Multiple positive solutions to a class of Kirchhoff equation on R3R3 with indefinite nonlinearity

We study the existence and multiplicity of positive solutions to a class of Kirchhoff equation −(1+b∫R3|∇u(x)|2dx)Δu(x)+u(x)=V(x)|u(x)|p−2u(x)+λk(x)u(x),−(1+b∫R3|∇u(x)|2dx)Δu(x)+u(x)=V(x)|u(x)|p−2u(x)+λk(x)u(x),u(x):=u∈H1(R3)u(x):=u∈H1(R3). When V(x)V(x) is sign changing in R3R3, we prove that there is δ̄>0 such that the Kirchhoff equation possesses at least one positive solution for 0<λ<λ1+δ̄ and at least two positive solutions for λ1<λ<λ1+δ̄. Here λ1λ1 is the first eigenvalue of −Δu+u=λk(x)u−Δu+u=λk(x)u, u∈H1(R3)u∈H1(R3). A positive eigenfunction corresponding to λ1λ1 is denoted by e1e1. An interesting phenomena is that the term b(∫R3|∇u|2dx)Δub(∫R3|∇u|2dx)Δu can be used to compete with the indefinite nonlinearity and we do not assume the condition ∫R3V(x)e1pdx<0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (Alama and Tarantello, 1993).