Kirchhoff-type Equation, Existence And Multiplicity

In this paper,we firstly study the existence and multiplicity of weak solutions for a class of Kirchhoff type problems with Dirichlet boundary value condition by using the mountain pass theorem,the local linking theorem,the fountain theorem and the symmetric mountain pass lemma in critical point theory. Secondly,we study the resonance problem for Kirchhoff type equations with Dirichlet boundary value condition by using the linking theorem and the least action principle.Firstly,we consider the follwing Kirchhoff type problems with Dirichlet bound-ary value conditions whereΩis a smooth bounded domain in RN(N=1,2,3),a,b>0,and f∈C(Ω×R) satisfying (f1)|f(x,t)|≤C(|t|q-1+1),for some 40,such that F(x,t)≤a/2λ1t2 for |t|<δ,whereλ1 donotes thefirst eigenvalue of(-â–³,H01(Ω))ï¼›(f5)limtâ†'+∞f(x,t)/t3=+∞uniformly for a.e.x∈Ω. Then problem(P1)has at least one positive solution.Theorem 2 Suppose that f(x.t)satisfies(f1),(f2)and the following condi-tions hold: (f6)There existsδ>0 and k∈N,such that a/2λkt2+b/4λk2|Ω|t4≤F(x,t)≤a/2λk+1t2+b/8μ(?)t4 for all |t|≤δ,where 0<λ10 are constant:g∈C(R,R)and h∈L2(Ω).A constant A is called to be an eigenvalue for the nonlinear eigenvalue problem if problem has nontrivial solutions,and we denote by Aλthe set of souutions(called 3igenfunc-tion)for a givenλ.We make the following assumptions: (g1)lim(?)â†'∞g(t)/t3=0. Define ant let F(+∞)=(?) F(f), F(+∞)=(?) F(t), F(-∞)=(?)F(t),F(-∞)=(?)F(t).Now we state our main results.Theorem 5 Suppose that a>0,g∈C(R,R)satisfies(g1),h∈L2(Ω)and F(-∞)>-∞,F(+∞)<+∞.Then problem(P2)has at least one solution.Theorem 6 Suppose that a=0,g∈C(R,R)satisfies(g1).h∈L2(Q)and F(+∞)>-∞F(-∞)<+∞.Assume that the following condition is satisfied for all v∈Aλ\={0}.where v+=max{v(x),0},v-=(-v)+.Then problem(P2) has at least one solution.Theorem 7 Suppose that a=0,g∈C(R,R)satisfies(g1),h∈L2(Q)and F(-∞)>-∞,F(+∞)<+∞.Assume that the following condition is satisfied for all v∈Aλ\{0},where v+=max{v,(x),0),v-=(-v)+.Then problem(P2) has at least one solution.