Electron. J. Diff. Equ., Vol. 2016 (2016), No. 150, pp. 1-16.

Positive least energy solutions of fractional Laplacian systems with critical exponent

Qingfang Wang

Abstract:
We study the fractional Laplacian system with critical exponent
$$\displaylines{ (-\Delta)^s u+\lambda_1u =\mu_1|u|^{2_s^*-2}u+\beta|u|^{\frac{2_s^*}{2}-2}u|v|^{\frac{2_s^*}{2}} , \quad x\in \Omega , \cr (-\Delta)^s v+\lambda_2v =\mu_2|v|^{2_s^*-2}v+\beta|v|^{\frac{2_s^*}{2}-2}v |u|^{\frac{2_s^*}{2}} , \quad x\in \Omega , \cr u=v= 0\,, \quad x\in \partial \Omega , }$$
where $\Omega\subset\mathbb{R}^N$ $(N>2s)$ is a smooth bounded domain, $s\in(0,1)$, $(-\Delta)^s$ stands for the fractional Laplacian, $2_s^*:=\frac{2N}{N-2s}$ is the critical Sobolev exponent, $-\lambda_1(\Omega)<\lambda_1,\lambda_2<0$, and $\mu_1, \mu_2>0$, here $\lambda_1(\Omega)$ is the first eigenvalue of $(-\Delta)^s$ with Dirichlet boundary condition. For each fixed $\beta\geq\frac{2s}{N-2s}\max\{\mu_1,\mu_2\}$, we show that this system has a positive least energy solution.

Submitted September 25, 2015. Published June 20, 2016.
Math Subject Classifications: 35R11, 35J50, 35B33.
Key Words: Positive least energy solution; critical exponent; fractional Laplacian system.

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Qingfang Wang
School of Mathematics and Computer Science
Wuhan Polytechnic University
Wuhan 430023, China
email: hbwangqingfang@163.com

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