一类非局部近共振问题多重解的存在性
Existence of Multiple Solutions for a Class of Nonlocal Near Resonance Problems

$通过变分方法在光滑有界域Ω上研究由常数a，b＞0，参数λ＞0及连续函数f(x，u)共同决定的非局部问题：$ \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.$
$In this paper, we use the variational method to study the following nonlocal problems in the smooth bounded domain Ω, which are determined by the constant a, b > 0, the parameter λ > 0 and the continuous function f(x, u): $ \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.