The Study Of The Sign-changing Solutions And Positive Solutions For P(x)-Laplacian Kirchhoff Type Problem

This article investigates the p(x)-Laplacian Kirchhoff type equation.Based on the variational methods,deformation lemma and other technique of analysis,it is proved that the problem possesses one least energy sign-changing solution which has precisely two nodal domains.Then,by using the variational method and Nehari manifold approach,the existence and multiplicity of positive solutions are studied.In chapter 2,the purpose is to consider the existence of sign-changing solutions for variable exponent Kirchhoff type problem in RN with N?2,namely,(?) where a>0,b>0,-?p(x)u?div(|?v|p(x)-2?u)is the p(x)-Laplacian operator,the function f(x,u)is the external force.When parameter b?0,then the equation(Pb)reduces to-a?p(x)u+V(x)|u|p(x)-2u=f(x,u).(PO)We will prove there exist convergence ub?u0 if the nonlinear term satisfies some conditions,where uo is the least energy sign-changing solution of equation(PO).Chapter 3 mainly consider the existence and multiplicity of positive solutions of the equation(?) where ?(?)RN,N?3 is bounded domain,the parameter ?>0,p(x),q(x),h(x),s1(x),s2(x)?C(?).We will prove the equation(P?)exists at least two positive solutions when A small enough.