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Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth

In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_Ω|\nabla u|^2dx)Δu=|u|^4u+λ|u|^{q-2}u, u=0\ \ \text{on}\ \ \partialΩ, \end{array}% \right. $$ where $1<q<2$, $λ,\ a,\ b>0$ are parameters and $Ω$ is a bounded domain in $\R^3$. We prove that there exists $λ_1=λ_1(q,Ω)>0$ such that for any $λ\in(0,λ_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: $$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)Δu+u=Q(x)|u|^4u+λf(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$ for any $a,\ b>0$ and $λ\in \big(0,λ_0(q,Q,f)\big)$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between $λ_0$ and $b$ to show that for any (large) $λ>0$, there exists a $b_0(λ)>0$ such that the above results hold when $b>b_0(λ)$ and $a>0$.