Abstract
In this paper, we consider the following problem involving fractional Laplacian operator \begin{equation*} (-\Delta)^{s} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{s}-2}u\quad \text{in } \Omega,\qquad u=0\quad \text{on } \partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $0<s<1$, $2^*_{s}={2N}/({N-2s})$, and $(-\Delta)^{s}$ is the fractional Laplacian. 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.
Citation
Jinguo Zhang. Xiaochun Liu. Hongying Jiao. "Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity." Topol. Methods Nonlinear Anal. 53 (1) 151 - 182, 2019. https://doi.org/10.12775/TMNA.2018.043