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arXiv 2103.12600math.AP
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分数阶$q(.)$ -Laplacian方程的多解性

Multiplicity of solutions for fractional $q(.)$-Laplacian equations

Abita Rahmoune, Umberto Biccari

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摘要：

本文研究了椭圆型方程$$ \begin{cases} (-Δ)_{q(.)}^{s(.)}u + λVu = α\left\vert u\right\vert^{p(.)-2}u+β\left\vert u\right\vert^{k(.)-2}u & \text{ in }Ω, \\[7pt] u =0 & \text{ in }\mathbb{R}^{n}\backslash Ω, \end{cases} $$，其中$q(.):\overlineΩ\times \overlineΩ\rightarrow \mathbb{R}$是可测函数，$s(.):\mathbb{R}^n\times \mathbb{R}^n\rightarrow (0,1)$是连续函数，$n>q(x,y)s(x,y)$对所有$(x,y)\in Ω\times Ω$，$(-Δ)_{q(.)}^{s(.)}$是变阶分数Laplace算子，$V$是正连续势.利用山路范畴定理和Ekeland变分原理，我们得到了所有$λ>0$至少两个不同解的存在性.此外，我们还证明了这些解收敛于一个极限问题$λ\rightarrow +\infty $的无穷多个解中的两个。

英文摘要：

In this paper, we deal with the following elliptic type problem $$ \begin{cases} (-Δ)_{q(.)}^{s(.)}u + λVu = α\left\vert u\right\vert^{p(.)-2}u+β\left\vert u\right\vert^{k(.)-2}u & \text{ in }Ω, \\[7pt] u =0 & \text{ in }\mathbb{R}^{n}\backslash Ω, \end{cases} $$ where $q(.):\overlineΩ\times \overlineΩ\rightarrow \mathbb{R}$ is a measurable function and $s(.):\mathbb{R}^n\times \mathbb{R}^n\rightarrow (0,1)$ is a continuous function, $n>q(x,y)s(x,y)$ for all $(x,y)\in Ω\times Ω$, $(-Δ)_{q(.)}^{s(.)}$ is the variable-order fractional Laplace operator, and $V$ is a positive continuous potential. Using the mountain pass category theorem and Ekeland's variational principle, we obtain the existence of a least two different solutions for all $λ>0$. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as $λ\rightarrow +\infty $.

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