  We establish the existence and multiplicity of positive solutions to the
problems involving the fractional Laplacian: \begin{equation*}
\left\{\begin{array}{lll} &(-\Delta)^{s}u=\lambda u^{p}+f(u),\,\,u>0 \quad
&\mbox{in}\,\,\Omega,\\ &u=0\quad
&\mbox{in}\,\,\mathbb{R}^{N}\setminus\Omega,\\ \end{array}\right.
\end{equation*} where $\Omega\subset \mathbb{R}^{N}$ $(N\geq 2)$ is a bounded
smooth domain, $s\in (0,1)$, $p>0$, $\lambda\in \mathbb{R}$ and $(-\Delta)^{s}$
stands for the fractional Laplacian. When $f$ oscillates near the origin or at
infinity, via the variational argument we prove that the problem has
arbitrarily many positive solutions and the number of solutions to problem is
strongly influenced by $u^{p}$ and $\lambda$. Moreover, various properties of
the solutions are also described in $L^{\infty}$- and
$X^{s}_{0}(\Omega)$-norms.
