Existence And Multiplicity Of Solutions For Kirchhoff Type Equations

This paper contains three chapters:the first chapter is the introduction, the second chapter studies the existence and multiplicity of Kirchhoff equation in a smooth bounded domain by using the mountain pass theorem and the fountain theorem. In the third chapter, we discuss Kirchhoff type equation by using cut-off technique and the mountain theorem. We get that the equation has at least a nontrivial non-negative solution and a nontrivial non-positive solution.The second chapter we consider the Kirchhoff problem where Ω is a smooth bounded domain in RN, λ> 0,a, b> 0, N≥3, q∈(4,6) and f∈ C(Ω×R,R) satisfies(f0) There exists C> 0 such that|f(x,t)|≤C(1+|t|p-1), (x,t) ∈Ω×R, where p∈(2,2*),2*=2N/(N-2);(f1) There exists R1>0 such that F(x,t)≥bμ1/2λ |t|4 for all|t|> R1, where μ1 is the first eigenvalues of the equation (2.2.2), F(x, t)=∫t0f(x, s)ds, (x,t)∈Ω xR;(f2) limtâ†'0 f(x, t)/t=0 uniformly in x∈Ω;(f3) There exists a>0,R2>1,τ∈[0,2] such that tf(x,t)-4F(x,t)>-α|t|Ï„,|t|> R2,x∈Ω;(f4)f(x,-t)=-f(x,t), (x, t)∈Ω×R.We have the following conclusion:Theorem 2.1.1 Assume f satisfies these conditions (fo)-(f3),0<λ<aλ1/α. Then the problem (1) has at least a nontrivial solution.Theorem2.2.2 Assume f satisfies these conditions (fo)-(f4), p<q and 0<λ<aλ1/α. Then the problem (1) has infinitely many solutions.The third chapter discusses the following Kirchhoff equation where Ω is a smooth bounded domain in RN,N≥3,a,b>0,and f satisfies(f0)f∈C(Ω×R,R);(f1)There exists q∈(2,2*),C>0 such that |f(x,t)|≤C(1+|t|q-1),(x,t)∈Ω×R, where 2’=2N/(N-2);(f2)llmtâ†'0 f(x,t)/t=0 uniformly in x∈Ω;(f3)There exists θ>4,r>0 such that 0<θF(x,t)≤tf(x,t)for all x∈Ω,|t|≥r, where F(x,t)=∫0t(x,s)ds,(x,t)∈Ω×R;(f4)f(x,0)=0,x∈Ω.We have the following conclusion:Theorem3.1.1 Assume f satisfies these conditions(f0)-(f4).Then the problem(2)has at least a nontrivial non-negative and a nontrivial non-positive solution.