On elliptic equations with N-independent stable operators

Abstract

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space $\R^{N}$, we establish the nonexistence of positive supersolutions for $1<p\leq\frac{N}{N-2s}$. Furthermore, the symmetry of positive solutions is obtained when $p>\frac{N}{N-2s}$. It is crucial for these solutions to exhibit suitable decay at infinity to compensate for the absence of the Kelvin transform. Notably, while these solutions are symmetric, they are not radially symmetric due to the non-rotational invariance of the operator involved. Next, in the half space $\R_{+}^{N}$, we observe the nonexistence of positive supersolutions in the region $1<p\leq\frac{N+s}{N-s}$. Additionally, we find that positive solutions with appropriate decay for the Dirichlet boundary problem do not exist. Finally, we present the symmetry of positive solutions in the unit ball $B_{1}$.