On finding solutions of a Kirchhoff type problem

TL;DR

This paper derives explicit solutions for a Kirchhoff type problem in a ball, revealing new phenomena and partially addressing open questions, especially in four dimensions, by building on known special cases.

Contribution

It introduces novel methods to explicitly express solutions of the Kirchhoff problem for all dimensions, connecting special cases and solving open problems in dimension four.

Findings

Explicit solutions expressed in parameters for all dimensions.

Revealed new phenomena in solution behavior.

Partially answered Neimen's open problems in dimension four.

Abstract

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\ u&=0,&\quad\text{on}\partial\mathbb{B}_R, \endaligned \right.\eqno{(\mathcal{P})} $$ where $\mathbb{B}_R\subset \bbr^N(N\geq3)$ is a ball, $2\leq q<p\leq2^*:=\frac{2N}{N-2}$ and $a$, $b$, $\lambda$, $\mu$ 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 $N\geq3$ 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 to point out that it seems to be the first time that the solutions of $(\mathcal{P})$ can be expressed precisely on the parameters $a$, $b$, $\lambda$, $\mu$, 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".