We obtain a uniform \(L^{\infty }(\Omega )\) a priori bound, for any positive weak solutions to elliptic problem with a \(C^1\) nonlinearity f slightly subcritical, slightly superlinear, and regularly varying. We isolate a structural condition, \(\frac{s^{2+N/2}\, |f'(s)|}{f(s)^{N/2}} \, \rightarrow +\infty \) as \(s\rightarrow \infty \) , that precludes concentration near the Sobolev threshold and yields a priori bounds for the critical case with \(q=2^*-1\) . To achieve our result, we first obtain a uniform estimate of an specific \(L^1(\Omega )\) weighted norm. This, combined with moving planes method and elliptic regularity theory, provides a uniform \(L^\infty \) bound in a neighborhood of the boundary of \(\Omega \) . Next, using Pohozaev’s identity, we obtain a uniform estimate of one weighted norm of the solutions. Joining now elliptic regularity theory, and Morrey’s Theorem, we estimate from below the radius of a ball where a solution exceeds the half of its \(L^\infty (\Omega )\) -norm. Finally, going back to the previous uniform weighted norm estimate, we conclude our result. We include Table 1 that summarizes a collection of examples satisfying our hypothesis.
