Geodesic
Existence of solutions for fractional \(p\)-Kirchhoff equations with critical nonlinearities
Electronic journal of differential equations, Tome 2015 (2015)
In this article, we show the existence of non-negative solutions of the fractional p-Kirchhoff problem

$\displaylines{ -M(\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy)\mathcal{L}_Ku =\lambda f(x,u)+|u|^{p^* -2}u\quad {in }\Omega,\cr u=0\quad {in }\mathbb{R}^{n}\setminus\Omega, }$

where $\mathcal{L}_K$ is a p-fractional type non local operator with kernel K, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, M and f are continuous functions, and $p^*$ is the fractional Sobolev exponent.
Classification : 34B27, 35J60, 35B05
Keywords: Kirchhoff non-local operators, fractional differential equations, critical exponent 
@article{EJDE_2015__2015__a94,
     author = {Mishra,  Pawan Kumar and Sreenadh,  Konijeti},
     title = {Existence of solutions for fractional {\(p\)-Kirchhoff} equations with critical nonlinearities},
     journal = {Electronic journal of differential equations},
     year = {2015},
     volume = {2015},
     zbl = {1433.35450},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a94/}
}
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Mishra,  Pawan Kumar; Sreenadh,  Konijeti. Existence of solutions for fractional \(p\)-Kirchhoff equations with critical nonlinearities. Electronic journal of differential equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a94/