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A critical Kirchhoff type problem involving a nonlocal operator. (English) Zbl 1283.35156

Summary: We show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is \[ -M(||u||_Z^2)\mathcal L_Ku={\lambda}f(x,u)+|u|^{2^\ast -2}u \text{ in } {\varOmega}, \quad u=0 \quad \text{ in } \mathbb R^n\setminus {\varOmega} \] where \(\mathrm{L}_K\) is an integrodifferential operator with kernel \(K, {\varOmega}\) is a bounded subset of \(\mathbb R^n\), \(M\) and \(f\) are continuous functions, \(||\cdot ||_Z\) is a functional norm and \(2^\ast \) is a fractional Sobolev exponent.

MSC:

35R11 Fractional partial differential equations
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