Unilateral global interval bifurcation for problem with mean curvature operator in Minkowski space and its applications

Abstract

In this paper, we establish a unilateral global bifurcation result from interval for a class problem with mean curvature operator in Minkowski space with non-differentiable nonlinearity. As applications of the above result, we shall prove the existence of one-sign solutions to the following problem

$$\left\{ {\matrix{{ - {\rm{div}}\left( {{{\nabla v} \over {\sqrt {1 - {{\left| {\nabla v} \right|}^2}} }}} \right) = \alpha (x){v^ + } + \beta (x){v^ - } + \lambda a(x)f(v),} \hfill & {{\rm{in}}\,{B_R}(0),} \hfill \cr {v(x) = 0,} \hfill & {{\rm{on}}\,\partial {B_R}(0),} \hfill \cr } } \right.$$

where λ ≠ 0 is a parameter, R is a positive constant and B_R(0) = {x ∈ ℝ^N: ∣x∣ < R} is the standard open ball in the Euclidean space ℝ^N (N ≥ 1) which is centered at the origin and has radius R. v⁺ = max{v, 0},v⁻ = − min{v, 0}, \(a(x) \in C(\overline {{B_R}(0)} \), a(x), α(x) and β(x) are radially symmetric with respect to x; f ∈ C(ℝ, ℝ), sf(s) > 0 for s ≠ 0, and f₀ ∈ [0, ∞], where f₀ = lim_∣s∣→0f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results. We also study the asymptotic behaviors of positive radial solutions as λ → +∞.