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Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

  • Franco Obersnel and Pierpaolo Omari EMAIL logo
Published/Copyright: October 30, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem:

− div ∇ v 1 + | ∇ v | 2 = f ( x , v , ∇ v ) in Ω , a 0 v + a 1 ∂ v ∂ ν = 0 on ∂ Ω ,

with Ω an open ball in ℝ N , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

Keywords: prescribed mean curvature equation; Dirichlet; Neumann; Robin boundary conditions; radial symmetry; classical solution; sub- and super-solutions
MSC 2010: 35J62; 35J93; 35J25; 34C25

1 Introduction and main results

This paper deals with the existence of classical solutions of the mean curvature problem:

(1.1) − t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 f ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 ,

in the presence of couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition, that is, the sub-solution is smaller than the super-solution. It is assumed throughout this work that

  • (h 1) R > 0 , N ∈ ℕ + , a 0 , a 1 ∈ ℝ are given constants, with a 0 ⋅ a 1 ≥ 0 , a 0 + a 1 > 0 ,

    and

  • (h 2) f : [ 0 , R ] × ℝ × ℝ → ℝ is continuous.

Definition 1.1

(Notion of solution) By a classical solution of (1.1) we mean a function u ∈ C 1 ( [ 0 , R ] ) ∩ C 2 ( ] 0 , R [ ) which satisfies the differential equation for all t ∈ ] 0 , R [ and the boundary condition.

It is easy to verify, arguing like, e.g., in [1, Lemma 3.1], that a classical solution u of (1.1) actually belongs to C 2 ( [ 0 , R ] ) and satisfies the equation:

− u ′ 1 + u ′ 2 ′ − N − 1 t u ′ 1 + u ′ 2 = f ( t , u , u ′ ) for all t ∈ [ 0 , R ] ,

provided that the convention u ′ ( t ) t = u ″ ( 0 ) if t = 0 is set. Accordingly, solutions of (1.1) will always be assumed to belong to C 2 ( [ 0 , R ] ) .

From [1, Remark 3.2] it also follows that if u is a solution of (1.1), then, setting v ( x ) = u ( | x | ) for all x ∈ Ω ¯ , with Ω the open ball in ℝ N of center 0 and radius R, the function v belongs to C 2 ( Ω ¯ ) and is a classical solution of

(1.2) − div ∇ v 1 + | ∇ v | 2 = f | x | , v , ∇ v ⋅ x | x | in Ω , a 0 v + a 1 ∇ v ⋅ x | x | = 0 on ∂ Ω .

The existence of classical solutions of boundary value problems associated with the mean curvature equation

(1.3) − div ∇ v 1 + | ∇ v | 2 = f ( x , v , ∇ v ) ,

in a bounded domain Ω , is a central and delicate topic in the field of the quasilinear elliptic partial differential equations and has been largely discussed in the literature, also due to the applications of (1.3) in numerous geometrical and physical issues. Although breakthrough results were proven between the late sixties and the early seventies, starting with [2,3,4,5,6,7,8], this study has been carried out further in the subsequent decades by many authors; for a wide, but far from exhaustive, overview we refer to the monographs [9,10,11,12] and references therein.

However, in spite of such a large amount of work, a general sub- and super-solution method – a quite powerful technique for the study of semilinear and quasilinear elliptic problems [13,14,15,16,17] – has not been established yet for the classical solutions of boundary value problems associated with Eq. (1.3). Actually, some partial results have been formulated in [18, Theorems 2.1 and 2.2] and [19, Theorem 1.1], but unfortunately the given proofs do not seem complete, as all of them rely on the unproven Lemma 3.2 in [20]. This substantial lack of results within the existing literature might be attributed to the fact that mean curvature problems are fraught with a number of difficulties which do not arise in dealing with other quasilinear problems, such as the possible occurrence of gradient blow up phenomena and the corresponding development of singularities for the solutions, up to the formation of discontinuities. In order to overcome these drawbacks and to include in the analysis such kind of singular solutions, it appeared natural, starting with [21,22,23,24,25,26], to settle the problem in the space of bounded variation functions. In particular, this approach allowed to develop in [27,28,29] a sub- and super-solution method for the bounded variation solutions of various boundary value problems associated with Eq. (1.3), according to the notion of solution introduced in [30]. The likely most relevant feature of these results is that, assuming the sub- and super-solutions satisfy the standard ordering condition, no bound and no monotonicity are imposed on the function f, contrary to what was required in all preceding studies.

When limiting the discussion to the radial problem (1.2) with homogeneous Dirichlet boundary conditions, which has also been the subject of extensive study since the late eighties [31,32,33,34,35,36], the existence of classical solutions can be inferred from [28, Theorem 2.4] if the right-hand side f of the equation is gradient independent, locally Lipschitz and non-increasing with respect to the state variable, and further the sub- and the super-solutions are regular and satisfy the boundary conditions. A complementary result for the same problem was obtained by a different approach – monotone iteration versus variational techniques – in [19, Theorem 3.1], assuming that f is gradient independent, continuous, but non-decreasing with respect to the state variable, and again the sub- and the super-solutions are regular and fulfill the Dirichlet boundary conditions.

Since these results require that f satisfies some monotonicity property, it seems interesting to investigate the situation where such assumptions fail. Nevertheless, simple one-dimensional examples, given, e.g., in [37,38], show that the sole existence of a couple of sub- and super-solutions, although yielding the existence of a bounded variation solution, cannot guarantee the existence of a classical one. Therefore, as some supplementary assumption should be added, in this paper we adopt an alternative approach that allows us to remove any monotonicity requirement on f at the expense of imposing a local growth condition, which is independent of the size of the interval [ 0 , R ] and, therefore, completely different from those considered in the literature, and in turn is almost necessary for guaranteeing the regularity of the solutions of the associated Cauchy problems. Namely, we introduce the following assumption:

(h3) there exist an interval I ⊆ ℝ and a continuous function g : I → ℝ such that, for all t ∈ [ 0 , R ] , s ∈ I and ξ ∈ ℝ ,

(1.4) f ( t , s , ξ ) ⋅ sgn ( ξ ) ≥ − g ( s )

and

(1.5) ∫ I g ( s ) d s < 1 .

It is apparent that the condition (1.4) in ( h 3 ) implies g ( s ) ≥ 0 for all s ∈ I .

Remark 1.1

One sample case where ( h 3 ) holds is the following. Suppose that the function f admits the decomposition

f ( t , s , ξ ) = f 1 ( t , s ) + f 2 ( ξ ) ⋅ ξ ,

with f 2 ( ξ ) ≥ 0 for all ξ ∈ ℝ . Then, f satisfies ( h 3 ) provided that f 1 does. This means that no growth restriction with respect to the gradient variable is required here.

It is worth observing that the condition ( h 3 ) guarantees that, for every u 0 ∈ I , any non-extendible solution u of the Cauchy problem:

− t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 f ( t , u , u ′ ) , u ( 0 ) = u 0 , u ′ ( 0 ) = 0 ,

with range u ⊆ I , exists on the whole of [ 0 , R ] . This is a direct consequence of the inequality

1 − 1 1 + u ′ ( t ) 2 = u ′ ( τ ) 2 1 + u ′ ( τ ) 2 t 0 t − ∫ t 0 t u ′ ( τ ) u ″ ( τ ) 1 + u ′ ( τ ) 2 d τ = − ∫ t 0 t N − 1 τ u ′ ( τ ) 2 1 + u ′ ( τ ) 2 d τ − ∫ t 0 t f ( τ , u ( τ ) , u ′ ( τ ) ) u ′ ( τ ) d τ ≤ ∫ t 0 t g ( u ( τ ) ) | u ′ ( τ ) | d τ ≤ ∫ range u g ( s ) d s , for all t ∈ dom u ,

where t 0 ∈ [ 0 , t ] is the critical point of u closest to t.

Conversely, in the one-dimensional case, the condition (1.5) in ( h 3 ) turns out to be necessary for the existence of solutions. Namely, if g : ℝ → ℝ is an arbitrary continuous function, any solution u ∈ C 2 ( [ a , b ] ) of the equation

− u ′ 1 + u ′ 2 ′ = g ( u )

satisfies the identity

1 1 + u ′ ( t 1 ) 2 − 1 1 + u ′ ( t 2 ) 2 = ∫ u ( t 2 ) u ( t 1 ) g ( s ) d s , for all t 1 , t 2 ∈ [ a , b ] ,

and hence, setting I = range u , condition (1.5) must hold.

On the other hand, it is clear that assumptions ( h 1 ) , ( h 2 ) , ( h 3 ) alone do not imply the solvability of the problem (1.1). A very simple example is given by

(1.6) − t N − 1 u ′ 1 + u ′ 2 ′ = N t N − 1 in ] 0 , 1 [ , u ′ ( 0 ) = 0 , a 0 u ( 1 ) + a 1 u ′ ( 1 ) = 0 .

Indeed, integrating the equation on [ 0 , 1 ] shows that (1.6) has no classical solutions although the constant function f ( t , s , ξ ) = N satisfies ( h 3 ) , for any interval I such that | I | < 1 N and for g = N .

The solvability of (1.1) can actually be achieved assuming the existence of couples of sub- and super-solutions, whose definition is given below taking into account the previously set convention: u ′ ( t ) t = u ″ ( 0 ) if t = 0 , for all u ∈ C 2 ( [ 0 , R ] ) such that u ′ ( 0 ) = 0 . We refer to [28,29,35,37,39,40,41] for the concrete construction of sub- and super-solutions in the frame of the mean curvature equation.

Definitions 1.1

(Notion of sub- and super-solution).

  1. A function α : [ 0 , R ] → ℝ is a sub-solution of the problem (1.1) if there exist α 1 , … , α p ∈ C 2 ( [ 0 , R ] ) such that α = max { α 1 , … , α p } and for each i ∈ { 1 , … , p }

    − t N − 1 α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ ≤ t N − 1 f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ ] 0 , R [ , α i ′ ( 0 ) ≥ 0 , a 0 α i ( R ) + a 1 α i ′ ( R ) ≤ 0 .

  2. A sub-solution α is strict if for each i ∈ { 1 , … , p } either

    − t N − 1 α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ < t N − 1 f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ ] 0 , R [ , α i ′ ( 0 ) > 0 , a 0 α i ( R ) + a 1 α i ′ ( R ) < 0 ,

    or

    − α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ − N − 1 t α i ′ ( t ) 1 + α i ′ ( t ) 2 < f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ [ 0 , R [ , α i ′ ( 0 ) = 0 , a 0 α i ( R ) + a 1 α i ′ ( R ) < 0 ,

    or

    − t N − 1 α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ < t N − 1 f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ ] 0 , R ] , α i ′ ( 0 ) > 0 , a 0 α i ( R ) + a 1 α i ′ ( R ) = 0 ,

    or

    − α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ − N − 1 t α i ′ ( t ) 1 + α i ′ ( t ) 2 < f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ [ 0 , R ] , α i ′ ( 0 ) = 0 , a 0 α i ( R ) + a 1 α i ′ ( R ) = 0 .

  3. A function β : [ 0 , R ] → ℝ is a super-solution of the problem (1.1) if there exist β 1 , … , β q ∈ C 2 ( [ 0 , R ] ) such that β = min { β 1 , … , β q } and for each j ∈ { 1 , … , q }

    − t N − 1 β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ ≥ t N − 1 f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ ] 0 , R [ , β i ′ ( 0 ) ≤ 0 , a 0 β j ( R ) + a 1 β j ′ ( R ) ≥ 0 .

  4. A super-solution β is strict if for each j ∈ { 1 , … , q } either

− t N − 1 β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ > t N − 1 f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ ] 0 , R [ , β j ′ ( 0 ) < 0 , a 0 β j ( R ) + a 1 β j ′ ( R ) > 0 ,

or

− β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ − N − 1 t β j ′ ( t ) 1 + β j ′ ( t ) 2 > f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ [ 0 , R [ , β j ′ ( 0 ) = 0 , a 0 β j ( R ) + a 1 β j ′ ( R ) > 0 ,

or

− t N − 1 β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ > t N − 1 f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ ] 0 , R ] , β j ′ ( 0 ) < 0 , a 0 β j ( R ) + a 1 β j ′ ( R ) = 0 ,

or

− β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ − N − 1 t β j ′ ( t ) 1 + β j ′ ( t ) 2 > f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ [ 0 , R ] , β j ′ ( 0 ) = 0 , a 0 β j ( R ) + a 1 β j ′ ( R ) = 0 .

Notation. For all u , v ∈ C 1 ( [ 0 , R ] ) , we write u ≤ v if u ( t ) ≤ v ( t ) for all t ∈ [ 0 , R ] ; u < v if u ≤ v and u ≠ v ; u ≪ v if u ( t ) < v ( t ) for all t ∈ ] 0 , R [ and, in addition, either u ( 0 ) < v ( 0 ) or else u ( 0 ) = v ( 0 ) and u ′ ( 0 ) < v ′ ( 0 ) , as well as either u ( R ) < v ( R ) or else u ( R ) = v ( R ) and u ′ ( R ) > v ′ ( R ) .

The following existence and localization result holds in the presence of one couple of sub- and super-solutions satisfying the standard ordering condition.

Theorem 1.1

Assume ( h 1 ) , ( h 2 ) , ( h 3 ) and

( h 4 ) there exist a sub-solution α and a super-solution β of the problem (1.1) satisfying

α ≤ β a n d range α , range β ⊆ I .

Then, the problem (1.1) has at least one classical solution u, with

α ≤ u ≤ β .

Furthermore, the multiplicity of solutions can be detected in the presence of two couples of sub- and super-solutions, if one of them does not satisfy the standard ordering condition. Indeed, the following version of the celebrated Amann three-solution theorem [13,16,42] holds true for the problem (1.1).

Theorem 1.2

Assume ( h 1 ) , ( h 2 ) , ( h 3 ) and ( h 4 ) . Suppose further that

( h 5 ) there exist a strict sub-solution α 1 and a strict super-solution β 1 of the problem (1.1) satisfying

α ≤ α 1 ≤ β , α ≤ β 1 ≤ β a n d α 1 ≰ β 1 .

Then, the problem (1.1) has at least three classical solutions u 1 , u 2 , u 3 , with

u 1 < u 3 < u 2 , α ≤ u 1 ≪ β 1 , α 1 ≪ u 2 ≤ β , u 3 ≱ α 1 a n d u 3 ≰ β 1 .

Remark 1.2

Although not explicitly stated here, the conclusions of Theorems 1.1 and 1.2 can be completed with the existence of extremal solutions. This easily follows from the adopted notion of sub- and super-solutions arguing like, e.g., in [43].

Theorems 1.1 and 1.2 are the main results obtained in this work and the entire Section 2 is devoted to their proofs. Finally, in Section 3 we provide some additional results for the Neumann and the periodic problems, where some alternative boundedness conditions are imposed on the function f (see ( h 6 ) , or ( h 6 ′ ) , below) that unlike ( h 3 ) involve the length of the interval [ 0 , R ] , or, respectively, the period T. It is clear that the complementary nature of the assumptions introduced in this paper allows us to deal with manifold situations, which can be treated by suitably combining our statements, but are not covered by other results available in the existing literature, such as those in [29,34,37,40,44,45,46,47,48,49].

2 Proofs

In this section, we deliver the proofs of Theorems 1.1 and 1.2. We first introduce a modified version of the problem (1.1), inspired from [16], and show that any solution of the modified problem is actually a solution of the original one. Next, we prove the existence of a solution of the modified problem, thus completing the proof of Theorem 1.1. We treat separately the Neumann case and the Dirichlet and the Robin cases since the proof, although basically the same, requires more care in the former situation. We also provide some information on the topological degree which will be crucial in detecting the existence of multiple solutions. The proof of Theorem 1.2 is then given in the final part of this section, relying on the previous degree calculations. For convenience, we henceforth set

φ ( ξ ) = ξ 1 + ξ 2 for all ξ ∈ ℝ .

2.1 A modified problem

In this section, we introduce a modified version of the problem (1.1) and prove that any solution of the modified problem is actually a solution of the original one. Assume ( h 1 ) , ( h 2 ) , ( h 3 ) and ( h 4 ) . Let us set

(2.1) K = ∫ I g ( s ) d s and L = K ( 2 − K ) 1 − K .

For any M > 0 , we define

(2.2) φ M ( ξ ) = φ ( ξ ) if 0 ≤ ξ ≤ M , φ ( M ) + φ ′ ( M ) ( ξ − M ) if ξ > M , − φ M ( − ξ ) if ξ < 0 .

We also write

Φ M ( ξ ) = ∫ 0 ξ φ M ( σ ) d σ .

Setting

ψ ( ξ ) = φ M ( ξ ) ξ − Φ M ( ξ ) − 1 + 1 1 + ξ 2

and observing that ψ is even, ψ ( 0 ) = 0 and ψ ′ ( ξ ) ≥ 0 for all ξ ≥ 0 , we see that

(2.3) φ M ( ξ ) ξ − Φ M ( ξ ) ≥ 1 − 1 1 + ξ 2 for all ξ ∈ ℝ .

Take L ˜ ∈ ℝ such that

(2.4) L ˜ > L + max { ∥ α i ′ ∥ ∞ , ∥ β j ′ ∥ ∞ : i = 1 , … , p , j = 1 , … , q } .

We introduce the following truncation of the function f:

(2.5) h ( t , s , ξ ) = f ( t , s , − L ˜ ) if ξ < − L ˜ , f ( t , s , ξ ) if | ξ | ≤ L ˜ , f ( t , s , L ˜ ) if ξ > L ˜ .

We set, for each i = 1 , … , p ,

γ i ( t , s ) = α i ( t ) if s < α i ( t ) , s if s ≥ α i ( t )

and, for each j = 1 , … , q ,

δ j ( t , s ) = β j ( t ) if s > β j ( t ) , s if s ≤ β j ( t ) .

Define, for each i = 1 , … , p , the operator ℋ i ̲ : C 1 ( [ 0 , R ] ) → L ∞ ( 0 , R ) by

ℋ i ̲ ( u ) ( t ) = h t , γ i ( t , u ( t ) ) , d d t γ i ( t , u ( t ) ) for a .e . t ∈ [ 0 , R ]

and, for each j = 1 , … , q , the operator ℋ j ¯ : C 1 ( [ 0 , R ] ) → L ∞ ( 0 , R ) by

ℋ j ¯ ( u ) ( t ) = h t , δ j ( t , u ( t ) ) , d d t δ j ( t , u ( t ) ) for a .e . t ∈ [ 0 , R ] .

Finally, we introduce the operator ℋ : C 1 ( [ 0 , R ] ) → L ∞ ( 0 , R ) by setting, for all u ∈ C 1 ( [ 0 , R ] ) and a.e. t ∈ [ 0 , R ] ,

(2.6) ℋ ( u ) ( t ) = max i = 1 , … , p ℋ i ̲ ( u ) ( t ) + arctan ( α ( t ) − u ( t ) ) if u ( t ) < α ( t ) , h ( t , u ( t ) , u ′ ( t ) ) if α ( t ) ≤ u ( t ) ≤ β ( t ) , min j = 1 , … , q ℋ j ¯ ( u ) ( t ) − arctan ( u ( t ) − β ( t ) ) if u ( t ) > β ( t ) .

Let us consider the modified problem

(2.7) − ( φ M ( u ′ ) t N − 1 ) ′ = t N − 1 ℋ ( u ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 .

By a solution of (2.7), we mean a function u ∈ W 2 , ∞ ( 0 , R ) which satisfies the equation a.e. in ] 0 , R [ as well as the boundary conditions.

We prove that, taking M > L ˜ , any solution u of (2.7) is a solution of (1.1) too. We first verify that u ≥ α by showing that u ≥ α i for each i ∈ { 1 , … , p } . Let us fix i ∈ { 1 , … , p } . By the monotonicity of φ M we have that

0 ≤ ∫ { t : u ( t ) < α i ( t ) } t N − 1 ( φ M ( u ′ ) − φ M ( α i ′ ) ) ( u ′ − α i ′ ) d t = − ∫ 0 R t N − 1 ( φ M ( u ′ ) − φ M ( α i ′ ) ) ( ( u − α i ) − ) ′ d t .

Integrating by parts and observing that, from Definitions 1.1 and from (2.7),

[ ( t N − 1 ( φ M ( u ′ ( t ) ) − φ M ( α i ′ ( t ) ) ) ) ( u ( t ) − α i ( t ) ) − ] 0 R ≤ 0 ,

we get

(2.8) 0 ≤ ∫ 0 R ( t N − 1 ( φ M ( u ′ ) − φ M ( α i ′ ) ) ) ′ ( u − α i ) − d t .

Note that, as M > L ˜ , α i satisfies the inequality

(2.9) − ( φ M ( α i ′ ) t N − 1 ) ′ ≤ t N − 1 f ( t , α i , α i ′ ) in ] 0 , R [ .

Hence we obtain, from (2.8), (2.7), (2.9) and from the definitions of ℋ and h,

0 ≤ ∫ 0 R t N − 1 ( − ℋ ( u ) + f ( t , α i , α i ′ ) ) ( u − α i ) − d t ≤ ∫ 0 R t N − 1 ( − ℋ i ̲ ( u ) − arctan ( α − u ) + f ( t , α i , α i ′ ) ) ( u − α i ) − d t = ∫ { t : u ( t ) < α i ( t ) } t N − 1 h t , γ i ( t , u ) , d d t γ i ( t , u ) + arctan ( α − u ) − f ( t , α i , α i ′ ) ( u − α i ) d t = ∫ { t : u ( t ) < α i ( t ) } t N − 1 arctan ( α − u ) ( u − α i ) d t ≤ ∫ { t : u ( t ) < α i ( t ) } t N − 1 arctan ( α i − u ) ( u − α i ) d t ≤ 0 .

Thus, we infer that ( u − α i ) − = 0 , that is, u ≥ α i . Since this is valid for each i ∈ { 1 , … , p } , we conclude that u ≥ α . In a symmetric way, we verify that u ≤ β . Therefore, u is a solution of the problem:

(2.10) − ( φ M ( u ′ ) t N − 1 ) ′ = t N − 1 h ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 .

To prove that u is a solution of (1.1) it remains to show that ∥ u ′ ∥ ∞ ≤ L ˜ . Assume first that max [ 0 , R ] u ′ = u ′ ( t 1 ) > 0 for some t 1 ∈ ] 0 , R ] and take t 0 = max { t ∈ [ 0 , t 1 ] : u ′ ( t ) = 0 } . We have u ′ ( t ) > 0 for all t ∈ ] t 0 , t 1 ] and hence, using also (2.3),

1 − 1 1 + u ′ ( t 1 ) 2 ≤ φ M ( u ′ ( t 1 ) ) u ′ ( t 1 ) − Φ M ( u ′ ( t 1 ) ) = φ M ( u ′ ( t ) ) u ′ ( t ) t 0 t 1 − ∫ t 0 t 1 φ M ( u ′ ( t ) ) u ″ ( t ) d t = ( 1 − N ) ∫ t 0 t 1 u ′ ( t ) t φ M ( u ′ ( t ) ) d t − ∫ t 0 t 1 h ( t , u ( t ) , u ′ ( t ) ) u ′ ( t ) d t ≤ ∫ t 0 t 1 g ( u ( t ) ) u ′ ( t ) d t = ∫ u ( t 0 ) u ( t 1 ) g ( s ) d s ≤ K .

Recalling (2.1) we find that u ′ ( t 1 ) ≤ L < L ˜ . Similarly, we show that if min [ 0 , R ] u ′ = u ′ ( t 1 ) < 0 for some t 1 ∈ ] 0 , R ] , then we have | u ′ ( t 1 ) | ≤ L < L ˜ .

2.2 Existence of a solution

In this section, we prove the existence of a solution of the modified problem (2.7) for any given M > 0 , treating separately the Neumann case and the Dirichlet and Robin cases.

2.2.1 The Neumann problem

Assume a 0 = 0 in (2.7), i.e., consider the Neumann problem:

(2.11) − ( φ M ( u ′ ) t N − 1 ) ′ = t N − 1 ℋ ( u ) in ] 0 , R [ , u ′ ( 0 ) = 0 , u ′ ( R ) = 0 ,

with ℋ defined by (2.6).

Existence of solutions

Set

X = { u ∈ C 1 ( [ 0 , R ] ) : u ′ ( 0 ) = 0 , u ′ ( R ) = 0 }

and

X ˜ = { w ∈ X : w ( R ) = 0 } .

We endow X with the norm

(2.12) ∥ u ∥ X = | u ( R ) | + ∥ u ′ ∥ ∞ .

It is clear that, with this norm, X is a Banach space and X ˜ is a closed subspace of X. As any u ∈ X admits a unique representation of the form u = r + w , with r = u ( R ) ∈ ℝ and w ∈ X ˜ , it follows that X = ℝ ⊕ X ˜ , algebraically and topologically. Define the operators:

P : L ∞ ( 0 , R ) → ℝ by

P v = N R − N ∫ 0 R t N − 1 v ( t ) d t ,

T ˜ : X → X ˜ by

T ˜ ( u ) ( t ) = ∫ t R φ M − 1 ∫ 0 s σ s N − 1 ℋ ( u ) ( σ ) − P ℋ ( u ) d σ d s for all t ∈ [ 0 , R ] ,

and T : X → X by

T ( r + w ) = r + P ℋ ( r + w ) + T ˜ ( r + w ) .

It is easy to verify that u = r + w ∈ X is a fixed point of T if and only if u is a solution of (2.11). Compute, for a.e. t ∈ [ 0 , R ] ,

d d t T ˜ ( u ) ( t ) = − φ M − 1 ∫ 0 t σ t N − 1 ( ℋ ( u ) ( σ ) − P ℋ ( u ) ) d σ

and

d 2 d t 2 T ˜ ( u ) ( t ) = − ( φ M − 1 ) ′ ∫ 0 t σ t N − 1 ( ℋ ( u ) ( σ ) − P ℋ ( u ) ) d σ ⋅ ( 1 − N ) t − N ∫ 0 t σ N − 1 ( ℋ ( u ) ( σ ) − P ℋ ( u ) ) d σ + ( ℋ ( u ) ( t ) − P ℋ ( u ) ) .

Since the operator ℋ : C 1 ( [ 0 , R ] ) → L ∞ ( 0 , R ) is continuous and has a bounded range, the operator P : L ∞ ( 0 , R ) → ℝ is linear and continuous, and the function φ M − 1 : ℝ → ℝ is of class C 1 and has a bounded derivative, it follows that T ˜ : X → X ˜ is continuous and there exists a constant C > 0 such that

∥ T ˜ ( u ) ∥ W 2 , ∞ < C for all u ∈ X .

For later use, it is convenient to assume further that C > L ˜ , where L ˜ is the constant introduced in (2.4). By the Ascoli-Arzelà theorem, T ˜ has a relatively compact range in X ˜ , i.e., range T ˜ ¯ is compact in X ˜ . Consequently, the operator T : X → X is completely continuous, i.e., T is continuous and maps bounded subsets of X onto relatively compact subsets of X.

Choose a constant

D > max i = 1 , … , p ∥ α i ∥ ∞ + max j = 1 , … , q ∥ β j ∥ ∞ + R C

and define in X the open bounded set

(2.13) O = { u = r + w ∈ X : | r | < D and ∥ w ∥ X < C } .

We prove that, for all λ ∈ [0, 1] , any u ∈ O ¯ such that

u = λ T ( u ) ,

or equivalently

w = λ T ˜ ( u ) and ( 1 − λ ) r = λ P ℋ ( u ) ,

satisfies u ∈ O . Since, for all λ ∈ [ 0 , 1 ] , we have that

∥ w ∥ X = ∥ λ T ˜ ( u ) ∥ X ≤ ∥ T ˜ ( u ) ∥ X < C ,

it is enough to show that | r | < D . Assume, by contradiction, that there is λ ∈ ] 0 , 1 ] such that, e.g., r ≤ − D . We have, for all t ∈ [ 0 , R ] ,

u ( t ) = r + w ( t ) ≤ − D + ∥ w ∥ ∞ < − max i = 1 , … , p ∥ α i ∥ ∞ − max j = 1 , … , q ∥ β j ∥ ∞ − R C + R C ≤ min i = 1 , … , p min [ 0 , R ] α i

and hence

R N N P ℋ ( u ) = ∫ 0 R t N − 1 ℋ ( u ) ( t ) d t = ∫ 0 R t N − 1 max i = 1 , … , p ℋ i ̲ ( u ) ( t ) + arctan ( α ( t ) − u ( t ) ) d t > ∫ 0 R t N − 1 max i = 1 , … , p ℋ i ̲ ( u ) ( t ) d t .

Thus we get, for each i = 1 , … , p ,

R N N P ℋ ( u ) > ∫ 0 R t N − 1 h t , γ i ( t , u ( t ) ) , d d t γ i ( t , u ( t ) ) d t = ∫ 0 R t N − 1 f ( t , α i ( t ) , α i ′ ( t ) ) d t ≥ − ∫ 0 R ( t N − 1 φ M ( α i ′ ( t ) ) ) ′ d t = − R N − 1 φ M ( α i ′ ( R ) ≥ 0 ,

as α i satisfies (2.9) and α i ′ ( R ) ≤ 0 , thus yielding the contradiction

0 ≥ ( 1 − λ ) r = λ P ℋ ( u ) > 0 .

Similarly, we can verify that r < D . Consequently, the homotopy invariance of the degree implies that

deg ( ℐ − T , O ) = 1 .

In particular, T has a fixed point u ∈ X which in turn is a solution of (2.11).

Degree calculations

Assume now that the sub-solution α and the super-solution β are strict. We first show that if u is a solution of (1.1) with α ≤ u ≤ β , then u satisfies α ≪ u ≪ β . We already know that α ≤ u ≤ β and, in particular, we have φ M ( u ′ ) = φ ( u ′ ) as M > L ˜ > ∥ u ′ ∥ ∞ . We shall limit ourselves to prove that u satisfies u ≫ α i for each i ∈ { 1 , … , p } , as showing that u ≪ β j for each j ∈ { 1 , … , q } is totally similar. Fix i ∈ { 1 , … , p } and set v = u − α i . Let us suppose, by contradiction, that . Hence, there exists t 0 ∈ [ 0 , R ] such that v ( t 0 ) = min v = 0 . By definition of sub-solution, we immediately realize that v ′ ( t 0 ) = 0 and thus α i ( t 0 ) = u ( t 0 ) and α i ′ ( t 0 ) = u ′ ( t 0 ) . In particular, if t 0 = 0 or t 0 = R , the definition of solution yields α i ′ ( t 0 ) = 0 . Recalling now the definition of strict sub-solution we observe that, if t 0 = R , only the third or the fourth alternative in Definition 1.1 may occur. Hence, if 0 < t 0 ≤ R , we get

− φ M ′ ( u ′ ( t 0 ) ) v ″ ( t 0 ) = − φ M ′ ( u ′ ( t 0 ) ) u ″ ( t 0 ) − N − 1 t 0 φ M ( u ′ ( t 0 ) ) + φ M ′ ( α i ′ ( t 0 ) ) α i ″ ( t 0 ) + N − 1 t 0 φ M ( α i ′ ( t 0 ) ) > f ( t 0 , u ( t 0 ) , u ′ ( t 0 ) ) − f ( t 0 , α i ( t 0 ) , α i ′ ( t 0 ) ) = 0 .

On the other hand, if t 0 = 0 , only the second or the fourth alternative in Definition 1.1 may occur and we find

− N v ″ ( 0 ) = − N φ M ′ ( 0 ) v ″ ( 0 ) > f ( 0 , u ( 0 ) , 0 ) − f ( 0 , α i ( 0 ) , 0 ) = 0 .

In all cases we infer v ″ ( t 0 ) < 0 , but this is impossible at a minimum point which is simultaneously a critical point.

Next we define in X the open bounded set

U = { u ∈ X : α ≪ u ≪ β , ∥ u ′ ∥ ∞ < C } .

It is clear that U ⊆ O , where O is defined in (2.13). Since any fixed point u of T is a solution of (2.11), it satisfies

α ≪ u ≪ β , ∥ u ′ ∥ ∞ < C ,

and hence O \ U ¯ is fixed point free. The excision property of the degree thus yields

deg ( ℐ − T , U ) = deg ( ℐ − T , O ) = 1 .

2.2.2 The Dirichlet and the Robin problems

Assume now a 0 > 0 in (2.7), i.e., consider either the Dirichlet problem

− ( φ M ( u ′ ) t N − 1 ) ′ = t N − 1 ℋ ( u ) in ] 0 , R [ , u ′ ( 0 ) = 0 , u ( R ) = 0 ,

in case a 1 = 0 , or the Robin problem

− ( φ M ( u ′ ) t N − 1 ) ′ = t N − 1 ℋ ( u ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 ,

in case a 1 > 0 .

Existence of solutions

Much like in the Neumann case, we set

X = { u ∈ C 1 ( [ 0 , R ] ) : u ′ ( 0 ) = 0 }

and we endow X with the norm defined by (2.12). We also introduce the operator T : X → X by setting

(77) T ( u ) ( t ) = a 1 a 0 φ M − 1 ∫ 0 R σ R N − 1 ℋ ( u ) ( σ ) d σ + ∫ t R φ M − 1 ∫ 0 s σ s N − 1 ℋ ( u ) ( σ ) d σ d s for all t ∈ [ 0 , R ] .

The fixed points u ∈ X of T are precisely the solutions of (2.7). It is easy to see that T is completely continuous and there exists a constant C > L ˜ such that

∥ T ( u ) ∥ W 2 , ∞ < C , for all u ∈ X .

Define in X the open ball

O = { u ∈ X : ∥ u ∥ X < C } .

Since T ( O ¯ ) ⊆ O , the Schauder fixed point theorem implies the existence of a fixed point u ∈ O , which in turn is a solution of (2.7). Moreover, a simple homotopy argument yields

deg ( I − T , O ) = 1 .

Degree calculations

We first show that if u is a solution of (1.1) with α ≤ u ≤ β , then u satisfies α ≪ u ≪ β , keeping the notation of Section 2.2.1. From the definition of strict sub-solution, we infer that v ( t ) > 0 for all t ∈ [ 0 , R [ . Assume that v ( R ) = 0 . Since v ( R ) = min v , we get either v ′ ( R ) = 0 or v ′ ( R ) < 0 . In the former situation, a contradiction follows arguing as we did in the Neumann case. In the latter situation, we automatically have that v ≫ 0 . The rest of the proof then proceeds exactly in the same way as for the Neumann problem.

2.3 Multiplicity of solutions

2.3.1 Degree calculations

In this section, we consider the Neumann case and the Dirichlet and Robin cases simultaneously. In what follows, the symbols X, C, T refer to the symbols introduced in Section 2.2.1, in the case of the Neumann problem, or in Section 2.2.2, in the case of the Dirichlet and the Robin problems. Let us consider the modified problem (2.7). Since all solutions u of (2.7) satisfy α ≤ u ≤ β , it is clear that α − 1 ≪ u ≪ β + 1 . Let us introduce the following open bounded subsets of X

U 0 = { u ∈ X : α − 1 ≪ u ≪ β + 1 and ∥ u ′ ∥ ∞ < C } , U 1 = { u ∈ X : α − 1 ≪ u ≪ β 1 and ∥ u ′ ∥ ∞ < C } , U 2 = { u ∈ X : α 1 ≪ u ≪ β + 1 and ∥ u ′ ∥ ∞ < C } .

Note that U 1 ⊂ U 0 , U 2 ⊂ U 0 , and, as α 1 ≰ β 1 , U 1 ∩ U 2 = ∅ . Let u ∈ X be a fixed point of T . Then it satisfies α − 1 ≪ u ≪ β + 1 . Suppose further that u ≤ β 1 . Since β 1 is a strict super-solution of the problem (1.1), we actually have u ≪ β 1 , as shown in Sections 2.2.1 and 2.2.2 when computing the degree. Similarly, if u ≥ α 1 , since α 1 is a strict sub-solution of the problem (1.1), we actually have u ≫ α 1 . Therefore, we conclude that

(2.14) 0 ∉ ( ℐ − T ) ( ∂ U 0 ∪ ∂ U 1 ∪ ∂ U 2 ) .

Define now the open bounded subset of X

U 3 = U 0 \ U 1 ∪ U 2 ¯ .

By (2.14), using the excision property of the degree, we get

deg ( ℐ − T , U 0 ) = deg ( ℐ − T , U 0 \ ( ∂ U 1 ∪ ∂ U 2 ) )

and, hence, the additivity property of the degree implies that

deg ( ℐ − T , U 0 ) = deg ( ℐ − T , U 1 ) + deg ( ℐ − T , U 2 ) + deg ( ℐ − T , U 3 ) .

Since we have that

deg ( ℐ − T , U 0 ) = deg ( ℐ − T , U 1 ) = deg ( ℐ − T , U 2 ) = 1 ,

we finally get

deg ( ℐ − T , U 3 ) = − 1 .

2.3.2 Existence of solutions

Since U 1 , U 2 , U 3 are mutually disjoint, the previous degree calculations imply that there are three distinct fixed points u 1 , u 2 , u 3 of the operator T , with

u 1 ∈ U 1 , u 2 ∈ U 2 , u 3 ∈ U 3 .

Since any fixed point u of T satisfies α ≤ u ≤ β , u 1 , u 2 , u 3 are all solutions of (1.1). Moreover, we have that

α ≤ u 1 ≪ β 1 , α 1 ≪ u 2 ≤ β , u 3 ≱ α 1 , u 3 ≰ β 1 .

Since min { u 3 , β 1 } is a super-solution of (1.1), there exists a solution u ˜ 1 of (1.1) satisfying

α ≤ u ˜ 1 ≤ min { u 3 , β 1 } < u 3 .

Similarly, as max { u 3 , α 1 } is a sub-solution of (1.1), there exists a solution u ˜ 2 of (1.1) satisfying

u 3 < max { u 3 , α 1 } ≤ u ˜ 2 ≤ β .

Then, replacing u 1 with u ˜ 1 and u 2 with u ˜ 2 , we can conclude that (1.1) has three solutions such that

u 1 < u 3 < u 2 .

3 Complementary results

In this section, we provide some additional results, complementing Theorems 1.1 and 1.2, where alternative bounds are imposed on the function f, involving, unlike assumption ( h 3 ) , the length of the interval [ 0 , R ] . We shall mainly focus on the Neumann problem as, in the case of Dirichlet or Robin boundary conditions, a suitable boundedness assumption on f is enough to guarantee alone the existence of solutions. In the last part, we briefly discuss also the periodic problem, which is closely related to the one-dimensional Neumann problem.

3.1 The Neumann problem

In this section, we consider the Neumann problem:

(3.1) − t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 f ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , u ′ ( R ) = 0 .

It has already been observed in [46, Lemma 2.5] that a global bound on f yields an estimate on the derivative of any solution of (3.1) and eventually, in [46, Theorem 2.6], the solvability of (3.1) if an asymptotic sign condition holds. The next result provides instead the existence of a solution of (3.1) assuming a local bound on f with a sharper constant, together with the existence of a couple of sub- and super-solutions satisfying the standard ordering condition. In the light of the non-existence results in [50] the bound on the function f is optimal in dimension N = 1 .

Proposition 3.1

Assume ( h 1 ) , ( h 2 )

( h 6 ) there exist an interval I ⊆ ℝ and a constant K, with

0 < K < N 2 N R ,

such that, for all t ∈ [ 0 , R ] , s ∈ I and ξ ∈ ℝ ,

(3.2) | f ( t , s , ξ ) | ≤ K

and

( h 7 ) there exist a sub-solution α and a super-solution β of the problem (3.1) satisfying

α ≤ β a n d range α , range β ⊆ I .

Then, the problem (3.1) has at least one classical solution u, with

α ≤ u ≤ β .

Proof

Suppose u is a solution of the problem (3.1) with range u ⊆ I . If t ∈ ] 0 , R 2 N ] , integrating the equation in (3.1) between 0 and t, we find

| φ ( u ′ ( t ) ) | ≤ t 1 − N ∫ 0 t s N − 1 | f ( s , u ( s ) , u ′ ( s ) ) | d s ≤ K t N ≤ K R N 2 N < 1 ,

whereas, if t ∈ ] R 2 N , R [ , integrating between t and R, we get

| φ ( u ′ ( t ) ) | ≤ t 1 − N ∫ t R s N − 1 | f ( s , u ( s ) , u ′ ( s ) ) | d s ≤ K R N − t N N t N − 1 ≤ K R N 2 N < 1 .

Set

(3.3) L = φ − 1 K R N 2 N .

Then any solution u of the problem (3.1), with range u ⊆ I , satisfies the estimate

(3.4) ∥ u ′ ∥ ∞ ≤ L .

Next we proceed as in the proof of Theorem 1.1, by introducing, for any M > 0 , the function φ M as in (2.2), the constant L ˜ as in (2.4), the truncated function h as in (2.5), the operator ℋ as in (2.6) and the modified problems (2.10) and (2.7) with a 0 = 0 . As in that proof we show that a solution u of the modified problem (2.7) exists and is a solution of the problem (2.10) too. To conclude we observe that the function h satisfies assumption ( h 6 ) and therefore, arguing as in the first part of this proof, the estimate (3.4) holds. Hence, u is a solution of (3.1), provided that M > L ˜ .□

In the following result, we consider the case where α and β do not satisfy the standard ordering condition. In this situation, we need to ask the bound in (3.2) to be global, in the sense that I = ℝ .

Proposition 3.2

Assume ( h 1 ) , ( h 2 ) , ( h 6 ) with I = ℝ , and

( h 8 ) there exist a sub-solution α and a super-solution β of the problem (3.1) satisfying

α ≰ β .

Then, the problem (3.1) has at least one classical solution u, with

u ( t 1 ) ≤ α ( t 1 ) a n d u ( t 2 ) ≥ β ( t 2 ) f o r s o m e t 1 , t 2 ∈ [ 0 , R ] .

Proof

We define L as in (3.3) and

D = max { ∥ α i ∥ C 1 , ∥ β j ∥ C 1 : i = 1 , … , p , j = 1 , … , q } + L ( 1 + R ) .

As noted in the proof of Proposition 3.1, condition ( h 6 ) implies that any solution u of the problem (3.1) satisfies (3.4). Therefore, all solutions u of (3.1) such that u ( t 1 ) ≤ α ( t 1 ) and u ( t 2 ) ≥ β ( t 2 ) , for some t 1 , t 2 ∈ [ 0 , R ] , satisfy

(3.5) ∥ u ∥ C 1 ≤ ∥ α ∥ ∞ + ∥ β ∥ ∞ + ( 1 + R ) ∥ u ′ ∥ ∞ ≤ D .

Choose L ˜ > D and define a continuous function η : ℝ 2 → [ 0 , 1 ] satisfying

η ( s , ξ ) = 1 if ( s , ξ ) ∈ [ − D , D ] 2 , 0 if ( s , ξ ) ∈ ℝ 2 \ [ − L ˜ , L ˜ ] 2 .

Choose any ε , with 0 < ε < 2 π K , and define the function h : [ 0 , R ] × ℝ 2 → ℝ by

h ( t , s , ξ ) = η ( s , ξ ) f ( t , s , ξ ) − ε arctan + ( s − L ˜ ) + ε arctan − ( s + L ˜ ) .

Consider the Neumann problem

(3.6) − t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 h ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , u ′ ( R ) = 0 .

Since for all t ∈ [ 0 , R ]

h ( t , α ( t ) , α ′ ( t ) ) = f ( t , α ( t ) , α ′ ( t ) ) and h ( t , β ( t ) , β ′ ( t ) ) = f ( t , β ( t ) , β ′ ( t ) ) ,

α is a sub-solution and β is a super-solution of the problem (3.6). Furthermore, as for all t ∈ [ 0 , R ] :

h ( t , − L ˜ − 1 , 0 ) = ε π 4 and h ( t , L ˜ + 1 , 0 ) = − ε π 4 ,

the constant function α 0 = − L ˜ − 1 is a strict sub-solution and the constant function β 0 = L ˜ + 1 is a strict super-solution of the problem (3.6). We have that

α 0 ≤ α ≤ β 0 , α 0 ≤ β ≤ β 0 and α ≰ β .

Let us distinguish three possible cases.

We first suppose that there exists a solution u of (3.6) such that u ≥ α but . In this case, there exists t 1 ∈ [ 0 , R ] with u ( t 1 ) = α ( t 1 ) . As α ≰ β there exists also t 2 ∈ [ 0 , R ] with u ( t 2 ) ≥ β ( t 2 ) .

Next, we suppose that there exists a solution u of (3.6) such that u ≤ β but . In this case, there exists t 2 ∈ [ 0 , R ] with u ( t 2 ) = β ( t 2 ) . As α ≰ β there exists also t 1 ∈ [ 0 , R ] with u ( t 1 ) ≤ α ( t 1 ) .

Finally, we suppose that, for any solution u of (3.6), if u ≥ α , then u ≫ α , and if u ≤ β , then u ≪ β . In this case, degree calculations similar to those performed in Sections 2.2 and 2.3 yield the existence of a solution u of (3.6) such that u ≱ α and u ≰ β , that is, u ( t 1 ) < α ( t 1 ) and u ( t 2 ) > β ( t 2 ) for some t 1 , t 2 ∈ [ 0 , R ] .

To conclude we observe that the function h satisfies assumption ( h 6 ) with I = ℝ and, therefore, arguing as in the first part of this proof, the estimate (3.5) holds and, hence, u is a solution of (3.1) as well.□

3.2 The Dirichlet and the Robin problems

In this section, we consider the Dirichlet and the Robin problems:

(3.7) − t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 f ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 ,

with a 0 > 0 . In this case, if f satisfies a bound similar to (3.2), which yields an estimate on the derivative, the solvability of (3.7) is always guaranteed, without requiring the existence of sub- and super-solutions.

Proposition 3.3

Assume ( h 1 ) , with a 0 > 0 , ( h 2 ) and

( h 9 ) there exists a constant K, with

0 < K < N R

such that, for all t ∈ [ 0 , R ] , s ∈ ℝ and ξ ∈ ℝ ,

| f ( t , s , ξ ) | ≤ K .

Then, the problem (3.7) has at least one classical solution u.

Proof

Let us consider, depending on λ ∈ [ 0 , 1 ] , the problem

(3.8) − t N − 1 u ′ 1 + u ′ 2 ′ = t N − 1 λ f ( t , u , u ′ ) in ] 0 , R [ , u ′ ( 0 ) = 0 , a 0 u ( R ) + a 1 u ′ ( R ) = 0 ,

with a 0 > 0 . Suppose u is a solution of (3.8) for some λ ∈ [ 0 , 1 ] . As in the proof of Proposition 3.1, integrating the equation in (3.8) between 0 and any t ∈ ] 0 , R ] , we find

| φ ( u ′ ( t ) ) | ≤ K R N < 1 .

Using the boundary condition we then infer the existence of a constant C > 0 such that

∥ u ∥ C 1 < C ,

for any such u and λ . A simple degree argument then yields the existence of a solution of the problem (3.7).□

3.3 The periodic problem

We discuss in this section the existence of classical periodic solutions of the equation:

(3.9) − u ′ 1 + u ′ 2 ′ = f ( t , u , u ′ ) .

For the sake of simplicity and to point out the symmetry with the previous sections, we shall denote the assumptions considered in this part with the symbols ( h i ′ ) , i = 1 , 2 , 3 … . It is assumed throughout this section that

  • ( h 1 ′ ) T > 0 is a given period

    and

  • ( h 2 ′ ) f : ℝ × ℝ × ℝ → ℝ is continuous and T-periodic with respect to the first variable.

Definition 3.1

(Notion of solution) A classical T-periodic solution of (3.9) is a function u ∈ C 2 ( ℝ ) , which is T-periodic and satisfies the equation for all t ∈ ℝ .

We note that, if u ∈ C 1 ( ℝ ) is any T-periodic function, then there exists t 0 ∈ [ 0 , T [ such that u ′ ( t 0 ) = 0 . Therefore, if u is any T-periodic solution of (3.9), then, for some t 0 , u satisfies the Neumann boundary conditions u ′ ( t 0 ) = u ′ ( t 0 + T ) = 0 . Consequently, all gradient estimates established in the previous sections for the one-dimensional Neumann problem extend to the periodic case. Taking this fact into account, we omit the proofs of the following propositions, as they parallel those of Theorems 1.1, 1.2 and Propositions 3.1, 3.2. Before stating our results we need to introduce the notion of sub-solution and super-solution for the periodic problem.

Definitions 3.1

(Notion of sub- and super-solution).

  1. A function α : [ 0 , T ] → ℝ is a sub-solution of the T-periodic problem for (3.9) if there exist α 1 , … , α p ∈ C 2 ( [ 0 , T ] ) such that α = max { α 1 , … , α p } and for each i ∈ { 1 , … , p }

    − α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ ≤ f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ ]0, T [ , α i (0) = α i ( T ) , α i ′ (0) ≥ α i ′ ( T ) .

  2. A sub-solution α is strict if for each i ∈ { 1 , … , p } either

    − α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ < f ( t , α i , ( t ) α i ′ ( t ) ) for all t ∈ ] 0 , T [ , α i ′ ( 0 ) > α i ′ ( T )

    or

    − α i ′ ( t ) 1 + α i ′ ( t ) 2 ′ < f ( t , α i ( t ) , α i ′ ( t ) ) for all t ∈ [ 0 , T ] .

  3. A function β : [ 0 , T ] → ℝ is a super-solution of the T-periodic problem for (3.9) if there exist β 1 , … , β q ∈ C 2 ( [ 0 , T ] ) such that β = min { β 1 , … , β q } and for each j ∈ { 1 , … , q }

    − β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ ≥ f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ ] 0 , T [ , β i ( 0 ) = β i ( T ) , β i ′ ( 0 ) ≤ β i ′ ( T ) .

  4. A super-solution β is strict if for each j ∈ { 1 , … , q } either

− β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ > f ( t , β j ( t ) , β j ′ ( t ) ) for all t ∈ ] 0 , T [ , β i ′ ( 0 ) < β i ′ ( T )

or

− β j ′ ( t ) 1 + β j ′ ( t ) 2 ′ > f ( t , β j , ( t ) β j ′ ( t ) ) for all t ∈ [ 0 , T ] .

Notation. For all u , v ∈ C 1 ( [ 0 , T ] ) satisfying u ( 0 ) = u ( T ) and v ( 0 ) = v ( T ) , we write u ≤ v if u ( t ) ≤ v ( t ) for all t ∈ [ 0 , T ] , u < v if u ≤ v and u ≠ v , u ≪ v if u ( t ) < v ( t ) for all t ∈ [ 0 , T ] .

Proposition 3.4

Assume ( h 1 ′ ) , ( h 2 ′ ) and

( h 3 ′ ) there exist an interval I ⊆ ℝ and a continuous function g : I → ℝ such that, for all t ∈ ℝ , s ∈ I and ξ ∈ ℝ ,

f ( t , s , ξ ) ⋅ sgn ( ξ ) ≥ − g ( s ) a n d ∫ I g ( s ) d s < 1

and

( h 4 ′ ) there exist a sub-solution α and a super-solution β of the T -periodic problem for (3.9) satisfying

α ≤ β a n d range α , range β ⊆ I .

Then, Eq. (3.9) has at least one classical T-periodic solution u, with

α ≤ u ≤ β .

Proposition 3.5

Assume ( h 1 ′ ) , ( h 2 ′ ) , ( h 3 ′ ) , ( h 4 ′ ) and

( h 5 ′ ) there exist a strict sub-solution α 1 and a strict super-solution β 1 of the T- periodic problem for (3.9) satisfying

α ≤ α 1 , β 1 ≤ β a n d α 1 ≰ β 1 .

Then, Eq. (3.9) has at least three classical T-periodic solutions u 1 , u 2 , u 3 , with

u 1 < u 3 < u 2 , α ≤ u 1 ≪ β 1 , α 1 ≪ u 2 ≤ β 1 , u 3 ≱ α 1 a n d u 3 ≰ β 1 .

Proposition 3.6

Assume ( h 1 ′ ) , ( h 2 ′ ) ,

( h 6 ′ ) there exist an interval I ⊆ ℝ and a constant K, with

0 < K < 2 T ,

such that, for all t ∈ [ 0 , R ] , s ∈ I and ξ ∈ ℝ ,

| f ( t , s , ξ ) | ≤ K

and

( h 7 ′ ) there exist a sub-solution α and a super-solution β of the T - periodic problem for (3.9) satisfying

α ≤ β a n d range α , range β ⊆ I .

Then, Eq. (3.9) has at least one classical T-periodic solution u, with

α ≤ u ≤ β .

Proposition 3.7

Assume ( h 1 ′ ) , ( h 2 ′ ) , ( h 6 ′ ) , with I = ℝ , and

( h 8 ′ ) there exist a sub-solution α and a super-solution β of the T - periodic problem for (3.9) satisfying

α ≰ β .

Then, Eq. (3.9) has at least one classical T-periodic solution u, with

u ( t 1 ) ≤ α ( t 1 ) a n d u ( t 2 ) ≥ β ( t 2 ) f o r s o m e t 1 , t 2 ∈ [ 0 , R ] .

Propositions 3.4–3.7, also in combination with Remark 1.1, complete and extend several results previously obtained in [29,37,40,44,45,48,49].

Acknowledgments

This research has been performed under the auspices of “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni – Istituto Nazionale di Alta Matematica.” The authors have been supported by “Università degli Studi di Trieste – Finanziamento di Ateneo per Progetti di Ricerca Scientifica – FRA 2018, FRA 2020.”

References

[1] C. Corsato, C. De Coster, N. Flora, and P. Omari, Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry, Nonlinear Anal. 181 (2019), 9–23.10.1016/j.na.2018.11.001Search in Google Scholar

[2] R. Finn, Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature, J. Analyse Math. 14 (1965), 139–160.10.1007/BF02806384Search in Google Scholar

[3] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math. 229 (1968), 170–187.Search in Google Scholar

[4] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496.10.1098/rsta.1969.0033Search in Google Scholar

[5] O. A. Ladyzhenskaya and N. N. Ural’tseva, Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pure Appl. Math. 23 (1970), 677–703.10.1002/cpa.3160230409Search in Google Scholar

[6] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268.10.1007/BF01404309Search in Google Scholar

[7] E. Bombieri, E. De Giorgi, and M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal. 32 (1969), 255–267.10.1007/BF00281503Search in Google Scholar

[8] R. Temam, Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Rational Mech. Anal. 44 (1971), no. 72, 121–156.10.1007/BF00281813Search in Google Scholar

[9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984.10.1007/978-1-4684-9486-0Search in Google Scholar

[10] U. Massari and M. Miranda, Minimal Surfaces of Codimension One, North-Holland Publishing Co., Amsterdam, 1984.Search in Google Scholar

[11] R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, New York, 1986.10.1007/978-1-4613-8584-4Search in Google Scholar

[12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.10.1007/978-3-642-61798-0Search in Google Scholar

[13] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709.10.1137/1018114Search in Google Scholar

[14] V. K. Le and K. Schmitt, Sub-supersolution theorems for quasilinear elliptic problems: a variational approach, Electron. J. Differ. Equ. 118 (2004), 7.Search in Google Scholar

[15] V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal. 28 (2006), no. 1, 87–103.Search in Google Scholar

[16] C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier B.V., Amsterdam, 2006.Search in Google Scholar

[17] K. Schmitt, Revisiting the method of sub- and supersolutions for nonlinear elliptic problems, Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations, vol. 15 of Electron. J. Differ. Equ. Conf., Southwest Texas State Univ., San Marcos, TX, 2007, pp. 377–385.Search in Google Scholar

[18] E. S. Noussair, C. A. Swanson, and J. F. Yang, A barrier method for mean curvature problems, Nonlinear Anal. 21 (1993), no. 8, 631–641.10.1016/0362-546X(93)90005-DSearch in Google Scholar

[19] H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions, J. Differ. Equ. 254 (2013), no. 3, 1464–1499.10.1016/j.jde.2012.10.025Search in Google Scholar

[20] E. S. Noussair and C. A. Swanson, Positive solutions of the prescribed mean curvature equation in RN, Indiana Univ. Math. J. 42 (1993), no. 2, 559–570.10.1512/iumj.1993.42.42025Search in Google Scholar

[21] E. Giusti, Superfici cartesiane di area minima, Rend. Sem. Mat. Fis. Milano 40 (1970), 135–153.10.1007/BF02923230Search in Google Scholar

[22] M. Emmer, Esistenza, unicità e regolarità nelle superfici di equilibrio nei capillari, Ann. Univ. Ferrara Sez. VII (N.S.) 18 (1973), 79–94.10.1007/BF02853418Search in Google Scholar

[23] M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math. 12 (1974), 73–86.10.1007/BF01166235Search in Google Scholar

[24] M. Miranda, Dirichlet problem with L1 data for the non-homogeneous minimal surface equation, Indiana Univ. Math. J. 24 (1974), no. 75, 227–241.10.1512/iumj.1975.24.24020Search in Google Scholar

[25] E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1976), no. 3, 501–548.Search in Google Scholar

[26] C. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature, J. Math. Pures Appl. 58 (1979), no. 1, 75–109.Search in Google Scholar

[27] V. K. Le, On a sub-supersolution method for the prescribed mean curvature problem, Czechoslovak Math. J. 58 (2008), no. 2, 541–560.10.1007/s10587-008-0034-7Search in Google Scholar

[28] F. Obersnel and P. Omari, Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions, Differ. Integral Equ. 22 (2009), no. 9–10, 853–880.10.57262/die/1356019512Search in Google Scholar

[29] F. Obersnel, P. Omari, and S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl. 13 (2012), no. 6, 2830–2852.10.1016/j.nonrwa.2012.04.012Search in Google Scholar

[30] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc. 290 (1985), no. 2, 483–501.10.1090/S0002-9947-1985-0792808-4Search in Google Scholar

[31] W.-M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Proceedings of the Conference commemorating the 1st Centennial of the Circolo Matematico di Palermo (Palermo, 1984), Rend. Circ. Mat. Palermo 8 (1985), no. 2, 171–185.Search in Google Scholar

[32] J. Serrin, Positive solutions of a prescribed mean curvature problem, Proceedings of the Conference on Calculus of Variations and Partial Differential Equations (Trento, 1986), vol. 1340 of Lecture Notes in Math., Springer-Verlag, Berlin, 1988, pp. 248–255.10.1007/BFb0082900Search in Google Scholar

[33] P. Clément, R. Manásevich, and E. Mitidieri, On a modified capillary equation, J. Differ. Equ. 124 (1996), no. 2, 343–358.10.1006/jdeq.1996.0013Search in Google Scholar

[34] C. Bereanu, P. Jebelean, and J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc. 137 (2009), no. 1, 161–169.10.1090/S0002-9939-08-09612-3Search in Google Scholar

[35] C. Corsato, C. De Coster, F. Obersnel, P. Omari, and A. Soranzo, A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 2, 213–256.10.3934/dcdss.2018013Search in Google Scholar

[36] C. Corsato, C. De Coster, F. Obersnel, and P. Omari, Qualitative analysis of a curvature equation modelling MEMS with vertical loads, Nonlinear Anal. Real World Appl. 55 (2020), 103123.10.1016/j.nonrwa.2020.103123Search in Google Scholar

[37] F. Obersnel and P. Omari, Multiple bounded variation solutions of a periodically perturbed sine-curvature equation, Commun. Contemp. Math. 13 (2011), no. 5, 863–883.10.1142/S0219199711004488Search in Google Scholar

[38] C. Corsato, P. Omari, and F. Zanolin, Subharmonic solutions of the prescribed curvature equation, Commun. Contemp. Math. 18 (2016), no. 3, 1550042.10.1142/S021919971550042XSearch in Google Scholar

[39] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differ. Equ. 249 (2010), no. 7, 1674–1725.10.1016/j.jde.2010.07.001Search in Google Scholar

[40] F. Obersnel and P. Omari, Multiple bounded variation solutions of a capillarity problem, Discrete Contin. Dyn. Syst. (Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. II) (2011), 1129–1137.Search in Google Scholar

[41] P. Omari and E. Sovrano, Positive solutions of indefinite logistic growth models with flux-saturated diffusion, Nonlinear Anal. 201 (2020), 111949.10.1016/j.na.2020.111949Search in Google Scholar

[42] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976), no. 3, 281–295.10.1007/BF01221152Search in Google Scholar

[43] C. De Coster, F. Obersnel, and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, in: Handbook of Differential Equations: Ordinary Differential Equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006, pp. 203–339.10.1016/S1874-5725(06)80007-6Search in Google Scholar

[44] C. Bereanu and J. Mawhin, Boundary-value problems with non-surjective ϕ-Laplacian and one-sided bounded nonlinearity, Adv. Differ. Equ. 11 (2006), no. 1, 35–60.Search in Google Scholar

[45] C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and ϕ-Laplacian, NoDEA Nonlinear Differ. Equ. Appl. 15 (2008), no. 1–2, 159–168.10.1007/s00030-007-7004-xSearch in Google Scholar

[46] C. Bereanu, P. Jebelean, and J. Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr. 283 (2010), no. 3, 379–391.10.1002/mana.200910083Search in Google Scholar

[47] C. Bereanu, P. Jebelean, and J. Mawhin, Radial solutions for Neumann problems with ϕ-Laplacians and pendulum-like nonlinearities, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 637–648.10.3934/dcds.2010.28.637Search in Google Scholar

[48] J. A. Cid, On the existence of periodic oscillations for pendulum-type equations, Adv. Nonlinear Anal. 10 (2021), no. 1, 121–130.10.1515/anona-2020-0222Search in Google Scholar

[49] A. Fonda and R. Toader, A dynamical approach to lower and upper solutions for planar systems, preprint (2020).10.3934/dcds.2021012Search in Google Scholar

[50] F. Obersnel, P. Omari, and S. Rivetti, Asymmetric Poincaré inequalities and solvability of capillarity problems, J. Funct. Anal. 267 (2014), no. 3, 842–900.10.1016/j.jfa.2014.03.011Search in Google Scholar
Received: 2020-07-31
Accepted: 2020-10-09
Published Online: 2020-10-30

© 2020 Franco Obersnel and Pierpaolo Omari, 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/math-2020-0097
Keywords for this article
prescribed mean curvature equation; Dirichlet; Neumann; Robin boundary conditions; radial symmetry; classical solution; sub- and super-solutions
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
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  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
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  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
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  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
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  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
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  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
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  28. On multivalued Suzuki-type θ-contractions and related applications
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  30. Two types of hypergeometric degenerate Cauchy numbers
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  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
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  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
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  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
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  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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