Existence Of Positive Solutions Of Kirchhoff Type Equations

In this paper,we mainly study the existence of positive solutions for two kinds of Kirchhoff type equations in bounded domain.Let ? be a bounded domain in RN(N?2)with smooth boundary (?)?.We first consider Dirichlet problem of inhomogeneous Kirchhoff type equation(?)(1)where b>0,p>1,?>0,??(0,2*-1/2)with 2*=+? for N=2,and 2*=N+2/N-2 for N?3.Denote by M the subset of C1(?)\{0} such that for any f(x)?M the following problem(?)(2)has a nontrivial solution.Main results we proved in the present paper can be summarized asTheorem 1 If 1<p<2?+1 and f(x)?M,then problem(1)has at least one positive solution for any ?>0.Theorem 2 If 1<p<2a+1 and b>b0 for some positive number b10 given by Chapter one,then problem(1)has positive solution for any ?>0 if and only if f(x)?M.Moreover,the solution is unique for ? small enough if in addition ??1/2.Theorem 3 If 2?1<p<2*and f(x)?M,then there are two positive constants ?f,?f<+? such that problem(1)has at least two positive solutions for??(0,?f),and has no positive solution for ?>?f.Theorem 4 If p>2*and ? is starshaped,then there are two positive constants?f,?f<+? such that problem(1)has positive solution for any ??(0,?f)if and only if f(x)?M,and has no positive solution for ?>?f.Secondly,we also concern the existence of positive solutions for the Dirichlet problem of the following p-kirchhoff type equations(?)(3)where ?pu=div{|?u|p-2?u)is said to be p-Laplacian operator?a>0,b>0.1<p<N,p-1<q<p*-1,0<?<p*-p/p,p*=Np/N-p.We first discuss the simple model of h(x,u,?u)=0,and analyze the influence of nonlocal term on the existence,number and gradualness of the positive solution of the problem(3),for more detailed explanation,see Chapter 3.1.Then we use the blow up method to obtain the prior estimate of the positive solution of the auxiliary problem.Finally,we use the continuity method to prove the existence of the positive solution of the problem(3).Add the following conditions to h(x,s,?):(H1)For s>0,h(x,s,?)is a continuous functions and h(x,s,?)?0;(H2)If p-1<q<p*-1,we assume that there exists a positive constant ?such that lim h(x,s,?)/sp-1=? uniformly in x and ?,and |h(x,s,?)|?C(sp-1+sq1)for s>0,p-1<q1<q<p*-1.We come to the following conclusion:Theorem 5 Assume that p(?+1)-1<q<p*-1,(H1),(H2)and ?<a?1(?).Then,for a>0,b>0,the problem(3)has at least one positive solution.