| We mainly studies the existence and multiplicity of sign-changing solutions of two types of nonlocal equations by using invariant sets of descending flow and variational methods.Firstly,we consider the following Kirchhoff-type problem:where 4<p<6,a,b>0,the potential Vλ(x)=λa(x)+a0(x)satisfies the following conditions:(V1)a0(x)∈ L∞(R3)and α0:=ess inf a0(x)>0.(V2)a(x)∈Lloc∞(R3),a(x)≥ 0 and there exists a closed subset Z?R3 with non-empty interior Ω=intZ such that a(x)=0 for x∈Z and a(x)>0 a.e.in R3\Z.(V3)There exists M0>0 such that {x∈R3:a(x)<M0} with non-empty interior has finite positive Lebesgue measure.Then there exists Λ>0 such that problem(0.0.1)possesses a sign-changing solution for λ>A.Next,we consider the following Schrodinger-Poisson system with two nonlocal nonlinearities:where 1<q<2<p<6,and the functions V(x),g(x)satisfy the following conditions:(V)V ∈C(R3,R)is radially symmetric and(?)V(x):=V0>0,(g)g(x)∈ Lp/p-q(R3)∩ C(R3,R)is a nonnegative function and g(x)(?)0 in R3.Using the variational method,through the invariant sets of descending flow and the abstract critical point theorems,assuming that the hypothesis(V)and(g)are established,when |g|p/p-q is small enough,there are three mountain-type nontrivial solutions of problem(0.0.2),which are a positive solution,a negative solution and a sign-changing solution.Still assuming that the conditions(V)and(g)are true,there exists Λ>0 such that problem(0.0.2)has infinitely many high energy sign-changing solutions for all 0<|g|p/p-q<A. |
