| This paper is concerned with existence of solutions for Kirchhoff type equation depending on variational method,Morse theorem and the property that critical groups are invariant under homotopies preserving isolateness critical points at a regional level and satisfying functional non-resonance and resonance conditions,we can obtain an isolated critical point under such conditions.Using this results,we can get at least three nontrivial solutions using variational method,three critical points theorem and mountain theorem when the functional of this equation is coercive.Firstly,we consider the following Kirchhoff type equationWhere a,b>0 are real constants.Moreover,if we assume that(f0)f ∈ C1(Ω × R,R),f(x,0)=0,and there is c>0 such that|f’(x,u)|≤c(1+|u|r-2),2≤r<2*= +∞,N=1,2;2N/N-2,N≥3,We impose on f the following non-resonance and resonance conditions:(f1)there exist λ ∈ R such thatlimf(x,u)/au=λ,uniformly in x∈Ω;(f2)there exist a>0 such thatug(x,u)≤0,for |u|≤α,x∈Ω,Where g(x,u)=f(x,u)-aλ1u,Our results read as follows;Theorem 1.1 Assume that(f0)and(f1)hold.If λ∈(λk,λk+l)then u=0 is an isolated critical point of I such thatC*(1,0)=δ*,kF.Theoreml.2 Assume that(fo),(f1)and(f2)hold.If λ=λ1,then u=0 is an isolated critical point of I such thatC*(I,0)=δ*,0F.(f3)there exist M>0 snd β<aλ1/2 such thatF(x,u)-b/4μ|u|4≤βu2,for|u|≥M,x∈Ω,Our next result read as follows;Theorem 1.3 Assume that N≤3,(f0),(f1)and(f3)hold.Ifλ ∈(λk,λk+1)with k≥2,then equation(1.1)has at least three nontrivial solutions. |
