Positive solution for the Kirchhoff‐type equation with supercritical concave and convex nonlinearities

We study the following Kirchhoff‐type equation 0.1 −a+b∫Ω|∇u|2dxΔu=|u|p−2u+λ|u|q−2u,x∈Ω,u=0,x∈∂Ω,$$\begin{equation}\hspace*{32pt} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll}-{\left(a+b\displaystyle \int _{\Omega }|\nabla u|^2dx\right)}\Delta u=|u|^{p-2}u+\lambda |u|^{q-2}u, \ & x\in \Omega,\\ u=0,\ & x\in \partial \Omega, \end{array} \right.} \end{equation}$$where a$a$, b>0$b>0$, λ>0$\lambda >0$ is a parameter, Ω⊂RN$\Omega \subset {\mathbb {R}}^N$ is a bounded domain with C2$\mathcal {C}^2$‐boundary and 1 2$p>2$, there exists λ∗>0$\lambda ^*>0$ such that for each λ∈(0,λ∗)$\lambda \in (0,\lambda ^*)$ Equation (0.1) has a positive solution with negative energy. Furthermore, by using the improved Clark theorem, we can obtain a sequence of solutions with negative energy converging to zero in L∞(Ω)$L^{\infty }(\Omega)$ without the restriction of λ$\lambda$.