Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator

Theorem A.

(See [10, Theorem 3.4].) Assume that

(a*) a is a T-periodic and locally integrable function with ∫ 0 T a ( x ) d x < 0 ;

(a _*) there exist m ≥ 1 intervals I 1 + , … , I m + , closed and pairwise disjoint in the quotient space R / T Z , such that

( g * ) g : R + → R + is a continuous function, g(0) = 0, g(s) > 0 for s > 0, and lim sup s → 0 + g ( s ) s < + ∞ ;

( g ∞ ) lim s → + ∞ , w → 1 g ( w s ) g ( s ) = 1 .

Then, there exists λ _* > 0 such that for every λ > λ _*, there exists at least a positive T-periodic solution to (1.3).

Remark 1.1

1. Condition (a*) guarantees that the periodic linear eigenvalue problem associated with (1.3) has a positive principal eigenvalue λ ₁ > 0 in addition to λ ₀ = 0.

2. In [10], (g _∞) is referred to as a regular-oscillation condition: for instance, the function g(u) = u ^p satisfies (g _∞) for every p > 1. Condition (g _∞) guarantees that the connected component branching from (λ ₀, 0) and the one branching from (λ ₁, 0) are separate.

3. Theorem A provides no any information about the existence of positive T-periodic solutions for λ ≤ λ _*.

4. Theorem A cannot deal with the case of m = ∞ in (a _*).

5. Theorem A cannot deal with the case where g in (1.3) contains the derivative term.

The purpose of this paper is to investigate the global structure of positive 2π-periodic solutions set for (1.1), where m in (a _*) is allowed to be equal to infinity and nonlinearity may depend on u′.

It is worth underlining that the solvability of the equations with derivative dependence of the form (1.1) has been explicitly raised as an open question in Mawhin [12]. On the other hand, the study of global behavior of the positive periodic solutions set is very useful for computing the numerical solutions of (1.1) as it can be used to guide the numerical work.

Assume that

1. a ∈ C ( R , R ) is a 2π-periodic sign-changing function that satisfies ∫ 0 2 π a ( x ) d x < 0 ;

2. f(0, p) = 0 for any p ∈ [−1, 1], f(s, p) > 0 for s > 0 and p ∈ [−1, 1];

3. lim s → 0 + f ( s , p ) s ≕ f 0 ∈ ( 0 , + ∞ ) uniformly for p ∈ [−1, 1];

4. lim ( s , ω , ν ) → ( + ∞ , 1,0 ) f ( ω s , ν s ) f ( s , 0 ) = 1 .

Let Y ≔ { u ∈ C ( R , R ) ∣ u is 2 π − periodic } and

Then Y and H are the Banach spaces endowed with the norm

respectively. Let

Remark 1.2.

If a ∈ Y and f ∈ C ( R × R , R ) , then any solution u ∈ C ¹[0, 2π] of

can be extended to a 2π-periodic function u ̃ defined on the whole of R .

By the same argument in the proof of [13, Theorem 2.1], with obvious changes, we may deduce that

has two simple principal eigenvalues λ ₀ = 0 and λ ₁ > 0. The eigenfunctions corresponding to λ ₀ and λ ₁ are a constant and a positive function respectively.

Theorem 1.1.

Assume that (H1)–(H4) hold. Then there exists an unbounded connected component C + ⊆ S + , such that

1. C + joins ( λ 1 f 0 , 0 ) with infinity in λ direction;

2. for any compact interval J ⊂ [0, + ∞), there exists a positive constant γ* < 1 such that for any ( λ , u ) ∈ C + ∩ ( J × H ) ,

   ‖ u ′ ‖ ∞ ≤ γ * ;

3. Proj R C + = [ λ * , ∞ ) for some λ * ∈ ( 0 , λ 1 f 0 ] .

Corollary 1.1.

Assume that (H1)–(H4) hold. Then there exists λ * ∈ ( 0 , λ 1 f 0 ] such that (1.1) has at least one positive 2π-periodic solution for λ > λ 1 f 0 , while it has no positive 2π-periodic solution for λ ∈ (0, λ _*].

Remark 1.3

1. It is the first time that a generalized regular-oscillation condition, i.e. (H4), has been introduced. A bounded connected component of positive solutions connecting two eigenvalues is called a “mushroom”. López-Gómez and Molina-Meyer [14] and Brown [15] proved the existence of “mushrooms” under appropriate conditions. It is worth underlining that, the generalized regular-oscillation condition guarantees that the “mushroom” can not occur in Theorem 1.1.

2. (H3) implies that f satisfies a generalized regular-oscillation condition at 0, i.e.,

   lim ( s , ω , ν ) → ( 0,1,0 ) f ( ω s , ν s ) f ( s , 0 ) = 1 .

3. We allow m = ∞ in (a _*).

Next, for the special case of (1.1) that f does not contain derivative term, we also study the global structure of 2π-periodic sign-changing solutions set to the equation

where a ∈ C ( R , R ) is a 2π-periodic function and f ∈ C ( R , R ) .

If λ _k , k ≥ 2, is the eigenvalue of (1.5), then the multiplicity of λ _k may be even. For example, if a ≡ 1, then (1.5) has a list of eigenvalues λ _k = (k − 1)², k = 1, 2, 3, …. The eigenspace corresponding to λ _k , k ≥ 2, is

Therefore, we cannot directly apply Rabinowitz global bifurcation theorem to (1.6), see [16], [17].

In order to overcome the above difficulties, we have to work in

and

which are the Banach spaces endowed with the norm

This idea of constructing invariant subspaces is similar to the one in Marlin [18] and Coron [19].

We will use the following assumptions

1. a ∈ C ( R , R ) is a 2π-periodic function that satisfies a(−x) = a(x), x ∈ R ;

2. f : R → R is an odd continuous function;

3. s f ( s ) ≥ 0 , s ∈ R \ { 0 } .

Theorem 1.2.

Assume that (H5)–(H6) hold and

Then for any k ∈ {2, 3, …}, ν ∈ {+, − },

1. there exists an unbounded connected component C k ν ⊂ R × E 1 of 2π-periodic sign-changing solutions of (1.6) such that, C k ν joins (0,0) with infinity in λ direction, and any solution ( λ , u ) ∈ C k ν satisfies ‖u‖_∞ < π;

2. if f satisfies (H7), then there exists an unbounded connected component D k ν ⊂ R × E 2 of 2π-periodic sign-changing solutions of (1.6) such that, D k ν joins (0,0) with infinity in λ direction, and any solution ( λ , u ) ∈ D k ν satisfies ‖u‖_∞ < 2π.

Corollary 1.2.

Assume that (H5)–(H6) and (1.7) hold. Then for any λ ∈ (0, + ∞), k ∈ {2, 3, …}, ν ∈ {+, − },

1. Equation (1.6) has 2π-periodic odd solution u k ν satisfies that u k + has exactly 2k − 3 zeros in (0, 2π) and is positive near 0⁺, u k − has exactly 2k − 3 zeros in (0, 2π) and is negative near 0⁺;

2. if f satisfies (H7), then (1.6) has 2π-periodic even solution v k ν satisfies that v k + has exactly 2k − 2 zeros in (0, 2π) and is positive near 0, v k − has exactly 2k − 2 zeros in (0, 2π) and is negative near 0.

Due to f is sublinear at 0, the global bifurcation techniques cannot be used directly in the case. Therefore, we refer to [20] and apply some properties of the superior limit of a certain infinity collection of connected sets.

Remark 1.4.

For other results of boundary value problems where the nonlinearity is odd, see Naito and Tanaka [21], Garcia-Huidobro and Ubilla [22].

Remark 1.5.

Bereanu and Torres [23], using critical point theory, proved that the existence of an infinite number of solutions for the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime under some appropriate assumptions. However, they provided no any information about the nodal properties of these solutions.

Proposition 2.1.

Let u ∈ W be a 2π-periodic solution of (1.1). Then u ∈ C ²[0, 2π].

Proof.

Since ϕ(u′) ∈ C ¹[0, 2π], it follows that there exists y ∈ C[0, 2π], such that

The boundary condition u(0) = u(2π) implies that

Thus

and subsequently,

where ψ = ϕ ⁻¹ which can be explicitly given by

Let P ( x ) ≔ ∫ x 0 x y ( τ ) d τ . Obviously ψ ( ⋅ ) , P ( ⋅ ) ∈ C 1 ( R , R ) . Then

Therefore, u ∈ C ²[0, 2π].□

Lemma 2.1.

(See [17, Theorem 1.3].) If μ ∈ r(L) is of odd multiplicity, then S possesses a maximal subcontinuum C _μ such that (μ, 0) ∈ C _μ , and C _μ either

1. meets infinity in R × X , or

2. meets ( μ ̂ , 0 ) , where μ ≠ μ ̂ ∈ r ( L ) .

Let l ∈ X ′ (the dual of X) be corresponding eigenvectors of L ^T , the transpose of L. Denote ⟨⋅, ⋅⟩ the duality between X and X′. Let B ϵ denote open balls in R × X of radius ϵ centered at (μ, 0). For ξ , η ∈ R , define

Lemma 2.2.

(See [17, Lemma 1.24].) There exists a ζ ₀ > 0 such that for all ζ < ζ ₀, ( C μ − { ( μ , 0 ) } ) ∩ B ζ ⊂ K ξ , η . If ( λ ̄ , v ) ∈ ( C μ − { ( μ , 0 ) } ) ∩ B ζ , then v = α v ̄ + w , where |α| > η‖v‖, v ̄ is the eigenfunction of L corresponding to μ. Moreover, | λ ̄ − μ | = ◦ ( 1 ) , w = ◦ (|α|) for α near 0.

Definition 2.1.

(See [24].) Let X be a Banach space and { C n ∣ n = 1,2 , … } be a family of subsets of X. Then the superior limit D of C n is defined by

Lemma 2.3.

(See [20, Lemma 2.2].) Let X be a Banach space, and let { C n } be a family of connected subsets of X. Assume that

1. there exist z n ∈ C n , n = 1,2 , … , and z* ∈ X, such that z _n → z*;

2. lim n → ∞ r n = ∞ , where r n = sup { ‖ x ‖ : x ∈ C n } ;

3. for every R > 0, ∪ n = 1 ∞ C n ∩ B R is a relatively compact set of X, where

Then there exists an unbounded connected component C in D and z * ∈ C .

Let us define h : R → R by setting

and consider the equation

where ρ : R 3 → R is assumed to be continuous and 2π-periodic with respect to the first variable.

Lemma 2.4.

(See [25, Proposition 2].) If (λ, u) is a 2π-periodic solution of (2.2), then u ∈ H satisfies

Lemma 2.5.

A function u ∈ H is a 2π-periodic solution of

if and only if u ∈ H is a 2π-periodic solution of (2.2).

Proof.

It is clear that a 2π-periodic solution u ∈ H of (2.3) is a 2π-periodic solution of (2.2) as well. Conversely, suppose that u ∈ H is a 2π-periodic solution of (2.2). Then ‖u′‖_∞ < 1 can be obtained by Lemma 2.4. Consequently, u ∈ H is a 2π-periodic solution of (2.3).□

Lemma 2.6.

If (λ, u) is a solution of (2.4) and u has a double zero, then u ≡ 0.

Proof.

Assume that there exists a 2π-periodic solution ( λ ̃ , u ) , λ ̃ > 0 , of (2.4) and u has a double zero. Let x _* ∈ [0, 2π] be a double zero of u. Then it is easy to check that u satisfies the Cauchy problem

It is easy to deduce that the function F(x, s, p) is locally Lipschitz with respect to x, s and p in some neighbourhood U of (x _*, 0, 0),

for some constant δ ₀ > 0. Thus, by the existence and uniqueness theorem of initial value problem [26, Page 127, Theorem III], it concludes that u ≡ 0 in a small neighbourhood U ̃ of x _*. Denote U ̃ = [ a , b ] ⊂ [ 0,2 π ] . If a = 0 and b = 2π, then u ≡ 0 in [0, 2π]. It suffices to repeat the above analysis if a ≠ 0 or b ≠ 2π.□

Lemma 2.7.

For any fixed λ > 0, if {u _i } is a sequence of solutions of (2.4) satisfying lim i → ∞ u i ( x ) = 0 uniformly for x ∈ [0, 2π], then lim i → ∞ u i ′ ( x ) = 0 uniformly for x ∈ [0, 2π].

Proof.

Integrating the first equation of (2.4) from 0 to x for any x ∈ [0, 2π], we obtain that

which together with lim i → ∞ u i ( x ) = 0 uniformly for x ∈ [0, 2π],

uniformly for x ∈ [0, 2π]. This suggests that lim i → ∞ u i ′ ( x ) = c uniformly for x ∈ [0, 2π], where c ∈ R is a constant. If c ≠ 0, then

and

This is a contradiction.□

Lemma 2.8.

For any 2π-periodic sign-changing solution u ∈ H of (1.6), if u is odd, then

Proof.

If u(0) = u(2π) = ‖u‖_∞, then ‖u‖_∞ = 0. The conclusion is clearly correct. If u(0) = u(2π) ≠ ‖u‖_∞, we know that there is at least one point x ₀ ∈ (0, 2π) such that u′(x ₀) = 0. Suppose instead |u(x ₀)| = ‖u‖_∞.

1. Assume that u(x ₀) > 0. Then there exist 0 ≤ x ₁ < x ₀ < x ₂ ≤ 2π such that u(x ₁) = u(x ₂) = 0 and u(x) > 0, x ∈ (x ₁, x ₂). From Lemma 2.4, |u′(x)| < 1 for any x ∈ [0, 2π] and

   u ( x 0 ) − u ( x 1 ) ≤ ∫ x 1 x 0 | u ′ ( τ ) | d τ < x 0 − x 1 ,

   u ( x 2 ) − u ( x 0 ) ≥ ∫ x 0 x 2 − | u ′ ( τ ) | d τ > x 0 − x 2 ,

   u ( x 0 ) < 0 + ( x 2 − x 1 ) 2 < π .

2. Assume that u(x ₀) < 0. Then there exist 0 ≤ x ₁ < x ₀ < x ₂ ≤ 2π such that u(x ₁) = u(x ₂) = 0 and u(x) < 0, x ∈ (x ₁, x ₂). Similarly,

   | u ( x 0 ) | < π .

   □

Lemma 2.9.

Assume that f : R → R is odd and f(s) ≥ 0, s ∈ [0, ∞). Then any 2π-periodic sign-changing solution u ∈ H of (1.6) satisfies

Proof.

Let x ₁, x ₂ ∈ [0, 2π] such that

Then u(x ₂) < 0 < u(x ₁) and moreover,

which implies

□

Lemma 3.1.

Let (H1)–(H4) hold and let Λ ⊂ (0, + ∞) be a closed and bounded interval. Then there exists a constant D* > 0 such that for any ( λ , u ) ∈ C + and λ ∈ Λ,

Proof.

Assume on the contrary that there exists a sequence

satisfies

and ‖y _i ‖_∞ → ∞. Set v i ( x ) = y i ( x ) ‖ y i ‖ ∞ and D _i = ‖y _i ‖_∞. Since ‖ y i ′ ‖ ∞ ≤ 1 , we easily find ‖ v i ′ ‖ ∞ → 0 . As a consequence, v _i → 1 uniformly in [0, 2π], since

where x ̂ i ∈ [ 0,2 π ] is such that y i ( x ̂ i ) = ‖ y i ‖ ∞ = D i . Integrate the equation in (3.4) on [0, 2π], we have

and

where ‖ a ‖ L 1 ≔ ∫ 0 2 π a ( x ) d x . Using (H4), a contradiction follows.□

Lemma 3.2.

Let (H1)–(H4) hold. Then there exists λ* > 0 such that

Proof.

Assume on the contrary that there exists

such that (μ _j , y _j ) → (0, c) for some constants c.

Claim. There exist two positive constants 0 < d _* ≤ D* such that d _* ≤ c ≤ D*.

Assume on the contrary that there exists a sequence

satisfies

and μ _i → 0, y _i → 0 uniformly for x ∈ [0, 2π]. Set v i ( x ) = y i ( x ) ‖ y i ‖ ∞ and d _i = ‖y _i ‖_∞. Passing to the absolute value we have