p z ′ z d t .

The left-hand side of (3.1) equals to zero. Hence, the right-hand side of (3.1) also equals to zero. By Young’s inequality, we obtain

∣ y ′ ∣ p + ( p − 1 ) y z ′ z p − p ϕ p ( y ) y ′ ϕ p z ′ z ≥ 0 ,

and the equality holds if and only if sgn y ′ = sgn z ′ and y ′ y p = z ′ z p . It follows that there exists a constant ν ≠ 0 , such that z ( t ) = ν y ( t ) and b 2 = b 1 .□

Since the bifurcation points of (1.2) are related to the eigenvalues of the problem

(3.2) ϕ p ( u ′ ( t ) ) ′ + λ h ( t ) ϕ p ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 ,

we summarize the eigenvalue properties of problem (1.1). We first notice that all eigenvalues of (1.1) are positive. Indeed, let u be the corresponding eigenfunction to μ satisfying the initial conditions

u ( 0 ) = A ′ , u ′ ( 0 ) = A , u ( 1 ) = B ′ , u ′ ( 1 ) = − B .

Multiplying by u both sides in equation (1.1) and integrating, we obtain

∫ 0 1 ( ϕ p ( u ′ ( s ) ) ) ′ u ( s ) d s = ϕ p ( u ′ ( t ) ) u ( t ) ∣ 0 1 − ∫ 0 1 ϕ p ( u ′ ( s ) ) u ′ ( s ) d s = − ∣ − B ∣ p − 2 B B ′ − ∣ A ∣ p − 2 A A ′ − ∫ 0 1 ∣ u ′ ( s ) ∣ p d s = − μ ∫ 0 1 h ( s ) ϕ p ( u ( s ) ) u ( s ) d s .

Thanks of A A ′ , B B ′ ≥ 0 , we have

μ = ∫ 0 1 ∣ u ′ ( s ) ∣ p d s + ∣ − B ∣ p − 2 B B ′ + ∣ A ∣ p − 2 A A ′ ∫ 0 1 h ( s ) ∣ u ( s ) ∣ p d s ≥ 0 .

If μ = 0 and A ≠ 0 , u ( t ) is explicitly given by u ( t ) = A ′ + A t , 0 ≤ t ≤ 1 . By using the conditions on A , A ′ , B , and B ′ , we easily see that the value μ = 0 is not an eigenvalue of (1.1).

Proposition 3.2

Assume h ∈ L 1 ( 0 , 1 ) . Then we have

the set of all eigenvalues of (1.1) is a countable set { λ k ( p ) ∣ k ∈ N } satisfying 0 < λ 1 ( p ) < λ 2 ( p ) < ⋯ < λ k ( p ) → ∞ ,
for each k, the solution space is a subspace of C 1 [ 0 , 1 ] and its dimension is 1,
let μ k be a corresponding eigenfunction to λ k ( p ) , then the number of interior zeros of μ k is k − 1 .

Proof

Let u ( t ; λ ) be the solution of (1.1) satisfying the initial condition

u ( 0 ) = A ′ , u ′ ( 0 ) = A .

By the underlying hypotheses A A ′ ≥ 0 and A 2 + A ′ 2 ≠ 0 , this solution u ( t ; λ ) is nontrivial. In what follows, we assume λ > 0 .

Let generalized sine function S = S ( τ ) be the unique solution of the specific half-linear equation

( ∣ S ′ ( t ) ∣ p − 2 S ′ ( t ) ) ′ + ( p − 1 ) ∣ S ( t ) ∣ p − 2 S ( t ) = 0

satisfying the initial condition

S ( 0 ) = 0 , S ′ ( 0 ) = 1 .

The generalized cosine function C is the derivative of S : C ( τ ) = S ′ ( τ ) . In particular, the functions C , S have the property that

(3.3) ∣ C ( τ ) ∣ p + ∣ S ( τ ) ∣ p = 1 .

Now, for the solution u ( t ; λ ) , we introduce the polar functions ρ ( t ; λ ) and θ ( t ; λ ) defined by

u ( t ; λ ) = ρ ( t ; λ ) S ( θ ( t ; λ ) ) , u ′ ( t ; λ ) = λ 1 ⁄ p ρ ( t ; λ ) C ( θ ( t ; λ ) ) .

Then

d d t u ( t ; λ ) = d d t ( ρ ( t ; λ ) S ( θ ( t ; λ ) ) ) = λ 1 ⁄ p ρ ( t ; λ ) C ( θ ( t ; λ ) ) , d d t ( u ′ ( t ; λ ) ) p − 1 = d d t ( λ 1 ⁄ q ρ p − 1 ( t ; λ ) C p − 1 ( θ ( t ; λ ) ) ) = − λ h ( t ) ϕ p ( u ( t ) ) .

Consequently, coupled with (3.3) and let ( ρ , θ ) = ( ρ ( t ; λ ) , θ ( t ; λ ) ) , we can obtain that (1.1) is equivalently written as follows:

(3.4) ρ ′ = λ 1 p ρ 1 − h ( t ) p − 1 ∣ S ( θ ) ∣ p − 1 C ( θ ) ,

(3.5) θ ′ = λ 1 p ∣ C ( θ ) ∣ p + h ( t ) p − 1 ∣ S ( θ ) ∣ p ≕ Θ ( t ; λ )

under the initial condition θ ( 0 ; λ ) = T − 1 ( λ 1 ⁄ p A ′ A ) = θ 0 , where T − 1 denotes the inverse of the generalized tangent function T = S ⁄ C .

Since A A ′ ≥ 0 , let π p = 2 π ( p − 1 ) 1 ⁄ p p sin ( π ⁄ p ) . We may obtain 0 ≤ θ ( 0 ; λ ) < π p 2 for the case A ≠ 0 , θ ( 0 ; λ ) = π p 2 for A = 0 and θ ( 0 ; λ ) = 0 if A ′ = 0 . Then, (3.5) has a unique solution θ ( t ; λ ) , and it can be extended on [ 0 , 1 ] , since

0 ≤ Θ ( t ; λ ) ≤ λ 1 p max h ( t ) p − 1 , 1 for all t ∈ [ 0 , 1 ] .

It is easy to see that λ > 0 is an eigenvalue of (1.1) if and only if λ satisfies

(3.6) θ ( 1 ; λ ) = T − 1 − λ 1 ⁄ p B ′ B + k π p

for some k ∈ N . Define h − ( t ) ≔ min 1 , h ( t ) p − 1 and h + ( t ) ≔ max 1 , h ( t ) p − 1 . Then by the continuity of θ with respect to λ and the inequalities

θ 0 + λ 1 ⁄ p ∫ 0 1 h − ( s ) d s ≤ θ ( 1 ; λ ) ≤ θ 0 + λ 1 ⁄ p ∫ 0 1 h + ( s ) d s ,

we obtain

lim λ → 0 + θ ( 1 ; λ ) = θ 0 , lim λ → + ∞ θ ( 1 ; λ ) = + ∞ .

By [16], it follows that θ ( t ; λ 1 ) ≥ θ ( t ; λ 2 ) for all t ∈ [ 0 , 1 ] if λ 1 > λ 2 > 0 . Furthermore, the function θ ( 1 ; λ ) is strictly increasing for λ ∈ ( 0 , ∞ ) .

On the other hand, by virtue of B B ′ ≥ 0 , the right-hand side of (3.6) is a nonincreasing function of λ ∈ ( 0 , λ ) for each k ∈ N . More precisely, in the case B B ′ > 0 , it is strictly decreasing and varies from k π p to ( k − 1 ⁄ 2 ) π p as λ varies from 0 to + ∞ ; in the case B ′ = 0 , it is the constant function k π p ; and in the case B = 0 , it is the constant function ( k − 1 ⁄ 2 ) π p .

This implies as in the aforementioned argument that, for each k = 1 , … , there exists a unique λ k ( p ) > 0 such that

θ ( 1 ; λ k ( p ) ) = T − 1 − λ 1 ⁄ p B ′ B + k π p .

Therefore, the corresponding eigenfunction u ( t , λ ) has exactly k − 1 zeros in the open interval ( 0 , 1 ) , where k = 1 , … . It is clear that

λ 1 ( p ) < λ 2 ( p ) < ⋯ < λ k ( p ) < ⋯ , lim k → ∞ λ k ( p ) = + ∞ .

This completes the proof of (i) and (iii).

The eigenfunctions are of C 1 [ 0 , 1 ] , since h ∈ L 1 ( 0 , 1 ) . Suppose that u 1 and u 2 are two eigenfunctions corresponding to the same eigenvalue λ k ( p ) . Without the loss of generality, we may assume that there exists an interval ( c , d ) ⊂ ( 0 , 1 ) , such that u 1 ( c ) = u 1 ( d ) = 0 , and u 1 , u 2 > 0 on ( c , d ) . It follows from Lemma 3.1 that ( u 1 u 2 ) ′ ≡ 0 and u 1 = ν u 2 on ( c , d ) for some ν ∈ R . Since u 1 , u 2 ∈ C 1 [ 0 , 1 ] and u 1 and ν u 2 share the same initial condition at c and d , we can extend the identity u 1 ≡ ν u 2 up to the interval ( 0 , 1 ) by the uniqueness of the initial value problem and this completes the proof of (ii).□

4 Proofs of Theorems

The aim of this section is to prove Theorems 1.1 and 1.2. Let E be a Banach space C [ 0 , 1 ] equipped with the norm ‖ u ‖ = max { ‖ u ‖ ∞ , ‖ u ′ ‖ ∞ } .

4.1 Superlinear growth at infinity

In this subsection, we consider the case of superlinear growth of f at infinity.

Following the similar procedure of [5] with obvious changes, for each n ∈ N , we define

f [ n ] ( s ) ≔ f ( s ) , s ∈ ( 0 , n ] , n ϕ p ( 2 n ) − f ( n ) n ( s − n ) + f ( n ) , s ∈ ( n , 2 n ) , n ϕ p ( s ) , s ∈ [ 2 n , + ∞ ) ,

and extend f [ n ] ( ⋅ ) to all of R \ { 0 } by setting F [ n ] ( s ) = f [ n ] ( ∣ s ∣ ) . Then

f ∞ [ n ] = lim s → ∞ f [ n ] ( s ) ϕ p ( s ) = n .

Let ζ : R → R be such that

F [ n ] ( s ) = f ∞ [ n ] ϕ p ( s ) + ζ ( s )

with lim ∣ s ∣ → + ∞ ζ ( s ) ϕ p ( s ) = 0 . From (A3), it follows that

c ( λ , s ) = c * s + ξ ( λ , s )

with lim s → + ∞ ξ ( λ , s ) s = 0 . Then we consider

(4.1) − ( ϕ p ( u ′ ( t ) ) ) ′ = λ f ∞ [ n ] h ( t ) ϕ p ( u ( t ) ) + λ ζ ( u ( t ) ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c * u ( 1 ) = 0

as a bifurcation problem from infinity.

We already know that for every given h ∈ L 1 ( 0 , 1 ) , there is a unique solution u to problem

(4.2) − ( ϕ p ( u ′ ( t ) ) ) ′ = h ( t ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c * u ( 1 ) = 0

(see [3]). Let T p ( h ) denote the unique solution to problem (4.2) and E B = { u ∈ E : u ( 0 ) = 0 , a u ′ ( 1 ) + c * u ( 1 ) = 0 } , then T p : L 1 ( 0 , 1 ) → E B is completely continuous. Obviously, problem (4.1) can be equivalently written as follows:

(4.3) u = T p ( λ Q p u + H ( λ , u ) ) ,

where Q p ( u ) = f ∞ [ n ] h ( t ) ϕ p ( u ) , H ( λ , ⋅ ) denotes the usual Nemitsky operator associated with λ ζ . Let L ( λ ) u = T p ( λ Q p u ) and H ˆ ( λ , u ) = T p ( λ Q p u + H ( λ , u ) ) − L ( λ ) u . Then it is easy to see that L ( ⋅ ) : E B → E B is homogeneous completely continuous. From hypotheses (A2) and (A3), the right-hand side of equation (4.3) defines a completely continuous operator from R × E B into E B . So H ˆ : R × E B → E B is completely continuous. Let

ζ ˜ ( u ) = max 0 ≤ ∣ s ∣ ≤ u ∣ ζ ( s ) ∣ ,

then ζ ˜ is nondecreasing. Further, we have

ζ ( u ) ‖ u ‖ p − 1 ≤ ζ ( ∣ u ∣ ) ‖ u ‖ p − 1 ≤ ζ ( ‖ u ‖ ∞ ) ‖ u ‖ p − 1 ≤ ζ ( ‖ u ‖ ) ‖ u ‖ p − 1 → 0 as ‖ u ‖ → ∞ .

It follows that H ˆ = o ( ‖ u ‖ ) near u = ∞ uniformly on bounded λ intervals.

By an argument similar to that of [5, Proposition 4.1], we can obtain

Lemma 4.1

Assume (A1)–(A3) and f ∞ = ∞ . Then for each n ∈ N , λ ∞ [ n ] ≔ λ 1 ( p ) f ∞ [ n ] is a bifurcation point from infinity for (4.1). More precisely, there exists a sequence of unbounded continua D n of solution set of problem (4.1) emanating from ( λ ∞ [ n ] , ∞ ) .

Lemma 4.2

Assume (A1)–(A3) and f ∞ = ∞ . Then for each n ∈ N , there exist two continuous functions γ n , β n : ( 0 , ∞ ) → ( 0 , ∞ ) , such that

γ n and β n are continuous in ( 0 , ∞ ) ;
γ n ( ρ ) ≤ λ ≤ β n ( ρ ) for any solution ( λ , u ) ∈ D n with ‖ u ‖ ∞ = ρ .

Proof

Let ( λ , u ) be a solution of auxiliary problem

(4.4) − ( ϕ p ( u ′ ( t ) ) ) ′ = λ h ( t ) f [ n ] ( u ( t ) ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ) = 0 .

It follows from Lemma 2.3 that u ( t ) > ‖ u ‖ ∞ d ( t ) ≥ 0 . By condition (A1), we may choose f ¯ ≔ max r ∈ [ 1 , s ] f ( r ) for s ≥ 1 . Then f ¯ is nondecreasing and there exists b > 0 such that f ( s ) ≤ b s α for s ≤ 1 . Thus, we can obtain

∣ f [ n ] ( s ) ∣ ≤ b s α + f [ n ] ¯ ( s ) , s ∈ [ 0 , ∞ )

and f ¯ ( s ) ϕ p ( s ) = f ∞ . This together with Lemma 2.3 implies

‖ u ‖ ∞ ≤ ‖ T λ u ‖ ∞ ≤ ∫ 0 t m ϕ p − 1 ∫ s t m λ h ( r ) b max { u , v λ } α + f [ n ] ¯ ( max { u , v λ } ) d r d s ≤ ∫ 0 t m ϕ q ∫ s t m λ h ( r ) b ‖ u ‖ ∞ α d ( r ) α + f [ n ] ¯ ( max { ‖ u ‖ ∞ , ‖ v λ ‖ ∞ } ) d r d s ≤ ϕ q λ b h * ‖ u ‖ ∞ α + ‖ h ‖ 1 f [ n ] ¯ ( max { ‖ u ‖ ∞ , ‖ v λ ‖ ∞ } ) ,

where h * = ∫ 0 1 h ( r ) d ( r ) α d r , and it follows from (A2) that h * < ∞ .

On the other hand, we can choose a nondecreasing continuous function f ̲ , which satisfies f ̲ ( s ) ≤ f ( s ) , inf s ∈ ( 0 , ∞ ) f ̲ ( s ) > 0 , and lim s → ∞ f ̲ ( s ) ⁄ ϕ p ( s ) = ∞ . Let t m ∈ ( 0 , 1 ) be the only critical point of u . Without loss of generality, we assume that t m ≥ 1 2 , then u satisfies

‖ u ‖ ∞ ≥ ‖ T λ u ‖ ∞ ≥ ∫ 0 1 4 ϕ p − 1 λ ∫ 1 4 1 2 h ( r ) f [ n ] ̲ ‖ u ‖ ∞ 4 d r d s ≥ 1 4 ϕ p − 1 λ h * f [ n ] ̲ ‖ u ‖ ∞ 4 ,

here, h * = ∫ 1 ⁄ 4 1 ⁄ 2 h ( r ) d r . If n is large enough, we may take

(4.5)□ γ n ( ρ ) ≔ ϕ p ( ρ ) b h * ρ α + ‖ h ‖ 1 f [ n ] ¯ ( max { ρ , ‖ v λ ‖ ∞ } ) , β n ( ρ ) ≔ ϕ p ( 4 ρ ) h * f [ n ] ̲ ( ρ 4 ) .

Lemma 4.3

Assume (A1)–(A3) and f ∞ = ∞ . Then for fixed ρ ∈ ( 0 , ∞ ) , we have

lim n → ∞ γ n ( ρ ) = ϕ p ( ρ ) b h * ρ α + ‖ h ‖ 1 f ¯ ( max { ρ , ‖ v λ ‖ ∞ } ) ≕ γ ( ρ ) lim n → ∞ β n ( ρ ) = ϕ p ( 4 ρ ) h * f ̲ ρ 4 ≕ β ( ρ ) .

Lemma 4.4

Assume (A1)–(A3) and f ∞ = ∞ . Then

lim ρ → + ∞ β ( ρ ) = 0 , lim ρ → 0 β ( ρ ) = 0 .

Proof of Theorem 1.1

Let X = R × ( E ∪ ∞ ) and taking z * = ( 0 , ∞ ) , clearly z * ∈ liminf n → + ∞ D n with ‖ z * ‖ = + ∞ . Define mapping K : X → X such that

K ( λ , u ) = λ , u ‖ u ‖ 2 if 0 < ‖ u ‖ < + ∞ , ( λ , 0 ) if ‖ u ‖ = + ∞ , ( λ , ∞ ) if ‖ u ‖ = 0 .

It is easy to verify that K is homeomorphism and ‖ K ( z * ) ‖ = 0 . It follows from [5] that K ( z * ) ∈ liminf n → + ∞ K ( D n ) . Let D = limsup n → + ∞ D n , clearly, z * ∈ D . We claim that D is unbounded closed connected. In fact, the unboundedness of z * shows that D is unbounded. From [14], we know that D is closed. So we only need to show that D is connected. Now Lemma 2.1 shows that D ˜ = limsup n → + ∞ K ( D n ) is unbounded closed connected. Then we can show that K ( D ) = D ˜ following the arguments in the proof of Theorem 1.2 in [5]. Suppose on the contrary that D is not connected, so there exist two subsets A and B of D such that A ∩ B ¯ = B ∩ A ¯ = ∅ . It is not difficult to verify that K ( A ) ∩ K ( B ¯ ) = K ( B ) ∩ K ( A ¯ ) = ∅ . That is to say K ( D ) is not connected which contradicts the connectedness of D ˜ .

On the other hand, from Lemma 4.2, it follows that D n joins ( λ ∞ [ n ] , ∞ ) with the set { ( μ , v ) ∣ γ n ( ρ ) ≤ μ ≤ β n ( ρ ) , ‖ v ‖ ∞ = ρ } . By (4.5), there exists Λ > 0 such that the set β n ( ρ ) ∈ ( 0 , Λ ] if n is large enough.

Then we can deduce that there exists w * ∈ { ( μ , v ) ∣ μ ∈ ( γ ( ρ ) , β ( ρ ) ) , ‖ v ‖ ∞ = ρ } such that D joins w * to ( 0 , ∞ ) . Moreover,

γ ( ρ ) ≤ μ ≤ β ( ρ ) , for ( μ , v ) ∈ D with ‖ v ‖ ∞ = ρ ∈ ( 0 , ∞ ) .

This completes the proof.□

4.2 Sublinear growth at infinity

In this section, we consider the case of sublinear growth of f at infinity.

For each n ∈ N , we define

f [ n ] ( s ) ≔ f ( s ) , s ∈ ( 0 , n ] , 1 n ϕ p ( 2 n ) − f ( n ) n ( s − n ) + f ( n ) , s ∈ ( n , 2 n ) , 1 n ϕ p ( s ) , s ∈ [ 2 n , + ∞ )

and extend f [ n ] ( ⋅ ) to all of R \ { 0 } by setting F [ n ] ( s ) = f [ n ] ( ∣ s ∣ ) . Then

f ∞ [ n ] = lim s → ∞ f [ n ] ( s ) ϕ p ( s ) = 1 n .

Let us consider the auxiliary family of the problems

(4.6) − ( ϕ p ( u ′ ( t ) ) ) ′ = λ h ( t ) f [ n ] ( u ( t ) ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 .

Following a similar argument of Lemma 4.1, we have

Lemma 4.5

Assume (A1)–(A3) and f ∞ = 0 . Then λ ∞ [ n ] is a bifurcation point from infinity for (4.6). More precisely, there exists a sequence of unbounded continua D n of solution set of problem (4.6) emanating from ( λ ∞ [ n ] , ∞ ) .

Proof of Theorem 1.2

It follows that lim n → ∞ f ∞ [ n ] = 0 and consequently lim n → ∞ λ ∞ [ n ] = ∞ . An argument similar to that of Theorem 1.1 can yield that there exists an unbounded component D of (1.2) joining ( ∞ , ∞ ) with the set { ( μ , v ) ∣ γ n ( σ ) ≤ μ ≤ β n ( σ ) , ‖ v ‖ ∞ = σ } .

Next we show the existence of three positive solutions. Let ( λ , u ) ∈ D be a solution of (1.2) with ‖ u ‖ ∞ = σ . From Lemma 2.3, we can obtain ‖ u ‖ ∞ ≥ ‖ v λ ‖ ∞ and u ( t ) ≥ ‖ u ‖ ∞ d ( t ) , respectively. Then by (A4), the following estimates hold

(4.7) ‖ u ‖ ∞ ≤ ∫ 0 t m ϕ q λ ∫ s t m h ( r ) g ( max { u , v λ } ) max { u , v λ } α d r d s ≤ ∫ 0 t m ϕ q λ ∫ s t m h ( r ) g ( ‖ u ‖ ∞ ) ‖ u ‖ ∞ α d ( r ) α d r d s ≤ ϕ q ( λ h * f ( ‖ u ‖ ∞ ) ) .

This implies λ ≥ ϕ p ( σ ) h * f ( σ ) .

Let ( λ , u ) ∈ D with ‖ u ‖ ∞ = 4 τ ( > σ ) . Assume that the unique critical of u satisfies t m ≥ 1 2 , then

(4.8) ‖ u ‖ ∞ ≥ ‖ T λ u ‖ ∞ ≥ ∫ 0 1 4 ϕ q λ ∫ s t m h ( r ) g ( max { u , v λ } ) max { u , v λ } α d r d s ≥ ∫ 0 1 4 ϕ q λ ∫ 1 4 1 2 h ( r ) g ‖ u ‖ ∞ 4 ‖ u ‖ ∞ α d r d s ≥ 1 4 ϕ q λ h * 4 α f ‖ u ‖ ∞ 4 .

The case when t m < 1 2 is proved in the same way. This implies λ ≤ 4 α ϕ p ( 16 τ ) h * f ( τ ) .

Define λ * = sup { λ : ( λ , u ) ∈ D with ‖ u ‖ ∞ = σ } and λ * = sup { λ : ( λ , u ) ∈ D with ‖ u ‖ ∞ = 4 τ } . Obviously, it follows from (A5) that

λ * ≤ 4 α ϕ p ( 16 τ ) h * f ( τ ) < ϕ p ( σ ) h * f ( σ ) ≤ λ * .

Combining (4.7)–(4.8), and using Lemma 4.5, it deduces that D is S * shaped. Then by standard arguments, (1.2) has a positive solution at λ = λ * and λ = λ * . There exist points u 1 , u 2 and u 3 for λ ∈ ( λ * , λ * ) such that 0 < ‖ u 1 ‖ ∞ < σ < ‖ u 2 ‖ ∞ < 4 τ < ‖ u 3 ‖ ∞ < ∞ . This completes the proof.□

Finally, we provide examples satisfying the hypotheses to illustrate Theorems 1.1–1.2.

Example 4.6

(Example 4.6) Let us consider the problem

(4.9) − ( ϕ p ( u ′ ( t ) ) ) ′ = λ h ( t ) u k u α + 1 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 ,

where α = 1 ⁄ 4 , a ≥ 0 , k > α + p − 1 , and h satisfies (A2). Then f ( s ) = s k s α + 1 satisfies (A1) and f ∞ = ∞ . By applying the result of Theorem 1.1, there exists λ * > 0 , such that problem (4.9) has at least two positive solutions for all λ ∈ ( 0 , λ * ) .

Example 4.7

Consider the problem

(4.10) − ( ϕ p ( u ′ ( t ) ) ) ′ = λ h ( t ) ( 256 2 ( 1 − p ) e ) 2 u 1 + u u α , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 ,

where α = 1 ⁄ 4 , a ≥ 0 and h satisfies (A2). Then f ( s ) = ( 256 2 ( 1 − p ) e ) 2 s 1 + s s α satisfies (A1)(A4) and f ∞ = 0 . Thus, (4.10) has a positive solution for all λ > 0 . Let σ = 1 and h * , h * be as before in (A5). Then we can chose τ = 1 ⁄ 3 and f ( 1 ⁄ 3 ) ϕ p ( 1 ⁄ 3 ) ⁄ f ( 1 ) ϕ p ( 1 ) = 3 α + p − 1 1 6 2 ( p − 1 ) > 4 α h * ϕ p ( 16 ) h * for p ≥ 2 . Hence, there exist λ * , λ * > 0 such that (4.10) has at least three positive solutions for λ ∈ ( λ * , λ * ) .

Acknowledgements

The authors are very grateful to the anonymous referees for their very valuable suggestions.

Funding information: This work was supported by Doctoral Research Foundation of Gansu Agricultural University (No. GAU-KYQD-2022-32), “Star of Innovation” Project of Outstanding Graduate Students of Gansu Provincial Department of Education (2023CXZX-689), and Gansu University Innovation Foundation (No. 2022B-107).
Conflict of interest: Authors state no conflict of interest.

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Received: 2023-01-15

Revised: 2023-08-08

Accepted: 2023-08-27

Published Online: 2023-09-26

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p-Laplacian; spectrum; nonlinear boundary conditions; bifurcation

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