Uniform a priori bounds for Slightly Subcritical Elliptic Problems

Abstract

We obtain a uniform $L^{\infty}(\Omega)$ a priori bound, for any positive weak solutions to elliptic problem with a nonlinearity $f$ slightly subcritical, slightly superlinear, and regularly varying. 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, by 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.