Uniform estimates for elliptic equations with Carath\'eodory nonlinearities at the interior and on the boundary

TL;DR

This paper derives explicit uniform a priori bounds for solutions to elliptic equations with nonlinearities at the interior and boundary, using advanced iterative and regularity techniques.

Contribution

It provides the first explicit $L^{ olinebreak\infty}$ a priori estimates for such elliptic problems with combined interior and boundary nonlinearities.

Findings

Established explicit $L^{\infty}$ bounds in terms of $H^{1}$ norms.

Combined De Giorgi-Nash-Moser iteration with elliptic regularity.

Applicable to slightly subcritical nonlinear elliptic problems.

Abstract

We establish an explicit uniform a priori estimate for weak solutions to slightly subcritical elliptic problems with nonlinearities simultaneously at the interior and on the boundary. Our explicit $L^{\infty}(\Omega )$ a priori estimates are in terms of powers of their $H^{1}(\Omega )$ norms. To prove our result, we combine a De Giorgi-Nash-Moser's iteration scheme together with elliptic regularity and the Gagliardo-Nirenberg's interpolation inequality.