Pei et al. Boundary Value Problems (28) 28:43 https://doi.org/.86/s366-8-963-5 R E S E A R C H Open Access Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space Minghe Pei, Libo Wang * and Xuezhe Lv * Correspondence: wlb_math@63.com School of Mathematics and Statistics, Beihua University, JiLin City, P.R. China Abstract In this paper, we consider a one-dimensional mean curvature equation in Minkowski space and the corresponding one-parameter problem. By using a fixed point theorem of cone expansion and compression of norm type, the existence and multiplicity of positive solutions for the above problems are obtained. Meanwhile, as applications of our results, some examples are given. MSC: 35J93; 34B6; 34B8 Keywords: Mean curvature equations; Positive solution; Existence; Multiplicity; Fixed point theorem of cone expansion and compression of norm type Introduction The problem ( ) v div = H(x, v) in v 2 arises from the study of prescribed mean curvature of a spacelike graph in the flat Minkowski space L N+ := { (x, t):x R N, t R } endowed with the Lorentzian metric N i= dx2 i dt2 []. Recently, these kinds of problems have been studied by many authors. In [2], Bereanu et al. considered the Dirichlet problem for the mean curvature equation in Minkowski space of the type v )+f( x, v)= inb R, v 2 v = on B R, where B R = {x R N : x < R} and f :[,R] [, α) R is positive on (, R] (, α). When α = R =,f is superlinear at and sublinear at with respect to φ(s) :=s/ s 2,the The Author(s) 28. This article is distributed under the terms of the Creative Commons Attribution 4. International License (http://creativecommons.org/licenses/by/4./), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Pei et al. Boundary Value Problems (28) 28:43 Page 2 of authors obtained the existence of classical positive radial solutions of the problem by using the Leray Schauder degree arguments. In [3], Bereanu et al. studied the positive radial solutions of the one-parameter problem involving the mean curvature operator in Minkowski space of the type v v 2 )+λ[μ( x )vq ]= inb R, v = on B R, where λ > is a parameter, q >,R >,μ :[,+ ) R is continuous, positive on (, ). Using upper and lower solutions together with Leray Schauder degree type arguments, the authors proved that there exists > such that the problem has zero, at least one or at least two positive radial solutions according to λ (, ), λ = or λ >.In[4], Pei and Wang considered the singular Dirichlet problem involving the mean curvature operator in Minkowski space of the form v )+f( x, v)= inb, v 2 v = on B, and the corresponding one-parameter problem v )+λf( x, v)= inb, v 2 v = on B, where f C([, ] [, ), [, + )). They provided sufficient conditions for the existence of triple and arbitrarily many positive solutions to the above problems by Leggett Williams fixed point theorem.in [5], Pei and Wang considered a strongly singular Dirichlet problem involving the mean curvature operator in Minkowski space of the form v )+f( x, v)= inb, v 2 v = on B, where f (r, u) is nonnegative and continuous on (, ) (, + ) andmaybesingularat r = and/or r = and strongly singular at u =. The authors presented sufficient conditions which guarantee the existence of positive solutions to the problem by applying the perturbation technique and nonlinear alternative of Leray Schauder type. In [6], Dai considered the solvability of the Dirichlet problem with mean curvature operator in the Minkowski space v )+λf(x, v)= in, v 2 v = on, where is a general bounded domain of R N. By bifurcation and topological methods, the author determined the interval of parameter λ in which the above problem has zero/one/two nontrivial nonnegative solutions according to sublinear/linear/superlinear
Pei et al. Boundary Value Problems (28) 28:43 Page 3 of nonlinearity at zero. We refer the reader to [7 5] forthen-dimensional mean curvature equation in Minkowski space. In particular, for one-dimensional mean curvature equation with Dirichlet/Neumann/periodic/mixed type boundary conditions in Minkowski space, we refer the reader to [6 24] and the references therein. Motivated by above work, in this paper, we will consider the one-dimensional mean curvature equations in Minkowski space of the type ( u u 2 ) + f (r, u)=, r (, ), (.) u () =, u() =, and the corresponding one-parameter problem ( u u 2 ) + λf (r, u)=, r (, ), u () =, u() =, (.2) where f C([, ] [, ), [, + )). Using a fixed point theorem of cone expansion and compression of norm type, we obtain the existence and multiplicity of positive solutions of the above problems. It is worth to note that, in our work, we do not assume that f is superlinear at and sublinear at with respect to φ(s). To the best of our knowledge, this is the first paper using a fixed point theorem of cone expansion and compression of norm type to study the above problem. The rest of the paper is organized as follows. By means of a fixed point theorem of cone expansion and compression of norm type (see [25]), in Sect. 2 we show the existenceand multiplicity of positive solutions of (.) and(.2). In Sect. 3, we give some examples to illustrate our results. 2 Main results In order to introduce our main theorem, we need some lemmas. Simple computations lead to the following lemma. Lemma 2. Let φ(s) be defined by above. Then φ (s)=s/ +s 2 and φ (s )φ (s 2 ) φ (s s 2 ) s s 2, s, s 2 [, + ). Lemma 2.2 ([25]) Let E be a Banach space and let K be a cone in E. Assume that and 2 are bounded open subsets of E such that 2, and let T : K ( 2 \ ) K be a completely continuous operator such that either (i) Tx x for x K and Tx x for x K 2, or (ii) Tx x for x K and Tx x for x K 2. Then T has a fixed point in K ( 2 \ ). Take E = C[, ] with the usual supremum norm andthecorrespondingopenball of center and radius ρ >willbedenotedbyb ρ.let K = { u E : u(r) is concave and nonincreasing on [, ], u() = }.
Pei et al. Boundary Value Problems (28) 28:43 Page 4 of Then K is a cone in E and, for all u K, u(r) ( r) u on [, ]. (2.) Now we define a nonlinear operator T on K B as follows: (Tu)(r)= φ f ( τ, u(τ) ) ) dτ ds, u K B. r Clearly, (Tu)(r), (Tu) (r) for all r [, ], (Tu) () = (Tu)() =, which implies that T(K B ) K. Moreover, it is easy to show by a standard argument that T is compact on K B ρ for all ρ (, ) (see [2]). In addition, it can easily be verified that u is a positive solution of problem (.)ifu K B is a fixed point of the nonlinear operator T. Now, we state and prove the existence and multiplicity of positive solutions of problem (.)and(.2) by using a fixed point theorem of cone expansion and compression of norm type. For convenience, we introduce some notations f J = lim f (r, s) min s + r J φ(s), f = lim max f (r, s) s + r [,] φ(s), f = lim max f (r, s) s r [,] φ(s), where J be a compact subinterval of [, ]. Theorem 2. Assume that there exists a compact subinterval J := [r, r ] [, ) such that f J > σ r and f <, where σ = (r r ) 2 ( +(r r ) 2 +).Then problem (.) has at least one positive solution. Proof Take a number L with lim Lφ(( r )δ) = L r >. δ + φ(σδ) σ σ r < L < f J.Then It follows that there exists δ (, ) with σδ <suchthat L φ(( r )δ) φ(σδ) >, δ (, δ ). Since f J > L,thereexistsρ (, δ )suchthat f (r, s) φ(s) > L, (r, s) J (, ρ]. Therefore, for each (r, s) J [( r )ρ, ρ]wehave f (r, s)>lφ(s) Lφ ( ( r )ρ ) > φ(σρ). (2.2)
Pei et al. Boundary Value Problems (28) 28:43 Page 5 of Now, let = {u E : u < ρ}.thenfrom(2.), (2.2) and Lemma 2.,weobtain Tu = φ f ( τ, u(τ) ) ) dτ ds r f ( τ, u(τ) ) ) dτ ds r ) > φ(σρ)dτ ds r φ r r φ r r σρ = σρ r r r φ (s r )ds φ (s)ds = ρ = u, u K. (2.3) Next, we prove that there exists ϱ (ρ,)suchthat Tu < u, u K B ϱ. (2.4) In fact, from f <,thereexistα (f,)andβ (, ) such that f (r, s) M + αφ(s), (r, s) [, ] [, ), (2.5) where M = max{f (r, s):(r, s) [, ] [, β]}. Select ϱ >with { ( )} M max ρ, φ < ϱ <. α Then from (2.5)weget Tu = φ f ( τ, u(τ) ) ) dτ ds ) φ ( ) M + αφ(ϱ) dτ ds = φ (( M + αφ(ϱ) ) s ) ds < φ ( M + αφ(ϱ) ) < φ ( ( α)φ(ϱ)+αφ(ϱ) ) = ϱ = u, u K B ϱ, that is, (2.4)holds. Let 2 = {u E : u < ϱ}.thenfrom(2.4)wehave Tu < u, u K 2. (2.6)
Pei et al. Boundary Value Problems (28) 28:43 Page 6 of Therefore, from (2.3), (2.6) and Lemma 2.2, the operator T has a fixed point u K ( 2 \ ), which is a positive solution of problem (.). This completes the proof of the theorem. Corollary 2. Assume that there exists a compact subinterval J := [r, r ] (, ) such that σ ( r )f J =: < 2 := f, where σ = ( +(r (r r ) 2 r ) 2 +).Then the one-parameter problem (.2) has at least one positive solution provided < λ < 2. Theorem 2.2 Assume that (i) f <and f <; (ii) there exist a compact subinterval [r, r ] [, ) and ρ (, ) such that σρ (, ) and f (r, s)>φ(σρ), (r, s) [r, r ] [ ( r )ρ, ρ ], where σ = ( +(r (r r ) 2 r ) 2 +). Then problem (.) has at least two positive solutions. Proof Since f <,thereexistsρ (, ρ)suchthat f (r, s)<φ(s), (r, s) [, ] [, ρ ]. Let = {u E : u < ρ },then,foru K, Tu = φ f ( τ, u(τ) ) ) dτ ds < φ φ ( u(τ) ) ) dτ ds ( ) φ φ(ρ )dτ ds = ρ = u. (2.7) Let = {u E : u < ρ}. Then from the proof of Theorem 2. we obtain Tu > u, u K. (2.8) Notice that f <, it follows from the proof of Theorem 2. that there exists ϱ (ρ,) such that Tu < u, u K 2, (2.9) where 2 = {u E : u < ϱ}. Therefore, it follows from (2.7), (2.8), (2.9) and Lemma 2.2 that the operator T has two fixed points u K ( \ )andu 2 K ( 2 \ ), which
Pei et al. Boundary Value Problems (28) 28:43 Page 7 of are two distinct positive solutions of problem (.). This completesthe proofof the theorem. Remark 2. In Theorem 2.2, the condition f < in (i) can be replaced by the following condition: there exists ρ (, ρ)suchthat f (r, s)<φ(ρ ), (r, s) [, ] [, ρ ]. The conclusion still holds. Corollary 2.2 Assume that (i) there exist a compact subinterval [r, r ] [, ) and ρ (, ) such that σρ (, ) and f (r, s)>, (r, s) [r, r ] [ ( r )ρ, ρ ] =: D, where σ = ( +(r (r r ) 2 r ) 2 +); φ(σρ) (ii) min (r,s) D f (r,s) =: < 2 := min{, }. f f Then the one-parameter problem (.2) has at least two positive solutions provided < λ < 2. Proof At first, from the assumption (i), we know that min (r,s) D f (r, s)>,andthus is a finite number. We now check that all conditions of Theorem 2.2 are satisfied. From the assumption (ii), for λ < 2,wehave λf i < 2 f i, i =,. On the other hand, from assumption (ii), for λ >,wehave λf (r, s)> f (r, s) > φ(σρ) f (r, s) min (r,s) D f (r, s) φ(σρ) min (r,s) D f (r, s) min f (r, s) (r,s) D = φ(σρ), (r, s) D. Therefore, by Theorem 2.2, the one-parameter problem (.2) has at least two positive solutions provided < λ < 2. Theorem 2.3 Assume that there exists a compact subinterval J := [r, r ] [, ) such that (i) f J = and f <; (ii) there exists ρ (, ) with σρ (, ) such that f (r, s)>φ(σρ), (r, s) J [ ( r )ρ, ρ ], where σ = (r r ) 2 ( +(r r ) 2 +);
Pei et al. Boundary Value Problems (28) 28:43 Page 8 of (iii) there exists ρ (, ρ) such that f (r, s)<φ(ρ ), (r, s) [, ] [, ρ ]. Then problem (.) has at least three positive solutions. Proof From the assumptions (i), (ii), (iii) and Remark 2., the conditions of Theorem 2.2 are all satisfied. Then from the proof of Theorem 2.2,thereexistρ and ϱ with < ρ < ρ < ϱ <suchthat Tu < u, u K B ρ, Tu > u, u K B ρ, and Tu < u, u K B ϱ, where B ρ = {u E : u < ρ }, B ρ = {u E : u < ρ}, B ϱ = {u E : u < ϱ}. Moreover, from f J = andtheproofoftheorem2., thereexistsρ >withρ < ρ, such that Tu > u, u K B ρ, where B ρ = {u E : u < ρ }. Then from Lemma 2.2, problem(.) has three distinct positive solutions u, u, u 2 with u K (B ρ \ B ρ ), u K (B ρ \ B ρ )andu 2 K (B ϱ \ B ρ ). Corollary 2.3 Assume that there exists a compact subinterval J := [r, r ] [, ) such that (i) f J = ; (ii) there exists ρ (, ) with σρ (, ) such that f (r, s)>, (r, s) J [ ( r )ρ, ρ ] =: D ρ, where σ = ( +(r (r r ) 2 r ) 2 +); (iii) there exists ρ (, ρ) such that { φ(σρ) min (r,s) Dρ f (r, s) =: < 2 := min f, φ(ρ } ), max (r,s) Dρ f (r, s) where D ρ := [, ] [, ρ ]. Then the one-parameter problem (.2) has at least three positive solutions provided < λ < 2. Proof SimilartotheproofofCorollary2.2,itiseasytocheckthat,for < λ < 2, λf (t, s) satisfies all conditions of Theorem 2.3. Then the conclusion hold.
Pei et al. Boundary Value Problems (28) 28:43 Page 9 of 3 Examples In this section, we give some examples to demonstrate the applications of the our results. Example 3. Consider the following one-dimensional mean curvature equations in Minkowski space of the type ( u u 2 ) +(r 2 )2 up ( u 2 ) q =, r (, ), u () =, u() =, (3.) where < p <andq < /2 are constants. Let f (r, s)=(r 2 )2 s p /( s 2 ) q on [, ] [, ). Take J =[ 3 4,],itiseasytoseethatf J = and f =.ByTheorem2.,theproblem(3.) has at least one positive solution. We note that Theorem of [2] cannot guarantee this conclusion since f (, s)=, s [, ). 2 Example 3.2 Consider the following one-parameter problem: ( u u 2 ) + λ r 37 48 p u q =, r (, ), u () =, u() =, (3.2) where λ > be a parameter, p, q > are constants. Let f (r, s)= r 37 48 p s q on [, ] [, ). It is easy to see that f = f =.Chooseρ =/2, r = /4 and r = 3/4. Then σρ =(2+ 5)/6 (, ), φ(σρ)= σρ = 2+ 5 (σρ) 2 27 4 5 and min f (r, s)= min (r,s) D (r,s) D r 37 48 p s q = 48 p+q, where D =[ 4, 3 4 ] [ 48, ]. Thus from Corollary 2.2, the one-parameter problem (3.2)has 2 at least two positive solutions provided λ > := 48 p+q 2+ 5 27 4. 5 Example 3.3 Consider the following one-parameter problem: ( u u 2 ) + λg(u)=, r (, ), u () =, u() =, (3.3)
Pei et al. Boundary Value Problems (28) 28:43 Page of where λ > is a parameter and s, s [, 96 ], 2 5 (36 6)(s 48 )+ 3 2, s [ 96, 48 ], g(s)= 3456s 2, s [ 48, 2 ], 276 5 (s 2 )+, s [ 2, 2 ], 4 2, s [ 2 s 2 2,). We choose ρ =/2,r = /4, r = 3/4 and ρ = /96. It is easy to check that the conditions for Corollary 2.3are all satisfied, where = 2(2 + 5) 3 27 4 5 <, 2 = 4 6 925 >. Hence from Corollary 2.3, problem(3.3) has at least three positive solutions when λ (, 2 ). Acknowledgements The authors thank the referee for valuable suggestions, which led to improvement of the original manuscript. Funding This work was supported by the Education Department of JiLin Province ([26]45). Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 27 November 27 Accepted: 2 March 28 References. Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 3 52. 982 83 2. Bereanu, C., Jebelean, P., Torres, P.J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264, 27 287 (23) 3. Bereanu, C., Jebelean, P., Torres, P.J.: Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J. Funct. Anal. 265, 644 659 (23) 4. Pei, M., Wang, L.: Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space. Appl. Math. Lett. 6, 5 55 (26) 5. Pei, M., Wang, L.: Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity. Proc.Am.Math.Soc.45, 4423 443 (27) 6. Dai, G.: Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain. Indiana Univ. Math. J. (in press). http://www.iumj.indiana.edu/iumj/preprints/7546.pdf 7. Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Am. Math. Soc. 37, 6 69 (29) 8. Bereanu, C., Jebelean, P., Mawhin, J.: Multiple solutions for Neumann and periodic problems with singular φ-laplacian. J. Funct. Anal. 26, 3226 3246 (2) 9. Bereanu, C., Jebelean, P., Mawhin, J.: The Dirichlet problem with mean curvature operator in Minkowski space a variational approach. Adv. Nonlinear Stud. 4, 35 326 (24). Coelho, I., Corsato, C., Rivetti, S.: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal. 44, 23 39 (24). Corsato, C., Obersnel, F., Omari, P., Rivetti, S.: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J. Math. Anal. Appl. 45, 227 239 (23) 2. Dai, G.: Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space. Calc. Var. Partial Differ. Equ. 55,72 (26)
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