On a nonlinear problem with zero Dirichlet boundary condition

Abstract

We are concerned with the following PDE: , in , , on , where is a bounded domain of , and is a given function. We provide a complete description of the loss of compactness of the equation under the assumption that K is strictly decreasing in the outward normal direction on and flat near its critical points. As product, we prove an existence result through an Euler–Poincaré characteristic argument.

Keywords:

• Nonlinear elliptic P.D.E
• lack of compactness
• critical points at infinity

AMS Subject Classifications:

• 35J20
• 35J60

Notes

No potential conflict of interest was reported by the author.