Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity

Jinguo Zhang, Xiaochun Liu, Hongying Jiao

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.

Keywords


Fractional Laplacian equation; critical Sobolev exponent; variational methods

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