Electron. J. Diff. Equ., Vol. 2015 (2015), No. 93, pp. 1-14.

Existence of solutions for fractional p-Kirchhoff equations with critical nonlinearities

Pawan Kumar Mishra, Konijeti Sreenadh

Abstract:
In this article, we show the existence of non-negative solutions of the fractional p-Kirchhoff problem
$$\displaylines{ -M(\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy)\mathcal{L}_Ku =\lambda f(x,u)+|u|^{p^* -2}u\quad \text{in }\Omega,\cr u=0\quad \text{in }\mathbb{R}^{n}\setminus\Omega, }$$
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.

Submitted September 2, 2014. Published April 12, 2015.
Math Subject Classifications: 34B27, 35J60, 35B05.
Key Words: Kirchhoff non-local operators; fractional differential equations; critical exponent.

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Pawan Kumar Mishra
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: pawanmishra31284@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: sreenadh@gmail.com

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