American Institute of Mathematical Sciences

We study the problem

$\left\{ \begin{align} &{{(-\Delta lta )}^{s}}u={{u}^{p}}-{{u}^{q}}\ \text{in}\ \text{ }{{\mathbb{R}}^{N}}, \\ &u\in {{{\dot{H}}}^{s}}({{\mathbb{R}}^{N}})\cap {{L}^{q+1}}({{\mathbb{R}}^{N}}), \\ &u>0\ \ \text{in}\ \ {{\mathbb{R}}^{N}}, \\ \end{align} \right.$

where

$s∈(0,1)$

is a fixed parameter,

$(-Δ)^s$

is the fractional Laplacian in

$\mathbb{R}^N$

,

$q>p≥q \frac{N+2s}{N-2s}$

and

$N>2s$

. For every

$s∈(0,1)$

, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all

$s∈(0,1)$

. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we show that the above problem does not have any solution when

$p=\frac{N+2s}{N-2s}$

. We also discuss radial symmetry and decreasing property of the solution and prove that when

$p>\frac{N+2s}{N-2s}$

, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every

$p≥q \frac{N+2s}{N-2s}$

and every solution is a classical solution.