A-priori bounds for a quasilinear problem in critical dimension

Abstract

We establish uniform a-priori bounds for solutions of the quasilinear problem $-{\Delta }_{N}u=f(u)$ in $\Omega$, with $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth and convex domain, and $f$ is a positive superlinear and subcritical function in the sense of the Trudinger-Moser inequality. The typical growth of $f$ is thus exponential. Finally, a generalization of the result for nonhomogeneous nonlinearities is given. Using a blow-up approach, this paper completes the results in [Damascelli-Pardo, Nonlinear Anal. Real World Appl. 41 (2018)] and [Lorca-Ruf-Ubilla, J. Differential Equations 246 no. 5 (2009)], enlarging the class of nonlinearities for which the uniform a-priori bound applies.

Major improvements in the proof of the main theorem

References in corpus (1)

• Uniform bounds for higher-order semilinear problems in conformal dimension

Cited by in corpus (1)

• Uniform bounds for higher-order semilinear problems in conformal dimension