A Constructive Approach About the Existence of Positive Solutions for Minkowski Curvature Problems

In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii’s theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far.

The research on problems associated with the mean curvature operator in Minkowski space is dated back to the 1960s, see [1][2][3][4][5] and the references therein. In the past decade, studies on nonexistence, existence and multiplicity of radial solutions on a bounded or an exterior domain and solutions for one-dimensional problems involving Minkowski curvature operator have attracted much attention. One may refer to [6][7][8][9][10] for radial solutions and [11][12][13][14][15][16][17] for one-dimensional problems.
Recently, Yang-Sim-Lee [17] studied nodal solutions for the following singular problem s , they proved a similar result for nodal solutions as in Coelho-Corsato-Obersnel-Omari [13].
It is interesting to notice that the studies of existence or multiple existence of positive solutions for problem (P) with condition m ∈ L 1 (0, 1) or m ∈ A have not been announced so far, which cannot be obtained by obvious modification from the results about one parameter family of problems (P λ ) in Coelho-Corsato-Obersnel-Omari [13] or Yang-Sim-Lee [17]. We also notice that the existence and nonexistence parameters λ * and λ * in the works of [13] or [17] were determined in a theoretical manner so that numerical information about these two parameters has not been known yet.
Thus the main goal of this paper is to introduce a new existence theorem for positive solutions of problem (P) and to obtain some numerical information about the ranges of nonexistence as well as existence of positive solutions with respect to parameters for problem (P λ ).
For the proof of existence theorem, we introduce newly defined integral operator on cone to apply the classical Krasnoselskii's fixed point theorem. For numerical information, we apply the existence theorem to problem (P λ ) and also investigate nonexistence of positive solutions for the problem. Modifying condition (F 2 ) in Sect. 3 for problem (P λ ), we obtain some possible intervals of existence and nonexistence but the global existence interval (i.e. when λ * = λ * ) could not be answered in this work. We note that the techniques in [15] used to estimate the norm of solutions cannot be applied directly to problem (P) due to the singularity of coefficient function m and the difference of boundary conditions. The rest of this paper is organized as follows. In Sect. 2, we introduce an integral operator which will be used as a fixed point operator for problem (P). In Sect. 3, we prove the existence of positive solutions for problem (P). In Sect. 4, we modify the existence condition given in Sect. 3 to derive numerical information about the range of existence for one parameter family of problems (P λ ) including problem (1.1).
In Sect. 5, we show the nonexistence of positive solutions for (P λ ) and give some examples to illustrate our main results by using the proposed numerical algorithm.

An Integral Operator
In this section, we introduce an integral operator which corresponds to a fixed point operator for problem (P) later. For this, we first consider Poisson-type Dirichlet boundary value problem where ξ ∈ A. Let u be a solution of problem (P O). Then reminding that ξ may not be integrable near 0 or 1, we fix σ ∈ (0, 1) in an arbitrary manner and integrate the equation in problem (P O) on the interval (t, σ ] for t ∈ (0, σ ] to obtain where we denote a (u (σ )). We claim that −1 a + σ t ξ(r )dr ∈ L 1 (0, σ ]. Indeed, by using changing order of integrations and ξ ∈ A, we get Thus we may integrate (2.1) on the interval (0, t) and get Similarly, for t ∈ [σ, 1), we obtain . For this, define a function H : R → R given as Then by boundedness and monotone property of −1 , we easily see that H is continuous, strictly increasing, and H (a) < 0 as a → −∞ and H (a) > 0 as a → ∞. Therefore H has a zero, say α unique up to σ and ξ . Define Then u a solution of problem (P O) can be equivalently written as u = T ξ .

Remark 2.1
Since α is uniquely determined, solution of problem (P O) exists uniquely.

An Existence Result
In this section, we study the existence of positive solutions of problem (P). For another type of function class B given by we note that A = B mainly by Fubini's theorem.
We now give some assumptions on nonlinear term f .
Remark 3. 1 We note from Theorem 2.1 in [16] that under assumption (F 1 ), all solutions u of (P) is of C 1 [0, 1] and u ∞ < 1. This implies that all solutions u of (P) satisfies We now state the Krasnoselskii's theorem of cone expansion and compression which will be used for the proof of our existence theorem.

Lemma 3.1 ([18]) Let E be a Banach space and K a cone in E. Assume that 1 and
2 are bounded open in E with 0 ∈ 1 ⊂ 1 ⊂ 2 , and let T : K ∩ ( 2 \ 1 ) → K be completely continuous such that either Then T has a fixed point in K ∩ ( 2 \ 1 ).

Remark 3.3 If m ∈
A and 0 ≤ f 0 < ∞, then any nontrivial solution of (P) is positive, mainly by concavity and double zero properties of solutions (see [9,17]).
We easily check that u is a solution of problem (P) if and only if u ∈ K satisfies u = T u. Moreover, we can easily check by a standard argument that T (K ) ⊂ K and T is completely continuous. For u ∈ K , T u is concave and satisfies the Dirichlet boundary condition. Thus we may assume that there exists t * ∈ (0, 1), a maximal point of T u satisfying T u ∞ = (T u)(t * ) and (T u) (t * ) = 0. We note that t * need not be unique. From (T u) (t * ) = 0, we obtain Since m ∈ L 1 (t * − δ, t * + δ) for any small δ, replacing σ with t * , we get α = 0 and T u can be written as If we find a nontrivial fixed point u of T in K , then u is a positive solution of problem (P), and we now give the existence result for problem (P). Then, by (F 1 ), there exists ρ 1 ∈ (0, 1 4 ρ) such that For t * given in (3.1), we first assume t * ∈ (0, 1 2 ], then by using (3.2), we derive If t * ∈ ( 1 2 , 1), then by an analogous argument, we also get the same inequality. Thus, On the other hand, define 2 = {u ∈ E : u ∞ < ρ} and consider u ∈ K ∩ ∂ 2 . If t * ∈ [ 1 2 , 1), then we obtain Since u ∈ K ∩ ∂ 2 , we see Thus, u(t) ≥ δ u ∞ for t ∈ [δ, t * ] and this implies u(t) ∈ [δρ, ρ] for t ∈ [δ, t * ]. By applying (F 2 ) and Remark 3.2, we get By a similar argument to the case t * ∈ (0, 1 2 ), we also get This implies that T u ∞ ≥ u ∞ for u ∈ K ∩ ∂ 2 . Therefore, by Lemma 3.1, operator T has a fixed point in K ∩ ( 2 \ 1 ), which is a positive solution of problem (P) and the proof is complete. Now we give an example to illustrate the applicability of Theorem 3.1.
where nonlinear term f is given by

Nonlinear Eigenvalue Problems
In this section, we apply Theorem 3.1 to study the existence of positive solutions for one parameter family of problems of the form ⎧ ⎨ ⎩ − (u (t)) = λm(t)g(u(t)), t ∈ (0, 1), Assume m ∈ A and g 0 = 0. For fixed δ ∈ (0, 1 2 ), define a function λ δ given by ρ δ < ∞ and δ is well-defined. Thus we see from Corollary 4.1 that problem (P λ ) has at least one positive solution for all λ > δ . Furthermore, we may prove that problem (P λ ) has at least one positive solution at λ = δ . Proposition 4.1 Assume m ∈ A and g 0 = 0. Then for each δ ∈ (0, 1 2 ), there exists at least one positive solution of problem (P λ ) at λ = δ .

A Nonexistence Result
In this section, we study nonexistence of positive solutions for problem (P λ ). Thus Similarly, we also obtain We note that the integration on the right-hand side is well-defined by (5.1) and (5.4). Combining (5.1), (5.4) and (5.5), we get It follows that λ > * . Therefore, we conclude that problem (P λ ) has no nontrivial solution for λ ∈ (0, * ] and the proof is complete.
As an immediate consequence of Corollary 4.2 and Theorem 5.1, we obtain the following result about existence and nonexistence.

Example 5.1 There is no nontrivial solution to problem
Acknowledgements The authors would like to thank the editor and reviewers for their useful suggestions which helped to improve the paper. The third author was supported by the National Research Foundation of Korea funded by the Korea Government (MEST) (NRF2016R1D1A1B04931741) and (MEST) (NRF2021R1A2C100853711).
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