Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent

Abstract

In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent $$\displaylines{ (-\Delta)^{s} u +\lambda u=u^p+\mu f(x), \quad u>0\quad \text{in } \Omega,\cr u =0, \quad \text{in } \mathbb{R}^N\setminus \Omega, }$$ where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $00$ are two parameters, $p=\frac{N+2s}{N-2s}$ and $f\in C^{0,\alpha}(\bar{\Omega})$, where $\alpha \in(0,1)$. $f\geq 0$ and $f\not \equiv 0$ in $\Omega$. For some $\lambda$ and N, by the barrier method and mountain pass lemma, we prove that there exists $0 \bar{\mu}$. Moreover, if $\mu=\bar{\mu}$, there is a unique solution ($\bar{\mu}; u_{\bar{\mu}}$), which means that ($\bar{\mu}; u_{\bar{\mu}}$) is a turning point for the above problem. Furthermore, in case $ \lambda > 0$ and $N \ge 6s$ if $\Omega$ is a ball in $\mathbb{R}^N$ and f satisfies some additional conditions, then a uniqueness existence result is obtained for $\mu>0$ small enough.

Keywords

• Non-homogeneous
• fractional Laplacian
• critical Sobolev exponent
• variational method