## 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

### Keywords:

Kirchhoff equation; vibrating string; fractional Laplacian
Full Text:

### References:

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