论文标题
关于分数$的广义特征值问题（P，Q）$ - 带有两个参数的拉普拉斯操作员
On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters
论文作者
论文摘要
对于$ s_1，s_2 \ in（0,1）$和$ p，q \ in（1，\ infty）$，我们研究了参数$α，β\ in \ mathbb {r} $驱动的两个非linelear eigenvalue问题$α，β\ in \ mathbb {r} \ begin {equation*} （-Δ）^{s_1} _p u+（ - δ）^{s_2} _q u =α| u | u |^{p-2} u+β| U | u |^{q-2} q-2} u \; \; \; \ \ \ \ \ \ \ \ \ text {in}ω， ω，\ \ \ \ \ qquad \ quad \ m马理{（p）} \ end {equation*}其中$ω\ subset \ mathbb {r}^d $是一个有限的开放集。根据$α，β$的值，我们完全描述了（p）阳性溶液的存在和不存在。我们在二维$（α，β）$ - 平面中构造连续阈值曲线，该平面分离了阳性溶液的存在区域和不存在的区域。此外，我们证明了$ p $ laplace和分数$ q $ -laplace运营商是线性独立的，这在曲线的形成中起着至关重要的作用。此外，我们确定（p）的每个非负解决方案都是全球界限的。
For $s_1,s_2\in(0,1)$ and $p,q \in (1, \infty)$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $α, β\in \mathbb{R}$ driven by the sum of two nonlocal operators: \begin{equation*} (-Δ)^{s_1}_p u+(-Δ)^{s_2}_q u=α|u|^{p-2}u+β|u|^{q-2}u\;\;\text{in }Ω, \quad u=0\;\;\text{in } \mathbb{R}^d \setminus Ω, \ \ \ \qquad \quad \mathrm{(P)} \end{equation*} where $Ω\subset \mathbb{R}^d$ is a bounded open set. Depending on the values of $α,β$, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(α, β)$-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$-Laplace and fractional $q$-Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.
