Topological Methods in Nonlinear Analysis

Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity

Jinguo Zhang, Xiaochun Liu, and Hongying Jiao

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 1 (2019), 151-182.

Dates
First available in Project Euclid: 20 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1550631833

Digital Object Identifier
doi:10.12775/TMNA.2018.043

Mathematical Reviews number (MathSciNet)
MR3939152

Citation

Zhang, Jinguo; Liu, Xiaochun; Jiao, Hongying. Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity. Topol. Methods Nonlinear Anal. 53 (2019), no. 1, 151--182. doi:10.12775/TMNA.2018.043. https://projecteuclid.org/euclid.tmna/1550631833


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