Electron. J. Differential Equations, Vol. 2017 (2017), No. 304, pp. 1-24.

Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent

Kun Cheng, Li Wang

In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent
 (-\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, $N\geq 1$, $0<2s<\min\{N,2\}$, $\lambda$ and $\mu>0$ 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 $0 <\bar{\mu}:= \bar{\mu}(s,\mu,N)< +\infty$ such that there are exactly two positive solutions if $\mu \in (0,\bar{\mu})$ and no positive solutions for $\mu>\bar{\mu}$. 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.

Submitted September 23, 2017. Published December 11, 2017.
Math Subject Classifications: 35A15,35J60, 46E35.
Key Words: Non-homogeneous; fractional Laplacian; critical Sobolev exponent; variational method.

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Kun Cheng
Department of Information Engineering
Jingdezhen Ceramic Institute
Jingdezhen 333403, China
email: chengkun0010@126.com
Li Wang
College of Science
East China Jiaotong University
Nanchang 330013, China
email: wangli.423@163.com

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