On finding solutions of a Kirchhoff type problem

Abstract:

Consider the Kirchhoff type problem \begin{equation}\tag {$\mathcal {P}$} \left \{ \begin {aligned} -\bigg (a+b\int _{\mathbb {B}_R}|\nabla u|^2dx\bigg )\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &&\text {in $\mathbb {B}_R$}, \\ u&>0, &&\text {in $\mathbb {B}_R$},\\ u&=0, &&\text {on $\partial \mathbb {B}_R$}, \end{aligned} \right . \end{equation} where $\mathbb {B}_R\subset \mathbb {R}^N(N\geq 3)$ is a ball, $2\leq q<p\leq 2^*:=\frac {2N}{N-2}$ and $a$, $b$, $\lambda$, $\mu$ are positive parameters. By introducing some new ideas and using the well-known results of the problem $(\mathcal {P})$ in the cases of $a=\mu =1$ and $b=0$, we obtain some special kinds of solutions to $(\mathcal {P})$ for all $N\geq 3$ with precise expressions on the parameters $a$, $b$, $\lambda$, $\mu$, which reveals some new phenomenons of the solutions to the problem $(\mathcal {P})$. It is also worth pointing out that it seems to be the first time that the solutions of $(\mathcal {P})$ can be expressed precisely on the parameters $a$, $b$, $\lambda$, $\mu$, and our results in dimension four also give a partial answer to Naimen’s open problems [J. Differential Equations 257 (2014), 1168–1193]. Furthermore, our results in dimension four seem to be almost “optimal”.