The Existence Of Nontrivial Solutions For Two Kinds Of Kirchhoff Type Problems In Whole Space

In this paper, firstly, the existence of positive solutions for a kind of Kirchhoff type systems with typical nonlinearity in M3 is studied by using variational methods and mountain pass lemma. Secondly, the existence of ground state solutions for p-Kirchhoff type equation without A-R conditions in RN is considered by using variational methods and constructing an appropriate manifold.Firstly, the existence of positive solutions for a kind of Kirchhoff type systems with typical nonlinearity is considered, where a> 0, b≥ 0 are constants,4<2q< 6, V∈C(M3, R), inf V (x)≥a0> 0, and (?)M> 0, meas{x∈M3:V(x)≤M}<+∞, where a0 is a constant and meas means Lebesgue measure in M3.The main result is as follows.Theorem 1 For any k> max{k1,k2}, the problem (P1) possesses at least a positive solution, whereSecondly,the existence of positive ground state solutions for the following p-Kirchhoff equation is investigated, where n>0,b)≥0 are constants,V∞>0,N≥3,and p satisfies f∈C1(R+,R),we define,(s)≡0 for s≤0 and p*=Np/N-p.f satisfies (f1)limf(s)/sp-1=p sâ†'0+ (f2)lim f’(s)sp&-2=0;sâ†'+∞ (f3)limf(s)sp-1=+∞;sâ†'+∞ (f4)f(s)sp-1 is striictly increasing in(0,+∞).The main result is as follows. (f1)-(f4),problem(P2)has a positive ground state solution.The structure of this paper is as follows.In the first chapter,we introduce the background of Kirchhoff type problems and give the main results of this paper.In the second chapter,firstly,we give some necessary prelliminaries of the problem(P1). Secondly,we give the proof of the main result.In the third chapter,firstly,we give some necessary pre’ liminaries of the problem(P2). Secondly,we give the proof of the main result.