Compactness and existence results for an elliptic PDE with zero Dirichlet boundary condition

Abstract

We prove compactness and existence results for a scalar curvature-type equation on a bounded domain of . The problem has a variational structure with lack of compactness. Using an asymptotic analysis and dynamical arguments, we characterize the accumulation points of all non-compact flow-lines of the gradient vector field, the so-called critical points at infinity. We then associate to each critical point at infinity a Morse index, which enables us to derive a Bahri–Coron index type criteria and to establish the existence.

Keywords:

• Non-linear elliptic PDE
• critical exponent
• lack of compactness
• critical points at infinity

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia [grant number 12-MAT2280-02].