Article Contents
Article Contents

# Unilateral global interval bifurcation for Kirchhoff type problems and its applications

The author is supported by NNSF of China (No. 11561038) and the National Science Foundation of Gansu (No.145RJZA087).
• 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.

Mathematics Subject Classification: Primary:35B32, 47J10;Secondary:35J65.

 Citation:

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