Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-02-16 DOI:10.1007/s13540-024-00242-y

引用次数: 0

Abstract

In this paper, we consider the following problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{} \text { in } \varOmega , \\ u>0, \text { in } \varOmega , \quad u=0, &{} \text { on } \partial \varOmega , \end{array}\right. } \end{aligned}$$ where $\varOmega \subset {\mathbb {R}}^{N}(N > 2s)$ is a smooth bounded domain, $s\in (0,1)$ , $\lambda $ is a positive constant, $0<\gamma <1$ , $2_{s}^{*}=\frac{2 N}{N-2s}$ and $(-\varDelta )^{s} $ is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity g.

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具有奇异非线性的临界分数拉普拉斯方程正解的存在性和多重性

Abstract In this paper, we consider following problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{}\text { in }\varOmega , \ u>0, \text { in }\varOmega , \quad u=0, \amp;{}\text { on }\partial \varOmega , \end{array}\right.}\end{aligned}$$ 其中 $\varOmega \subset {\mathbb {R}}^{N}(N > 2s)$ 是一个光滑的有界域, $s\in (0,1)$(\lambda)是一个正常数，$0<\gamma<1$，$2_{s}^{*}=\frac{2 N}{N-2s}$ 和$(-\varDelta )^{s}$是谱分数拉普拉奇。基于 Nehari 流形并使用变分法，我们将正解的数量与临界非线性系数 g 的全局最大值联系起来。

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