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 Ω(?)RN(N=1,2,3)is a bounded domain with smooth boundary(?)Ω,a>0,b>0,and α,β are two real parameters.By using Mountain Pass Lemma and the Nehari manifold,we provide a description of a two-dimensional set in the(α,β)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(α,β)lies in the different range.Finally,we consider the following Kirchhoff-type problem with concave-convex nonlinearities(?) where Ω is a bounded domain with smooth boundary(?)Ω in ΔΨ(V=1,2,3),a>0,b>0,0<q<1,3<p<5,and λ is a real parameter.By constraining the energy functional of the above problem on a subset Mλ*of the Nehari manifold corresponding to the problem,we show that there exists a constant λ*>0 such that for any λ∈(-∞,λ*),the above problem has a sigh-changing solution uλ∈Mλ*with positive energy.