Multiple solutions to the nonhomogeneous pp-Kirchhoff elliptic equation with concave–convex nonlinearities

In this paper, we study the multiplicity of solutions for the nonhomogeneous pp-Kirchhoff elliptic equation equation(0.1)−M(‖∇u‖pp)Δpu=λh1(x)|u|q−2u+h2(x)|u|r−2u+h3(x),x∈Ω, with zero Dirichlet boundary condition on ∂Ω∂Ω, where ΩΩ is the complement of a smooth bounded domain DD in RN(N≥3)RN(N≥3). λ>0λ>0, M(s)=a+bsk,a,b>0,k≥0M(s)=a+bsk,a,b>0,k≥0, h1(x),h2(x)h1(x),h2(x) and h3(x)h3(x) are continuous functions which may change sign on ΩΩ. The parameters p,q,rp,q,r satisfy 1