%0 Journal Article
%T 一类非局部问题解的存在性与多重性<br>Existence and Multiplicity of Solutions for a Class of Nonlocal Problems
%A 唐之韵
%A 欧增奇&lt
%A br&gt
%A TANG Zhi-yun
%A OU Zeng-qi
%J 西南大学学报(自然科学版)
%D 2018
%R 10.13718/j.cnki.xdzk.2018.04.008
%X $考虑一类非局部问题


    
    
        
$
\left\{ {\begin{array}{*{20}{c}}
\begin{array}{l}
 - \left( {a - b\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = \lambda g\left( x \right)\\
u = 0
\end{array}&\begin{array}{l}
x \in \mathit{\Omega }\\
x \in \partial \mathit{\Omega }
\end{array}
\end{array}} \right.
$
        
    


其中<i>a</i>＞0，<i>b</i>＞0，<inline-formula>$ \mathit{\Omega } \subset {{\mathbb{R}}^N} $</inline-formula>是有界开集，<i>λ</i>＞0且<i>g</i>∈<i>H</i><sup>-1</sup>(<i>Ω</i>)\{0}，这里<i>H</i><sup>-1</sup>(<i>Ω</i>)是Sobolev空间<i>H</i><sub>0</sub><sup>1</sup>(<i>Ω</i>)的对偶空间.应用Ekeland变分原理和山路引理证明了：存在<i>λ</i><sub>*</sub>＞0，使得：(ⅰ)当<i>λ</i>∈(0，<i>λ</i><sub>*</sub>)时，该非局部问题至少有3个不同的解；(ⅱ)当<i>λ</i>=<i>λ</i><sub>*</sub>时，该非局部问题至少有2个不同的解；(ⅲ)当<i>λ</i>＞<i>λ</i><sub>*</sub>时，该非局部问题至少有1个解.$<br>$Consider a class of nonlocal problems


    
    
        
$
\left\{ {\begin{array}{*{20}{c}}
\begin{array}{l}
 - \left( {a - b\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = \lambda g\left( x \right)\\
u = 0
\end{array}&\begin{array}{l}
x \in \mathit{\Omega }\\
x \in \partial \mathit{\Omega }
\end{array}
\end{array}} \right.
$
        
    


where <i>a</i> > 0, <i>b</i> > 0, <inline-formula>$ \mathit{\Omega } \subset {{\mathbb{R}}^N} $</inline-formula> is a bounded open set, <i>λ</i> > 0 and <i>g</i>∈<i>H</i><sup>-1</sup>(<i>Ω</i>)\{0}. The Ekeland's variational principle and the mountain pass lemma are applied to proved that there exists <i>λ</i><sub>*</sub> > 0 such that

(ⅰ)The problem has at least three solutions if <i>λ</i>∈(0, <i>λ</i><sub>*</sub>);

(ⅱ)The problem has at least two solutions if <i>λ</i>=<i>λ</i><sub>*</sub>;

(ⅲ)The problem has at least one solution if <i>λ</i> > <i>λ</i><sub>*</sub>.
%K 非局部问题
%K Ekeland变分原理
%K 山路引理
%K (PS)&lt
%K sub&gt
%K &lt
%K i&gt
%K c&lt
%K /i&gt
%K &lt
%K /sub&gt
%K 条件&lt
%K br&gt
%K nonlocal problems
%K Ekeland's variational principle
%K Mountain pass lemma
%K the (PS)&lt
%K sub&gt
%K &lt
%K i&gt
%K c&lt
%K /i&gt
%K &lt
%K /sub&gt
%K condition
%U http://xbgjxt.swu.edu.cn/jsuns/html/jsuns/2018/4/201804008.htm