Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.2 ) Pub Date : 2020-08-20 , DOI: 10.1007/s40840-020-00995-8 Hui Zhang , Fubao Zhang
We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator \({\mathcal {L}}_K\)
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$where \(\Omega \) be an open-bounded set of \({\mathbb {R}}^n\) with continuous boundary, \(\lambda \in {\mathbb {R}}\) is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with \(\lambda <\lambda _1\), where \(\lambda _1\) is the first eigenvalue of the nonlocal operator \(-{\mathcal {L}}_K\) with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with \(\lambda \ge \lambda _1\). As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$where \(0<\alpha <1\). The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.
中文翻译：
具有非局部积分微分算子的方程解的存在性和多重性
我们关注以下带有一般非局部积分微分运算符\（{\ mathcal {L}} _ K \）的椭圆方程
$$ \ begin {aligned} \ begin {aligned} \ left \ {\ begin {array} {ll}-{\ mathcal {L}} _ Ku = \ lambda u + f（x，u），＆{} \ quad \ text {in} \ quad \ Omega，\\ u = 0，＆{} \ quad \ text {in} \ quad {\ mathbb {R}} ^ n {\ setminus} \ Omega，\ end {array} \对。\ end {aligned} \ end {aligned} $$其中\（\ Omega \）是具有连续边界的\（{\ mathbb {R}} ^ n \）的无界集合，\（\ lambda \ in {\ mathbb {R}} \}是一个实参，f是具有次临界增长的非线性项。我们显示了一个基态的存在和无限多对解。该证明基于Nehari流形的方法，用于\（\ lambda <\ lambda _1 \）的方程，其中\（\ lambda _1 \）是非局部算子\（-{\ mathcal {L} } _K \）具有齐次Dirichlet边界条件，以及带有（（\ lambda \ ge \ lambda _1 \）的方程的广义Nehari流形方法。作为一个具体的例子，我们导出分数拉普拉斯算子驱动的方程的解的存在性和多重性
$$ \ begin {aligned} \ begin {aligned} \ left \ {\ begin {array} {ll}（-\ Delta）^ \ alpha u = \ lambda u + f（x，u），＆{} \ quad \ text {in} \ quad \ Omega，\\ u = 0，＆{} \ quad \ text {in} \ quad {\ mathbb {R}} ^ n {\ setminus} \ Omega，\ end {array} \对。\ end {aligned} \ end {aligned} $$其中\（0 <\ alpha <1 \）。这里介绍的结果可以看作是拉普拉斯算子到非局部分数设置的一些经典结果的扩展。
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