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 $\left(P_{\varepsilon }\right)$: $-\Delta u+V\left(x\right)u=u^{p-\varepsilon }$, $u>0$ in $\Omega$, $\partial u/\partial \nu =0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $R^n$, $n\ge 6$, $p+1=2n/(n-2)$ is the critical Sobolev exponent, $\varepsilon$ is a small positive real and $V$ is a smooth positive function defined on $\overline{\Omega }$. 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.