S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

Abstract In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem − div(ϕN(∇y))=λa(|x|)f(y)inA,∂y∂ν=0 on Γ1,y=0 onΓ2, $$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$ where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2}, ϕN(s)=s1−|s|2, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $ s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞), ∂y∂ν $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.


Introduction
Hypersurfaces of prescribed mean curvature in at Minkowski space L N+ = {(x, t) : x ∈ R N , t ∈ R}, with the Lorentzian metric N i= dx i − dt , where (x, t) = (x , x , · · · , x N , t), are of interest in di erential geometry and in general relativity. It is well-known that the study of spacelike submanifolds of codimension one in L N+ with prescribed mean extrinsic curvature leads to Dirichlet problems of the type where Ω is a bounded domain in R N and the nonlinearity f : Ω × R → R is continuous, see [1,2]. The existence, multiplicity and qualitative properties of solutions of (1) have been extensively studied by many authors in recent years, see Coelho et al. [3], Treibergs [4], Cano-Casanova et al. [5], Pan et al. [6], López [7], Corsato et al. [8,9], Korman [10] as well as Ma et al. [11,12], and the references therein. It is worth pointing out that the starting point of this type of problems is the seminal paper [13], and from Bartnik and Simon [2] as well as Bereanu and Mawhin [14], we know (1) has a solution whatever f is. This can be seen as a universal existence result for the above problem. However, in our study problem (1) generally admits the null solution, it may be interesting to investigate the existence of non-trivial solutions, especially the positive solutions. However, there are few works on positive solutions of (1), see Coelho et al. [15], Bereanu et al. [16,17], Ma et al. [18] and Dai [19].
Speci cally, depending on the behaviour of f = f (x, s) near s = , Coelho et al. [15] discussed the existence of either one, or two, or three, or in nitely many positive solutions of the quasilinear two-point boundary value problem where f is L p -Carathéodory function, and the proof of main results are based upon the variational and topological methods. Bereanu et al. [16,17] obtained some important existence, nonexistence and multiplicity results for the positive radial solutions of problem (1) in a ball by using Leray-Schauder degree argument and critical point theory. Recently, Ma et al. [18] are concerned with the global structure of radial positive solutions for the problem (1) in a ball by using global bifurcation techniques, and extended the results of [16,17] to more general cases, all results, depending on the behavior of nonlinear term f near . Dai [19] investigated the intervals of the parameter λ in which the problem (1) has zero, one or two positive radial solutions corresponding to sublinear, linear, and superlinear nonlinearities f at zero, respectively. However, [18,19] only give a full description of the set of radial positive solutions of (1) for certain classes of nonlinearities f , and give no any information about the directions of a bifurcation.
In 2015, Sim and Tanaka [20] proved the existence of S-shaped connected component in the set of positive solutions for the one-dimensional p-Laplacian problem with sign-changing weight , f ∈ C[ , ∞) and µ is a positive parameter. They obtained the following result by bifurcation techniques.
Of course, the natural question is whether or not the similar result can be established for the prescribed mean curvature problem (1)?
The purpose of this paper is to show the existence of the S-shaped connected component in the set of radial positive solutions for a prescribed mean curvature problem in an annular domain ∂y ∂ν denotes the outward normal derivative of y and | · | denotes the Euclidean norm in R N . To the best of our knowledge, for problem (3), such bifurcation curve is completely new and has not been practically described before.
Setting, as usual |x| = r and y(x) = u(r), the problem (3) reduces to the mixed boundary value problem , s ∈ R, ϕ : (− , ) → R is an odd, increasing homeomorphism and ϕ ( ) = . To nd a positive radial solution of (3), it is enough to nd a positive solution of (4). We say that a function Let λ k (a, R ) be the k-th eigenvalue of the eigenvalue problem and φ k be the eigenfunction corresponding to λ k (a, R ). It is well-known that < λ (a, R ) < λ (a, R ) < · · · < λ k (a, R ) < · · · → +∞ as k → +∞, and no other eigenvalues. Moreover, the algebraic multiplicity of λ k (a, R ) is , and the eigenfunction φ k has exactly k − zeros in (R , R ), see [21]. The rst eigenvalue λ (a, R ) is the minimum of the Rayleigh quotient, namely, Assume that: η is the rst positive eigenvalue of the problem The main result of this paper is the following.
Remark 1.1. Let (λ, u) be a solution of (4), then it follows from |u (r)| < that This leads to the bifurcation diagrams mainly depend on the behavior of f = f (s) near s = . This is a signi cant di erence between the Minkowski-curvature problems and the p-Laplacian problems.

Remark 1.2.
In the special case p = , (F1') reduces to where χ > is a su ciently small constant. It is easy to see that condition (F2) is weaker than (6), in fact f s +α is a special case of g(s).
The main result is obtained by reducing the problem (4) to an equivalent problem and use the Rabinowitz global bifurcation techniques [22]. Indeed, under (F1) and (F2) we get an unbounded connected component which is bifurcating from ( λ (a,R ) f , ), and condition (F2) pushes the bifurcation to the right near u = . Condition (F3) leads the bifurcation curve to the left at some point, and nally to the right near λ = ∞.
For other results concerning the existence of an S-shaped connected component in the set of solutions for diverse boundary value problems, see [23][24][25][26] for the semilinear boundary value problems, and [27] for the p-Laplacian boundary value problems.
The rest of the paper is organized as follows. In Section 2, we give an equivalent formulation of problem (4) and some preliminary results to show the change of direction of a bifurcation. Section 3 is devoted to proving the main result.

Some preliminary results . An equivalent formulation
Let us de ne a function f : R → R by setting Notice that, within the context of positive solutions, problem (4) is equivalent to the same problem with f replaced by f . In the sequel, we shall replace f with f , however, for the sake of simplicity, the modi ed function f will still be denoted by f . Next, let us de ne h as follows ( ) is a positive solution of (4) if and only if it is a positive solution of the problem ( )

Proof.
It is clear that a positive solution u ∈ C [R , R ] of (4) is a positive solution of (9). Conversely, assume that u ∈ C [R , R ] is a positive solution of (9). We aim to show that ||u ||∞ < . Assume on the contrary that this is not true. for every r ∈ [c, d]. Since ϕ − (C ) < , taking the limit as r → d − we obtain the contradiction |u (d)| < . Therefore ||u ||∞ < and accordingly, u is a positive solution of (4).

Lemma 2.2.
Assume that (A1) and (F1) hold. Let u be a nontrivial solution of (4). Then u > on [R , R ) and u is strictly decreasing.

Proof.
From it follows u ≤ because (A1) and (F1), so u is decreasing. Since u(R ) = , we have u ≥ on [R , R ]. As u is not identically zero, one has u(R ) > and, from (10) we deduce that u < on (R , R ], which ensures that actually u is strictly decreasing and u > on [R , R ).
Next, we give some property of concave functions. Proof. Assume on the contrary that for any R , R + R −R n , there exist sequence un ∈ E with un is concave and strictly decreasing on (R , R ), that is to say −u n (xn) > − ν.

. The direction of bifurcation
Let h ∈ X be given. It is well-known that the solution u of problem − (r N− u ) = r N− h(r), r ∈ (R , R ), can be expressed by where the Green's function of (11) for N = is explicitly given by and the Green's function of (11) for N ≥ is explicitly given by Let L : X → E be de ned by L(u) = K(au). Both K and L are completely continuous and (5) where ν(r) = N− r . Clearly, H is completely continuous and, by (12) and (13) Notice that ||u ||∞ < , (λ, u) ∈ S, which implies that ||u||∞ < R − R , (λ, u) ∈ S.
We notice that u is a solution of Proof. We divide the proof into four steps.
Then, similar to the proof of Lemma 2.9, we have u n converges to as n → ∞. From this fact and (19), after taking a subsequence and relabeling, if necessary, we have vn → v * in X for some v * ∈ X, and, − (r N− v * (r)) = r N− µn a(r)f v * (r), r ∈ (R , R ), v * (R ) = v * (R ) = , that is µn → λ (a,R ) f . This contradicts with the fact µn → ∞. Therefore, the claim is valid.