Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity

Jinguo Zhang, Xiaochun Liu, Hongying Jiao



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.


Fractional Laplacian equation; critical Sobolev exponent; variational methods

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