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 ? is an open ball in RN(N=1,2,3),?,??R,u+=max{u,0} u-=min {u,0},and u=u++u-.The nonlinear term f? C(? × 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 ? is an open ball in RN(N=1,2,3),a>0,b>0,?,??R,u+=max{u,0},u-=min{u,0},and u=u+u-.The nonlinear term f?C(?× R,R).By using Mountain Pass Theorem,Deformation lemma and ?-Linking Theorem,we obtained the existence of nontrivial solutions for problem(0.2)on two trivial and nontrivial curves of Fucik spectrum.