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

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.

Keywords

• Elliptic equation
• critical Sobolev exponent
• variational method
• critical point at infinity