On finding solutions of a Kirchhoff type problem

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)Δu&= λu^{q-1} + μu^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\ u&=0,&\quad\text{on}\partial\mathbb{B}_R, \endaligned \right.\eqno{(\mathcal{P})} $$ where $\mathbb{B}_R\subset \bbr^N(N\geq3)$ is a ball, $2\leq q<p\leq2^*:=\frac{2N}{N-2}$ and $a$, $b$, $λ$, $μ$ are positive parameters. By introducing some new ideas and using the well-known results of the problem $(\mathcal{P})$ in the cases of $a=μ=1$ and $b=0$, we obtain some special kinds of solutions to $(\mathcal{P})$ for all $N\geq3$ with precise expressions on the parameters $a$, $b$, $λ$, $μ$, which reveals some new phenomenons of the solutions to the problem $(\mathcal{P})$. It is also worth to point out that it seems to be the first time that the solutions of $(\mathcal{P})$ can be expressed precisely on the parameters $a$, $b$, $λ$, $μ$, and our results in dimension four also give a partial answer to Neimen's open problems [J. Differential Equations, 257 (2014), 1168--1193]. Furthermore, our results in dimension four seems to be almost "optimal".

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