Asymptotic behavior of interior peaked solutions for a slightly subcritical Neumann problem

Abstract

In this paper, we study the asymptotic behavior of solutions of the Neumann problem (Pɛ): −Δu+V(x)u=up−ɛ, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥6, p+1=2n/(n−2) is the critical Sobolev exponent, ɛ is a small positive real and V is a smooth positive function defined on Ω¯. We give a precise location of interior blow up points and blow up rates when the number of concentration points is less than or equal to 2. The proof strategy is based on a refined blow up analysis in the neighborhood of bubbles.

Keywords

• Partial differential equations
• Neumann elliptic problems
• Blow up analysis
• Critical Sobolev exponent