On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms
Juntao Sun, Kuan-Hsiang Wang, Tsung-fang Wu
Abstract
We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) Δu+μV(x)u=λf(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C(\mathbb{R}^{N})$ is a potential well with the bottom $Ω:=int\{x\in \mathbb{R}^{N}\ |\ V(x)=0\}$. When $N=3$ and $4<p<6$, for each $a>0$ and $μ$ sufficiently large, we obtain that at least one positive solution exists for $% 0<λ\leqλ_{1}(f_Ω) $ while at least two positive solutions exist for $λ_{1}(f_{Ω})< λ<λ_{1}(f_Ω)+δ_{a}$ without any assumption on the integral $% \int_{Ω}g(x)φ_{1}^{p}dx$, where $λ_{1}(f_{Ω})>0$ is the principal eigenvalue of $-Δ$ in $H_{0}^{1}(Ω)$ with weight function $f_{Ω}:=f|_{Ω}$, and $φ_{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2<p<\min \{4,2^{\ast }\}$, for $% μ$ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<λ<λ_{1}(f_{Ω})$; $% (ii)$ under the classical assumption $\int_{Ω}g(x)φ_{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $λ_{1}(f_{Ω})\leq λ<λ_{1}(f_Ω)+\overline{δ}% _{a} $; $(iii)$ under the assumption $\int_{Ω}g(x)φ_{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $λ^{+}_{a}< λ<λ_{1}(f_Ω)$ for some $a_{0}(p)>0$ and $λ^{+}_{a}\geq0$.