Unilateral global interval bifurcation for Kirchhoff type problems and its applications

In this paper, we establish a unilateral global bifurcation result from interval for a class of Kirchhoff type problems with nondifferentiable nonlinearity. By applying the above result, we shall prove the existence of one-sign solutions for the following Kirchhoff type problems.

$\left\{ {\begin{array}{*{20}{l}} { - M(\int_\Omega {|\nabla u{|^2}dx} )\Delta u = \alpha (x){u^ + } + \beta (x){u^ - } + ra(x)f(u),}&{{\text{in}}{\mkern 1mu} \;\Omega ,} \\ {u = 0,}&{{\text{on}}{\mkern 1mu} \;\partial \Omega ,} \end{array}} \right.$

where Ω is a bounded domain in $ \mathbb{R}^{N} $ with a smooth boundary $ \partial $Ω, $ M $ is a continuous function, r is a parameter, $ a(x) \in C(\overline \Omega ) $ is positive, $ u^{+} = \max\{u, 0\}, u^{-}= -\min\{u, 0\} $, $ \alpha ,\beta \in C\left( {\overline \Omega } \right) $; $ f \in C\left( {\mathbb{R},\mathbb{R}} \right) $, $ sf(s)>0 $ for $ s \in {\mathbb{R}^ + }, $ and $ {f_0} \in \left( {0,\infty } \right) $ and $ {f_\infty } \in \left( {0,\infty } \right] $ or $ {f_0} \in \infty $ and f_∞∈[0, ∞], where $ {f_0} = {\lim _{\left| s \right| \to 0}}f\left( s \right)/s,{f_\infty } = {\lim _{\left| s \right| \to + \infty }}f\left( s \right)/s $. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.