Interior Bubbling Solutions for an Elliptic Equation with Slightly Subcritical Nonlinearity

Abstract

In this paper, we considered the Neumann elliptic equation (Pε): −Δu+K(x)u=u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥6, ε is a small positive real and K is a smooth positive function on Ω¯. Using refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional, we constructed simple and non-simple interior bubbling solutions of (Pε) which allowed us to prove multiplicity results for (Pε) provided that ε is small. The existence of non-simple interior bubbling solutions is a new phenomenon for the positive solutions of subcritical problems.