Existence And Non-existence Of Positive Solutions And Sign-changing Solutions For Two Classes Of Kirchhoff-type Problems

In this paper,we study the existence and non-existence of solutions for two classes of Kirchhoff-type equations with different nonlinearities.Firstly,we consider the following Kirchhoff-type problem:(?) where ?(?)RN(N=1,2,3)is a bounded domain with smooth boundary(?)?,a>0,b>0,and ?,? are two real parameters.By using Mountain Pass Lemma and the Nehari manifold,we provide a description of a two-dimensional set in the(?,?)plane,which corresponds to the existence and non-existence of positive solutions for the above Kirchhoff type equation.Next,Combining Nehari mainfold and deformation lemma,we prove the exis-tence and non-existence of sign-changing solutions for problem(0.1)if(?,?)lies in the different range.Finally,we consider the following Kirchhoff-type problem with concave-convex nonlinearities(?) where ? is a bounded domain with smooth boundary(?)? in ??(V=1,2,3),a>0,b>0,0<q<1,3<p<5,and ? is a real parameter.By constraining the energy functional of the above problem on a subset M?*of the Nehari manifold corresponding to the problem,we show that there exists a constant ?*>0 such that for any ??(-?,?*),the above problem has a sigh-changing solution u??M?*with positive energy.