Multiplicity Of Solutions For Two Classes Of Kirchhoff Type Equations With Concave Convex Nonlinearity

Using the variational method and some analytical techniques,we study the multiplicity of solutions for two kinds of Kirchhoff-type equations with concave and convex nonlinear terms.Firstly,we study the Kirchhoff-type problem of concave-convex nonlinear terms with critical exponents:-(a+b ∫Ω|▽u|2dx)△u=|u|4u+μ|u|q-2u x∈Ω,u=0,x∈(?)Ωwhere Ω(?)R3 is a mooth bounded domain,a,b>0,1<q<2,μ>0.Applying the principle of compactness and the dual fountain theorem,we obtain infinite so-lutions to the equation.The main conclusions are as follows:Theorem 1 Suppose that Ω(?)R3 is bounded and a,b>0,1<q<2,then there exists μ*>0,such that problem(0.0.1)has a sequence of solutions(un)such that φμ(un)<0,and φμ(un)→n→∞,for all 0<μ<μ*.Secondly,we consider the following kirchhoff problem with a subcritical critical point nonlinearity term:-(a+b∫Ω|▽u|2dx)△u=λf(x)|u|p-2u+g(x)|u|q-2u,x∈Ω,u=0,(0.0.2)where Ω(?)R3 is a smooth bounded domain,a,b≥ 0.a+b>0,2<p<4<q<6 and the parameter λ>0.The coefficient functions f,g are continuous functions and satisfy the following conditions:(H0)f∈ L6/6-p(Ω),g ∈L6/6-q(Ω)with the sets {x∈Ω:f(x)>0} and{x∈Ω:g(x)>0} of positive measures,that is,f,g≥0 or f,g change sign on Ω.(H1)f ∈ L°°(Ω)with the set {x∈Ω:f(x)>0} of positive measure and g ∈ L°°(Ω)with g(x)≥0,g≠0.(H2)f∈L6/6-p(Ω),g∈L6/6-q(Ωl)are nonzero and nonnegative functions.Applying the Nehari manifold and the fibering maps,we can prove the exis-tence of two nonnegative solutions.Under some stronger conditions and the strong maximum principle,this problem admits two positive solutions and one of them is a ground-state solution.The main conclusions are as follows:Theorem 2 Suppose that a,b>0 and 2<p<4<q<6.Then(ⅰ)if(H0)holds,then there exist δ,λ>0 for all 0<a<δ,0<λ<λ,problem(0.0.2)has at least two nonnegative solutions u+∈ N+and u-∈ N-and one of them is a ground state solution.(ⅱ)if(H1)holds,then the same conclusions of(ⅰ)hold.Moreover,the two nonneg-ative solutions are positive solutions.(ⅲ)if(H2)holds,then the same conclusions of(ⅱ)hold.Moreover,the positive ground state solution belongs to N+.Corollary 1 Suppose that a=0.b>0 and 2<p<4<q<5.Then(ⅰ)if(H0)holds,then there exists λ1>0,problem(0.0.2)has at least two nonneg-ative solutions u+and u-such that u±∈ N± for all 0<λ<λ1 and one of them is a ground state solution.(ⅱ)if(H1)holds,then the same conclusions of(⒋)hold.Moreover,the two nonneg-ative solutions are positive solutions(iii)if(H2)holds,then the same conclusions of(ii)hold.Moreover,the positive ground state solution belongs to N+...