Electron. J. Differential Equations, Vol. 2018 (2018), No. 60, pp. 1-32.

Critical second-order elliptic equation with zero Dirichlet boundary condition in four dimensions

Zakaria Boucheche, Hichem Chtioui, Hichem Hajaiej

Abstract:
We are concerned with the nonlinear critical problem $-\Delta u=K(x)u^{3}$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain of $\mathbb{R}^4$. Under the assumption that $K$ is strictly decreasing in the outward normal direction on $\partial\Omega$ and degenerate at its critical points for an order $\beta \in (1,4)$, we provide a complete description of the lack of compactness of the associated variational problem and we prove an existence result of Bahri-Coron type.

Submitted October 22, 2017. Published March 2, 2018.
Math Subject Classifications: 35J60, 58E30.
Key Words: Elliptic equation; critical Sobolev exponent; variational method; critical point at infinity.

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Zakaria Boucheche
Department of mathematics
Faculty of Sciences of Sfax
3018 Sfax, Tunisia
email: Zakaria.Boucheche@ipeim.rnu.tn
Hichem Chtioui
Department of mathematics
Faculty of Sciences of Sfax
3018 Sfax, Tunisia
email: Hichem.Chtioui@fss.rnu.tn
Hichem Hajaiej
California State University Los Angeles
5151 University Drive
Los Angeles, CA 90032, USA
email: hichem.hajaiej@gmail.com

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