Interior Bubbling Solutions with Residual Mass for a Neumann Problem with Slightly Subcritical Exponent

In this paper, we consider the Neumann elliptic problem ( I ε ) : − Δ u + V ( x ) u = u p − ε , u > 0 in Ω , ∂ u / ∂ ν = 0 on ∂ Ω , where Ω is a bounded smooth domain in R n , n ≥ 3 , p + 1 = 2 n / ( n − 2 ) is the critical Sobolev exponent, ε is a small positive parameter and V is a smooth positive function on Ω ¯ . First, in contrast to the case where solutions converge weakly to zero, we rule out the existence of interior bubbling solutions with a non-zero weak limit in small-dimensional domains. Second, for large-dimensional domains, we construct both simple and non-simple interior bubbling solutions with residual mass. This construction allows us to establish the multiplicity results. The proofs of these results are based on refined asymptotic expansions of the gradient of the associated Euler–Lagrange functional.