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On strongly indefinite systems involving fractional elliptic operators. (English) Zbl 1388.35051

Summary: In this paper, we discuss the existence and regularity of solutions of the following strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain \(\Omega\) in \(\mathbb{R}^n\): \[ \begin{cases} \mathcal{L}u=\mu\nu + \nu^p,&\text{in }\Omega,\\ \mathcal{L}\nu=\lambda u+u^q,&\text{in }\Omega,\\ u=\nu=0,&\text{on }\Sigma,\end{cases} \] where \(\mathcal{L}\) refer to any of the two types of operators \(\mathcal{A}^s\) or \((-\Delta)^s\), \(0<s<1\), \(p,q>1\), \(\lambda\) and \(\mu\) are fixed real numbers and
\(\bullet\)
\(\Sigma=\partial\Omega\) for the spectral fractional Laplace operator \(\mathcal{A}^s\),
\(\bullet\)
\(\Sigma=\mathbb{R}^n\backslash\Omega\) for the restricted fractional Laplace operator \((-\Delta)^s\).

MSC:

35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
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