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

D’Ambrosio, Lorenzo; Mitidieri, Enzo; Pohozaev, Stanislav

doi:10.1090/S0002-9947-05-03717-7

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
DOI: https://doi.org/10.1090/S0002-9947-05-03717-7
Published electronically: April 22, 2005
MathSciNet review: 2177044
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Abstract: Let be a possibly degenerate second order differential operator and let $\Gamma_\eta=d^{2-Q}$ be its fundamental solution at $\eta$ ; here 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 at infinity. On the other hand, if 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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