Asymptotic Behavior and Construction of Blowing-Up Solutions for a Subcritical Elliptic Problem

In this paper, we study energy bounded solutions uε converging weakly to 0 of the subcritical problem −Δu+gu=hun+2n−2−ε,u>0inΩ,u=0on∂Ω, where Ω is a C2 bounded domain in Rn with n≥4, g is a C1 positive function on Ω¯, h is a C3 positive function on Ω¯, and ε is a small positive parameter. Assuming that the normal derivative of h is negative on the boundary, we prove that uε must blow up in the interior of the domain. Moreover, we determine the precise location of the blow-up points and the corresponding blow-up rates. Conversely, for sufficiently small ε, we construct blowing-up solutions that converge weakly to zero, which allows us to obtain a multiplicity result for the problem. In contrast, when the normal derivative of h is positive at a boundary point b, we show that it is possible to construct solutions converging to zero and blowing up precisely at b.