论文标题
在线性和超线性项的综合效果下，无限期的基尔chhoff型方程
On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms
论文作者
论文摘要
我们研究了一类Kirchhoff类型方程，涉及线性和超线性术语的组合如下：\ begin {equination*} - \ left（a \ int _ {\ mathbb {\ mathbb {r}^{n}}}} | \ nabla ΔU+μV（x）u =λf（x）u+g（x）| u |^{p-2} u \ quad \ text {in} \ mathbb {r}^{n}，\ end \ end {equation*}％ $ v \在c（\ mathbb {r}^{n}）$中是一种潜在的良好，对于底部$ω：= int \ {x \ in \ mathbb {r}^{r}^{n}^{n} \ | \ | \ | \ v（x）= 0 \} $。当$ n = 3 $和$ 4 <p <6 $，对于每个$ a> 0 $和$μ$足够大，我们得到至少存在一个阳性解决方案，$％0 <λ\leqλ_{1}（f_Ω）$，而至少有两个正面解决方案，而至少存在$λ_{1}（f_}（f_ {ω}）<{f_ {ω} <ω{ω） λ<λ_{1}（f_Ω）+δ_{a} $在积分$％\ int_ {ω} g（x）G（x）ϕ_ {1}^{p} dx $，$λ_{1}（f_ {ω}> 0 $ inmal eigenvalue）中$ h_ {0}^{1}（ω）$具有权重函数$ f_ {ω}：= f | _ {ω} $，$ ϕ_ {1}> 0 $是相应的主eigenFunction。当$ n \ geq 3 $和$ 2 <p <\ min \ {4,2^{\ ast} \} $，对于$％μ$，我们得出的结论是，$（i）$至少存在两个正面解决方案，$ a> $ a> $ a> $ a> $ a> $ a> $ a> $ a> $ 0 <λ__{1}}（1}}（f _}（f _}）$; $％（ii）$在经典假设$ \ int_ {ω} g（x）ϕ_ {1}^{p} dx <0 $中，至少存在三个正面解决方案，对于$ a> 0 $ a> $ a> $ a> $ small和$λ_{1}（f_ {ω}）（f_ {ω}） _ {a} $; $（iii）$在假设$ \ int_ {ω} g（x）ϕ_ {1}^{p} dx> 0 $中，至少存在两个积极解决方案，对于$ a> a_> a_> a_ {0}（p）$ and $λ^{+} {+} _ {+} _ {a} _ {a}<λ_<λ_{1} $ a $和$λ^{+} _ {a} \ geq0 $。
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 $λ^{+}_{a}\geq0$.
