Existence Of Non-trivial Solutions For The Kirchhoff-type Equations With Fu(?)ik-type Resonance At Infinity

In this paper,we obtain the existence of non-trivial solutions for the Kirchhoff-type equations with Fucik-type resonance at infinity by variational method.Firstly,we consider the following Kirchhoff-type problem:(?)where Ω is an open ball in RN(N=1,2,3),α,β∈R,u+=max{u,0} u-=min {u,0},and u=u++u-.The nonlinear term f∈ C(Ω × R,R)satisfies f(x,0)=0,namely problem(0.1)has a trivial solution u≠0.By using Mountain Pass Theorem with(Ce)condition,we obtain the existence of nontrivial solutions for problem(0.1)on two trivial curves of Fucik spectrum.Next,we consider the existence of nontrivial solutions for the Kirchhoff-type equation with Fucik-type resonance at infinity where Ω is an open ball in RN(N=1,2,3),a>0,b>0,α,β∈R,u+=max{u,0},u-=min{u,0},and u=u+u-.The nonlinear term f∈C(Ω× R,R).By using Mountain Pass Theorem,Deformation lemma and Γ-Linking Theorem,we obtained the existence of nontrivial solutions for problem(0.2)on two trivial and nontrivial curves of Fucik spectrum.