Blowing-Up Solutions with Residual Mass in a Slightly Subcritical Dirichlet Problem

In this paper, we study the Dirichlet elliptic problem $\left({\mathcal{P}}_{\epsilon }\right)$: $-\Delta u+Vu=u^{p-\epsilon }$, $u>0$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^n$ ( $n\ge 3$) is a bounded domain, V is a smooth positive function on $\overline{\Omega }$, $p+1=2n/(n-2)$ is the critical Sobolev exponent, and $\epsilon >0$ is a small parameter. First, we show that, unlike the case of weak convergence to zero, interior blowing-up solutions with a nonzero weak limit cannot occur in low dimensions. We then treat the general setting by removing the restriction that blow-up points are confined to the interior. Using delicate asymptotic expansions of the gradient of the associated functional, we prove that in dimensions $n=4$ and $n=5$, a single blow-up point cannot coexist with residual mass. We further elucidate the role of the sign of the normal derivative of the potential V on the boundary: if it is positive, any single blow-up solution with residual mass must occur in the interior; if it is negative at some boundary point, boundary blow-up solutions with residual mass can be constructed. Finally, we construct both simple and non-simple interior blow-up solutions exhibiting residual mass, without any assumption on the sign of the normal derivative of V. These results provide new insights into the interaction between the potential, the geometry of the domain, and the critical nonlinearity.