%0 Journal Article
%T 一类非局部近共振问题多重解的存在性<br>Existence of Multiple Solutions for a Class of Nonlocal Near Resonance Problems
%A 王跃
%A 梁金平
%A 索洪敏&lt
%A br&gt
%A WANG Yue
%A LIANG Jin-ping
%A SUO Hong-min
%J 西南大学学报(自然科学版)
%D 2018
%R 10.13718/j.cnki.xdzk.2018.04.009
%X $通过变分方法在光滑有界域<i>Ω</i>上研究由常数<i>a</i>，<i>b</i>＞0，参数<i>λ</i>＞0及连续函数<i>f</i>(<i>x</i>，<i>u</i>)共同决定的非局部问题：$
\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 + b\lambda {u^3} = f\left( {x,u} \right)\\
u = 0
\end{array}&\begin{array}{l}
x \in \mathit{\Omega }\\
x \in \partial \mathit{\Omega }
\end{array}
\end{array}} \right.
$
        
    


利用Ekeland变分原理和山路引理得到该问题近共振情形多重解的存在性.$<br>$In this paper, we use the variational method to study the following nonlocal problems in the smooth bounded domain <i>Ω</i>, which are determined by the constant <i>a</i>, <i>b</i> > 0, the parameter <i>λ</i> > 0 and the continuous function <i>f</i>(<i>x</i>, <i>u</i>):


    
    
        
$
\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 + b\lambda {u^3} = f\left( {x,u} \right)\\
u = 0
\end{array}&\begin{array}{l}
x \in \mathit{\Omega }\\
x \in \partial \mathit{\Omega }
\end{array}
\end{array}} \right.
$
        
    


The existence and multiple solutions are obtained for this class of problems with near resonance by the Ekeland variational principle and a mountain pass lemma.
%K 非局部问题
%K 近共振
%K 变分方法
%K Ekeland变分原理
%K 多重解&lt
%K br&gt
%K nonlocal problem
%K near resonance
%K variational method
%K Ekeland's variational principle
%K multiple solution
%U http://xbgjxt.swu.edu.cn/jsuns/html/jsuns/2018/4/201804009.htm