## Existence and multiplicity of solutions for a fractional $$p$$-Laplacian problem of Kirchhoff type via Krasnoselskii’s genus.(English)Zbl 1463.35493

Summary: We use the genus theory to prove the existence and multiplicity of solutions for the fractional $$p$$-Kirchhoff problem $\begin{cases}\displaystyle -\left[ M\left( \int_Q\frac{| u(x)-u(y)|^p}{| x-y|^{N+ps}}dx\, dy\right)\right]^{p-1}(-\Delta)_p^su=\lambda h(x,u)\quad\text{in}\;\Omega,\\ u=0\quad\text{on}\;\mathbb{R}^N\setminus\Omega,\end{cases}$ where $$\Omega$$ is an open bounded smooth domain of $$\mathbb{R}^N$$, $$p>1$$, $$N>ps$$ with $$s\in(0,1)$$ fixed, $$Q=\mathbb{R}^{2N}\setminus(C\Omega\times C\Omega)$$, $$\lambda >0$$ is a numerical parameter, $$M$$ and $$h$$ are continuous functions.

### MSC:

 35R11 Fractional partial differential equations 35R09 Integro-partial differential equations 35A15 Variational methods applied to PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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### References:

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