Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity
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초록
In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity $$ (-\Delta)^{s}u(x)=\lambda u^{\beta}(x)+a_{0}u^{-\gamma}(x), ~ x\in \mathbb{R}^{n}, $$ where $0<s<1$, $\gamma>0$, $1<\beta\leq\frac{n+2s}{n-2s}$, $\lambda>0$ are constants and $a_{0}\geq0$. We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions $u(x)$ must be radially symmetric and monotone increasing about some point in $\mathbb{R}^{n}$.
Abstract
In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity $$ (-\Delta)^{s}u(x)=\lambda u^{\beta}(x)+a_{0}u^{-\gamma}(x), ~ x\in \mathbb{R}^{n}, $$ where $0<s<1$, $\gamma>0$, $1<\beta\leq\frac{n+2s}{n-2s}$, $\lambda>0$ are constants and $a_{0}\geq0$. We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions $u(x)$ must be radially symmetric and monotone increasing about some point in $\mathbb{R}^{n}$.
참고문헌 (12)
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