Properties of the positive solutions of fractional p&q-Laplace equations with a sign-changing potential

In this paper, we consider the nonlinear equations involving the fractional p&q-Laplace operator with a sign-changing potential. This model is inspired by the De Giorgi Conjecture. There are two main results in this paper. First, in the bounded domain, we use the moving plane method to show that the solution is radially symmetric. Second, for the unbounded domain, in view of the idea of the sliding method, we find the existence of the maximizing sequence of the bounded solution, then obtain that the solution is strictly monotone increasing in some direction.