A priori estimates for positive solutions of subcritical elliptic equations
Ponente(s): Edgar Alejandro Antonio MartÃnez
Let us consider the following elliptic problem
\begin{equation} \label{PN1}
\left\{ \begin{array}{rcll}
-\Delta u +u &=& f(x,u), \quad &x\in \Om\,,\\%
\frac{\partial u}{\partial \eta} &=& f_{B}(x,u),\quad &x\in \partial \Omega ,
\end{array} \right.
\end{equation}
where $\Om \subset \mathbb{R}^{N}$, ($N>2$), is an open, connected, bounded domain with $C^{2}$ boundary, $\p /\p \eta =\eta \cdot\na$ is the (unit) outer normal derivative, and the functions $f:\Om\times \mathbb{R}\rightarrow \mathbb{R}$, and $f_{B}:\p \Om \times \mathbb{R} \rightarrow \mathbb{R}$, are both slightly subcritical Carath\'eodory functions.\\
Through a De Giorgi-Nash-Moser iteration scheme, it is known that weak solutions to \eqref{PN1} with critical growth are in $L^\infty(\Om)$. \\
Our contribution is to provide an explicit $L^\infty(\Om)$-estimate of weak solutions with slightly subcritical growth, in terms of powers of their $H^1(\Omega)$-norms. Our method combines elliptic regularity of weak solutions with Gagliardo--Nirenberg interpolation inequality. \\
\textbf{Keywords:} Brezis-Kato estimate; De Giorgi-Nash-Moser estimate; Sobolev embedding; H\"older inequality.