    December  2020, 19(12): 5591-5608. doi: 10.3934/cpaa.2020253

## Solutions of nonlocal problem with critical exponent

 School of Mathematics and computer science, Wuhan Polytechnic University, Wuhan 430023, China

* Corresponding author

Received  May 2020 Revised  July 2020 Published  October 2020

Fund Project: The first author is supported by NSF grant 11701439

This paper deals with the system with linearly coupled of nonlocal problem with critical exponent,
 $\begin{equation*} \begin{cases} (-\Delta)^\alpha u+\lambda_1u = |u|^{2_\alpha^*-2}u+\beta v , \quad x\in \Omega , \\ (-\Delta)^\alpha v+\lambda_2v = |v|^{2_\alpha^*-2}v+\beta u , \,\quad x\in \Omega , \\ u = v = 0,\ \ \qquad \qquad \qquad \quad \quad \qquad \,x\in \partial\Omega. \end{cases} \end{equation*}$
Here
 $\Omega$
is a smooth bounded domain in
 ${\mathbb{R}}^N(N>4\alpha)$
,
 $0<\alpha<1$
,
 $\lambda_1,\lambda_2>-\lambda_1(\Omega)$
are constants,
 $\lambda_1(\Omega)$
is the first eigenvalue of fractional Laplacian with Dirichlet boundary,
 $2_\alpha^* = \frac{2N}{N-2\alpha}$
is the Sobolev critical exponent and
 $\beta\in {\mathbb{R}}$
is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some
 $\beta>0$
. Via a perturbation argument, by doing some delicate estimates for the nonlocal term, we overcome some difficulties and find a positive higher energy solution when
 $|\beta|$
is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as
 $\beta\rightarrow 0$
are analyzed.
Citation: Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
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##### References:
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