English

Kirchhoff equations with Choquard exponential type nonlinearity involving the fractional Laplacian

Analysis of PDEs 2019-08-30 v1

Abstract

In this article, we deal with the existence of non-negative solutions of the class of following non local problem {M(RnRnu(x)u(y)nsxy2n dxdy)(Δ)n/ssu=(ΩG(y,u)xyμ dy)g(x,u)  in  Ω,u=0inRnΩ, \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-\Delta)^{s}_{n/s} u=\left(\displaystyle\int_{\Omega}\frac{G(y,u)}{|x-y|^{\mu}}~dy\right)g(x,u) \; \text{in}\; \Omega,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus \Omega, \end{array} \right. where (Δ)n/ss(-\Delta)^{s}_{n/s} is the n/sn/s-fractional Laplace operator, n1n\geq 1, s(0,1)s\in(0,1) such that n/s2n/s\geq 2, ΩRn\Omega\subset \mathbb R^n is a bounded domain with Lipschitz boundary, M:R+R+M:\mathbb R^+\rightarrow \mathbb R^+ and g:Ω×RRg:\Omega\times\mathbb R\rightarrow \mathbb R are continuous functions, where gg behaves like exp(unns)\exp({|u|^{\frac{n}{n-s}}}) as u|u|\rightarrow\infty.

Keywords

Cite

@article{arxiv.1908.11285,
  title  = {Kirchhoff equations with Choquard exponential type nonlinearity involving the fractional Laplacian},
  author = {Sarika Goyal and Tuhina Mukherjee},
  journal= {arXiv preprint arXiv:1908.11285},
  year   = {2019}
}

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23 pages