一类非局部问题解的存在性与多重性
Existence and Multiplicity of Solutions for a Class of Nonlocal Problems

$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a＞0，b＞0，$ \mathit{\Omega } \subset {{\mathbb{R}}^N} $是有界开集，λ＞0且g∈H^-1(Ω)\{0}，这里H^-1(Ω)是Sobolev空间H₀¹(Ω)的对偶空间.应用Ekeland变分原理和山路引理证明了：存在λ_*＞0，使得：(ⅰ)当λ∈(0，λ_*)时，该非局部问题至少有3个不同的解；(ⅱ)当λ=λ_*时，该非局部问题至少有2个不同的解；(ⅲ)当λ＞λ_*时，该非局部问题至少有1个解.$
$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 a > 0, b > 0, $ \mathit{\Omega } \subset {{\mathbb{R}}^N} $ is a bounded open set, λ > 0 and g∈H^-1(Ω)\{0}. The Ekeland's variational principle and the mountain pass lemma are applied to proved that there exists λ_* > 0 such that (ⅰ)The problem has at least three solutions if λ∈(0, λ_*); (ⅱ)The problem has at least two solutions if λ=λ_*; (ⅲ)The problem has at least one solution if λ > λ_*.