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*} (-Δ)^{\fracα{2}} u=λf(x)|u|^{q-2}u+|u|^{2^{*}_α-2}u, \quad\text{in}\,\,Ω, u=0,\text{on}\,\,\partialΩ, \end{equation*} where $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ 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 ${\lambda }_{\ast }>0$ such that the problem has at least two positive solutions for each $\lambda \in (0,{\lambda }_{\ast })$. In addition, the concentration behavior of the solutions are investigated.

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• Multiplicity results of fractional-Laplace system with sign-changing and singular nonlinearity