On positive solutions to the Lane-Emden problem with Neumann boundary conditions

In this talk we will examine how the number of positive solutions to -Δu + u = |u|ᵖ⁻² u on Ω, with Neumann boundary conditions, changes as the exponent p increases. In particular, when Ω is a ball of radius R, we will prove that this problem possesses as many positive solutions as one wants provided that R or p is large enough. This multiplicity of solutions will be established both for the sub-critical and the super-critical ranges. We will also show that the problem possesses degenerate solutions.