On a Kirchhoff Equation in Bounded Domains

Abstract

Abstract In this paper, we consider the following Kirchhoff equation: { - ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ u + | u | p - 2 u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}\bigg{(}a+b\int_{\Omega}\lvert\nabla u% |^{2}\,dx\bigg{)}\Delta u=\lambda u+|u|^{p-2}u&&\displaystyle\text{in }\Omega,% \\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) is a bounded domain with smooth boundary ∂ ⁡ Ω {\partial\Omega} , 2 < p < 2 * = 2 ⁢ N N - 2 {2<p<2^{*}=\frac{2N}{N-2}} is the Sobolev exponent and a, b, λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of Ω = 𝔹 R {\Omega=\mathbb{B}_{R}} ) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as b → 0 + {b\to 0^{+}} , the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.