Study On The Solutions Of One Kind Of Kirchhoff Type Equations With Negative Modulus

In this paper,we study the existence,multiplicity and nonexistence of two kinds of Kirchhoff equations with different nonlinear growth terms when Young’s modulus is negative by using the mountain pass lemma,Pohozaev identity,minimax critical point theorem and concentration compactness principle.In this paper,we respectively to consider the following p-Kirchhoff type equations with subcritical growth case and critical growth case:(?)where Ω?RN(N≥3)is a smooth bounded region,the constants a,b>0,r>0 and p>1.According to the minimax critical point theorem and the mountain pass lemma,we prove the existence and multiplicity of nontrivial solutions of equation(1).At the same time,through a new Pohozaev identity,we obtain the nonexistence of nontrivial solution of equation(1)in the critical case.We find that when the nonlinear growth term at the right side of the equation is critical,the Sobolev embedding is no longer compact and the energy functional related to the Palais-Smale condition cannot obtain the exact range.Therefore,in order to overcome this difficulty,we obtain the existence of nontrivial solutions to equation(2)by using the concentration compactness principle.For the convenience of description,let’s assume that the following conditions for f:(f1)There exist constants c0>0,p>1 and q∈(p,p*),such that (?) for all x∈Ω,t∈R;(f2)There exist p<μ<p(r+1),M≥0 such that 0<μF(x,t)≤tf(x,t)for any|t|>M and for almost every x∈Ω,where (?) uniformly in x∈Ω.