$L^{\infty }$ estimates for solutions to elliptic equations in the presence of gradient terms

Abstract

We consider an elliptic problem with slightly subcritical nonlinearities at the interior and on the boundary; the nonlinearity at the interior is also depending on a gradient term. For any $u$ weak solution, we provide explicit $L^{\infty }$ {\it a priori} estimates depending only on both nonlinearities, on the $H^1(\Omega )$ norm of $u,$ and on the domain $\Omega$. To obtain our results, we combine De Giorgi-Nash-Moser iteration procedure, elliptic regularity when the gradient term appears, Lebesgue interpolation and the Gagliardo-Nirenberg interpolation inequalities.