Consider the problem \[ -\Delta u + au = u^p \text{ in } \Omega,\;u > 0 \text{ in } \Omega,\;u = 0 \text{ on } \partial\Omega \] where \(p\) is the critical Sobolev exponent, and \(a\) is continuous and \(-\Delta + a\) is coercive. Denote \[ S_a = \inf\left\{\frac{\int_\Omega \| \nabla u\| ^2 + a(x)u^2\,dx}{\big(\int_\Omega\| u\| ^{p+1}\,dx\big)^{2/(p+1)}} : u \in H^1_0(\Omega)\backslash\{0\}\right\}. \] Let \(G_a\) be the Green function of \(-\Delta + a\) with the Dirichlet boundary condition and let \(H_a\) its regular part. Consider the properties
(i) \(\exists y \in \Omega \) such that \(H_a(y,y) > 0\), (ii) \(S_a < S_0\), (iii) \(S_a\) is achieved.
The implications (i) \(\implies\) (ii) \(\implies\) (iii) are known. In [Commun. Pure Appl. Math. 39, Suppl., S17–S39 (1986; Zbl 0601.35043)], Brezis conjectured the two following results (iii) \(\implies\) (ii) and (ii) \(\implies\) (i). O. Druet [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, 125–142 (2002; Zbl 1011.35060)] established these two conjectures. In this paper, the author presents a different and simpler approach in the blow-up analysis in exploiting integral estimates and by a careful expansion of \(S_{a+\delta}\).