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We establish the existence and multiplicity of positive solutions to theproblems 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 boundedsmooth 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 atinfinity, via the variational argument we prove that the problem hasarbitrarily many positive solutions and the number of solutions to problem isstrongly influenced by $u^{p}$ and $\lambda$. Moreover, various properties ofthe solutions are also described in $L^{\infty}$- and$X^{s}_{0}(\Omega)$-norms.

Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian