Continuum of one-sign solutions of one-dimensional Minkowski-curvature
problem with nonlinear boundary conditions
Abstract
In this work, we investigate the continuum of one-sign solutions of the
nonlinear one-dimensional Minkowski-curvature equation
$$-\big(u’/\sqrt{1-\kappa
u’^2}\big)’=\lambda
f(t,u),\ \ t\in(0,1)$$
with nonlinear boundary conditions $u(0)=\lambda
g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral
global bifurcation techniques, where
$\kappa>0$ is a constant,
$\lambda>0$ is a parameter
$g_1,g_2:[0,\infty)\to
(0,\infty)$ are continuous functions and
$f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$
is a continuous function. We prove the existence and multiplicity of
one-sign solutions according to different asymptotic behaviors of
nonlinearity near zero.
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