Regularity and nonexistence of solutions for a system involving the fractional Laplacian
De Tang Yanqin Fang
We consider a system involving the fractional Laplacian \begin{eqnarray} \left\{ \begin{array}{ll} (-\Delta)^{\alpha_{1}/2}u=u^{p_{1}}v^{q_{1}} & \mbox{in}\ \mathbb{R}^N_+,\\ (-\Delta)^{\alpha_{2}/2}v=u^{p_{2}}v^{q_{2}} &\mbox{in}\ \mathbb{R}^N_+,\\ u=v=0,&\mbox{in}\ \mathbb{R}^N\backslash\mathbb{R}^N_+, \end{array} \right. \end{eqnarray} where $\alpha_{i}\in (0,2)$, $p_{i},q_{i}>0$, $i=1,2$. Based on the uniqueness of $\alpha$-harmonic function [9] on half space, the equivalence between (1) and integral equations \begin{eqnarray} \left\{ \begin{array}{ll} u(x)=C_{1}x_{N}^{\frac{\alpha_{1}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{1}_{\infty}(x,y)u^{p_{1}}(y)v^{q_{1}}(y)dy,\\ v(x)=C_{2}x_{N}^{\frac{\alpha_{2}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{2}_{\infty}(x,y)u^{p_{2}}(y)v^{q_{2}}(y)dy. \end{array} \right. \end{eqnarray} are derived. Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12], we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.
keywords: Kelvin transform. moving planes in integral forms nonexistence integral equations equivalence Fractional laplacians
Method of sub-super solutions for fractional elliptic equations
Yanqin Fang De Tang
By applying the method of sub-super solutions, we obtain the existence of weak solutions to fractional Laplacian
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$
$f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$
is a Caratheódory function.
be a Radon measure. Based on the existence result in (1), we derive the existence of weak solutions for the semilinear fractional elliptic equation with measure data
$ \left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u)+\nu ,&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right. $
Some results in[7] are extended.
In addition, we generalize some results to systems of fractional Laplacian equations by constructing subsolutions and supersolutions.
keywords: Fractional Laplacian Radon measure Caratheódory function subsolution supersolution

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