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

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′=λa(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 ∫02πa(x)dx<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.

Keywords

• minkowski-curvature operator
• bifurcation
• periodic problem
• sign-changing solutions
• positive solutions
• 34b15
• 34c23
• 34c25