Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space

The problem arises from the study of prescribed mean curvature of a spacelike graph in the flat Minkowski space endowed with the Lorentzian metric \(\sum_{i=1}^{N}dx_{i}^{2}-dt^{2}\) [1].

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 where \(B_{R}=\{x\in\mathbb{R}^{N}:|x|< R\}\) and \(f:[0,R]\times[0,\alpha )\rightarrow\mathbb{R}\) is positive on \((0,R]\times(0,\alpha)\). When \(\alpha=R=1\), f is superlinear at 0 and sublinear at 1 with respect to \(\phi(s):=s/\sqrt{1-{s}^{2}}\), the 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 where \(\lambda>0\) is a parameter, \(q>1\), \(R>0\), \(\mu:[0,+\infty)\to R\) is continuous, positive on \((0,\infty)\). Using upper and lower solutions together with Leray–Schauder degree type arguments, the authors proved that there exists \(\Lambda> 0\) such that the problem has zero, at least one or at least two positive radial solutions according to \(\lambda\in(0,\Lambda)\), \(\lambda=\Lambda\) or \(\lambda>\Lambda\). In [4], Pei and Wang considered the singular Dirichlet problem involving the mean curvature operator in Minkowski space of the form and the corresponding one-parameter problem where \(f\in C([0,1]\times[0,1), [0,+\infty))\). 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 where \(f(r,u)\) is nonnegative and continuous on \((0,1)\times(0,+\infty )\) and may be singular at \(r=0\) and/or \(r=1\) and strongly singular at \(u=0\). 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 where Ω is a general bounded domain of \(\mathbb{R}^{N}\). 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 nonlinearity at zero. We refer the reader to [7, 8, 9, 10, 11, 12, 13, 14, 15] for the N-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 [16, 17, 18, 19, 20, 21, 22, 23, 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 and the corresponding one-parameter problem where \(f\in C([0,1]\times[0,1), [0,+\infty))\). 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 0 and sublinear at 1 with respect to \(\phi(s)\). 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.

Take \(E=C[0,1]\) with the usual supremum norm \(\|\cdot\|\) and the corresponding open ball of center 0 and radius \(\rho>0\) will be denoted by \(B_{\rho}\). Let Then K is a cone in E and, for all \(u\in K\),

Now we define a nonlinear operator T on \(K\cap B_{1}\) as follows: Clearly, \((Tu)(r)\geq0\), \((Tu)''(r)\leq0\) for all \(r\in[0,1]\), \((Tu)'(0)=(Tu)(1)=0\), which implies that \(T(K\cap B_{1})\subset K\). Moreover, it is easy to show by a standard argument that T is compact on \(K\cap\overline{B}_{\rho}\) for all \(\rho\in(0,1)\) (see [2]). In addition, it can easily be verified that u is a positive solution of problem (1.1) if \(u\in K\cap B_{1}\) is a fixed point of the nonlinear operator T.

For convenience, we introduce some notations where J be a compact subinterval of \([0,1]\).