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

We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mspace width="0.3333em"/> <m:mspace width="0.3333em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>,</m:mo> </m:math> -(u√1-u²)=λ a(x)f(u,u), x∈ R, where λ > 0 is a parameter, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> a∈ C(R,R) is a 2π-periodic sign-changing function with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>π</m:mi> </m:mrow> </m:msubsup> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo><</m:mo> <m:mn>0</m:mn> </m:math> ∫ ₀2πa(x)dx< 0 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> f∈ C(R×R,R) 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.