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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Representation formulae and inequalities for solutions of a class of second order partial differential equations

Authors: Lorenzo D'Ambrosio, Enzo Mitidieri and Stanislav I. Pohozaev
Journal: Trans. Amer. Math. Soc. 358 (2006), 893-910
MSC (2000): Primary 35H10, 35C15, 26D10
Published electronically: April 22, 2005
MathSciNet review: 2177044
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Abstract: Let $L$ be a possibly degenerate second order differential operator and let $\Gamma_\eta=d^{2-Q}$ be its fundamental solution at $\eta$; here $d$ is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of $-Lu\ge f(\xi,u)\ge 0$ on ${\mathbb{R}}^N$ to satisfy the representation formula

\begin{displaymath}(\mbox R)\qquad\qquad\qquad\qquad\qquad u(\eta)\ge\int_{\mat... ...amma_\eta f(\xi,u) \,d\xi.\qquad\qquad\qquad\qquad\qquad\qquad \end{displaymath}

We prove that (R) holds provided $f(\xi,\cdot)$ is superlinear, without any assumption on the behavior of $u$ at infinity. On the other hand, if $u$ satisfies the condition

\begin{displaymath}\liminf_{R\rightarrow\infty} {-\int}_{R\le d(\xi)\le 2R}\vert u(\xi)\vert d\xi =0,\end{displaymath}

then (R) holds with no growth assumptions on $f(\xi,\cdot)$.

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Additional Information

Lorenzo D'Ambrosio
Affiliation: Dipartimento di Matematica, via E. Orabona 4, Università degli Studi di Bari, I-70125 Bari, Italy

Enzo Mitidieri
Affiliation: Dipartimento di Scienze Matematiche, via A. Valerio 12/1, Università degli Studi di Trieste, I-34127 Trieste, Italy

Stanislav I. Pohozaev
Affiliation: Steklov Institute of Mathematics, Gubkina Str. 8, 117966 Moscow, Russia

Received by editor(s): April 19, 2004
Published electronically: April 22, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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