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.


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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