Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity

Abstract

In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation \begin{equation*} (-\Delta)^{\frac{\alpha}{2}} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{\alpha}-2}u, \quad\text{in}\,\,\Omega, u=0,\text{on}\,\,\partial\Omega, \end{equation*} where $\Omega\subset \mathbb{R}^{N}(N\geq 2)$ is a bounded domain with smooth boundary, $0<\alpha<2$, $(-\Delta)^{\frac{\alpha}{2}}$ stands for the fractional Laplacian operator, $f\in C(\Omega\times\mathbb{R},\mathbb{R})$ may be sign changing and $\lambda$ is a positive parameter. We will prove that there exists $\lambda_{*}>0$ such that the problem has at least two positive solutions for each $\lambda\in (0\,,\,\lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.