Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator

Abstract

Abstract We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator − u ′ 1 − u ′ 2 ′ = λ a ( x ) f ( u , u ′ ) , x ∈ R , $-{\left(\frac{{u}^{\prime }}{\sqrt{1-{u}^{\prime 2}}}\right)}^{\prime }=\lambda a\left(x\right)f\left(u,{u}^{\prime }\right), x\in \mathbb{R},$ where λ > 0 is a parameter, a ∈ C ( R , R ) $a\in C\left(\mathbb{R},\mathbb{R}\right)$ is a 2π-periodic sign-changing function with ∫ 0 2 π a ( x ) d x < 0 ${\int }_{0}^{2\pi }a\left(x\right)\mathrm{d}x{< }0$ , f ∈ C ( R × R , R ) $f\in C\left(\mathbb{R}{\times}\mathbb{R},\mathbb{R}\right)$ satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2π-periodic odd/even sign-changing solutions set under some parity conditions. The proof of our main results are based upon bifurcation techniques.