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# Solutions of nonlocal problem with critical exponent

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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.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33, 35B65.

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