# American Institute of Mathematical Sciences

## Journals

CPAA
Communications on Pure & Applied Analysis 2017, 16(5): 1741-1766 doi: 10.3934/cpaa.2017085
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.
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